Subgroups — C₆₇×#22551 #30701802

Order 17152.

1058 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3480	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1980	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1963	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1962	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1806	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1759	C₂	1	2	yes	maximal, complemented
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₆₇	C₄	1	2	yes	complemented
13	#559×C₆₇	C₄	1	1	yes	characteristic, complemented
14	#557×C₆₇	C₄	1	1	yes	characteristic, complemented
15	#556×C₆₇	C₄	1	1	yes	characteristic, complemented
16	#556×C₆₇	C₄	1	2	yes	complemented
17	#553×C₆₇	—	2	2	no	—
18	#553×C₆₇	—	2	1	no	—
19	#550×C₆₇	C₄	1	1	yes	characteristic, complemented
20	#548×C₆₇	—	2	1	no	—
21	#532×C₆₇	C₂²	1	1	yes	characteristic
22	#532×C₆₇	C₂²	1	2	yes	—
23	#529×C₆₇	—	2	1	no	—
24	#529×C₆₇	—	2	2	no	—
25	C₄×#41×C₆₇	—	2	1	no	—
26	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×#41×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	#446×C₆₇	—	2	1	no	—
30	#446×C₆₇	—	2	2	no	—
31	C₈×Q₈×C₆₇	—	2	1	no	—
32	#423×C₆₇	C₂²	1	1	yes	characteristic
33	#423×C₆₇	C₂²	1	2	yes	—
34	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#104×C₆₇	C₂²	1	2	yes	—
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	C₂×#104×C₆₇	C₂²	1	2	yes	—
39	C₂×#104×C₆₇	C₂²	1	2	yes	—
40	C₂×#104×C₆₇	C₂²	1	2	yes	—
41	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
43	#391×C₆₇	—	2	1	no	—
44	#389×C₆₇	—	2	1	no	—
45	#387×C₆₇	—	2	2	no	—
46	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	#376×C₆₇	C₂²	1	2	yes	—
49	C₄³×C₆₇	C₂²	1	1	yes	characteristic
50	#339×C₆₇	C₂²	1	1	yes	characteristic
51	#339×C₆₇	C₂²	1	1	yes	characteristic
52	#339×C₆₇	C₂²	1	2	yes	—
53	C₂×#41×C₆₇	—	2	1	no	—
54	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×#41×C₆₇	C₂³	1	2	yes	—
56	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
59	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
62	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
63	#127×C₆₇	C₂×C₄	1	2	yes	—
64	#126×C₆₇	—	2	2	no	—
65	#126×C₆₇	—	2	2	no	—
66	#126×C₆₇	—	2	2	no	—
67	#125×C₆₇	—	2	2	no	—
68	#125×C₆₇	—	2	1	no	—
69	#124×C₆₇	—	2	1	no	—
70	#124×C₆₇	—	2	2	no	—
71	#124×C₆₇	C₂×C₄	1	2	yes	—
72	#124×C₆₇	C₂×C₄	1	2	yes	—
73	#124×C₆₇	C₂×C₄	1	2	yes	—
74	#123×C₆₇	—	2	2	no	—
75	#123×C₆₇	—	2	2	no	—
76	#123×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	1	no	—
78	C₂²⋊Q₈×C₆₇	—	2	2	no	—
79	#121×C₆₇	—	2	1	no	—
80	#121×C₆₇	—	2	2	no	—
81	#121×C₆₇	—	2	1	no	—
82	#121×C₆₇	—	2	2	no	—
83	#121×C₆₇	—	2	2	no	—
84	#121×C₆₇	—	2	1	no	—
85	C₄×Q₈×C₆₇	—	2	1	no	—
86	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
88	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
90	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₆₇	D₄	1	2	yes	—
96	C₄×D₄×C₆₇	—	2	1	no	—
97	C₄×D₄×C₆₇	—	2	2	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	—	2	2	no	—
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₄×D₄×C₆₇	—	2	1	no	—
102	C₄×D₄×C₆₇	—	2	1	no	—
103	C₄×D₄×C₆₇	—	2	2	no	—
104	C₄×D₄×C₆₇	—	2	2	no	—
105	C₄×D₄×C₆₇	—	2	1	no	—
106	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
108	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
110	#117×C₆₇	—	2	2	no	—
111	#117×C₆₇	—	2	1	no	—
112	#117×C₆₇	—	2	2	no	—
113	#117×C₆₇	—	2	1	no	—
114	#117×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	—	2	1	no	—
120	C₂×C₄²×C₆₇	—	2	2	no	—
121	C₂×C₄²×C₆₇	—	2	1	no	—
122	C₂×C₄²×C₆₇	—	2	1	no	—
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₇	—	2	1	no	—
129	C₄⋊C₈×C₆₇	—	2	2	no	—
130	C₄⋊C₈×C₆₇	—	2	2	no	—
131	C₄⋊C₈×C₆₇	—	2	1	no	—
132	#104×C₆₇	—	2	2	no	—
133	#104×C₆₇	—	2	4	no	—
134	#104×C₆₇	—	2	2	no	—
135	#104×C₆₇	—	2	4	no	—
136	#104×C₆₇	—	4	2	no	—
137	#104×C₆₇	—	4	2	no	—
138	#104×C₆₇	—	4	2	no	—
139	#97×C₆₇	—	2	2	no	—
140	#97×C₆₇	—	2	1	no	—
141	C₄×C₈×C₆₇	—	2	1	no	—
142	C₄×C₈×C₆₇	—	2	2	no	—
143	#95×C₆₇	—	2	2	no	—
144	#95×C₆₇	—	2	2	no	—
145	#95×C₆₇	—	2	1	no	—
146	#95×C₆₇	—	2	2	no	—
147	#41×C₆₇	—	2	1	no	—
148	#41×C₆₇	—	2	2	no	—
149	#41×C₆₇	—	2	1	no	—
150	#41×C₆₇	—	2	2	no	—
151	#41×C₆₇	—	2	2	no	—
152	#41×C₆₇	—	2	1	no	—
153	#41×C₆₇	—	2	1	no	—
154	#41×C₆₇	—	2	2	no	—
155	#41×C₆₇	—	4	4	no	—
156	C₂×Q₈×C₆₇	—	4	2	no	—
157	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
158	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
160	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×D₄×C₆₇	—	2	1	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	1	no	—
166	C₂×D₄×C₆₇	—	2	2	no	—
167	C₂×D₄×C₆₇	—	2	1	no	—
168	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	M₁₆×C₆₇	—	2	1	no	—
185	M₁₆×C₆₇	—	2	1	no	—
186	M₁₆×C₆₇	—	2	2	no	—
187	M₁₆×C₆₇	—	2	2	no	—
188	M₁₆×C₆₇	—	2	1	no	—
189	M₁₆×C₆₇	#41	1	2	yes	complemented
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	2	2	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	1	no	—
195	C₄⋊C₄×C₆₇	—	2	1	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	2	1	no	—
199	C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₄⋊C₄×C₆₇	—	2	1	no	—
201	C₄⋊C₄×C₆₇	—	4	2	no	—
202	C₄⋊C₄×C₆₇	—	4	4	no	—
203	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
204	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
206	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
208	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
210	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
211	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
212	C₂²⋊C₄×C₆₇	—	2	1	no	—
213	C₂²⋊C₄×C₆₇	—	2	2	no	—
214	C₂²⋊C₄×C₆₇	—	2	2	no	—
215	C₂²⋊C₄×C₆₇	—	2	2	no	—
216	C₂²⋊C₄×C₆₇	—	2	2	no	—
217	C₂²⋊C₄×C₆₇	—	2	1	no	—
218	C₂²⋊C₄×C₆₇	—	2	1	no	—
219	C₂²⋊C₄×C₆₇	—	2	2	no	—
220	C₂²⋊C₄×C₆₇	—	2	2	no	—
221	C₂²⋊C₄×C₆₇	—	2	2	no	—
222	C₂²⋊C₄×C₆₇	—	2	2	no	—
223	C₄²×C₆₇	—	2	1	no	—
224	C₄²×C₆₇	—	2	2	no	—
225	C₄²×C₆₇	—	2	2	no	—
226	C₄²×C₆₇	—	2	1	no	—
227	C₄²×C₆₇	—	2	1	no	—
228	C₄²×C₆₇	—	2	1	no	—
229	C₄²×C₆₇	—	2	1	no	—
230	C₄²×C₆₇	—	2	2	no	—
231	C₄²×C₆₇	—	2	1	no	—
232	C₄²×C₆₇	—	2	2	no	—
233	C₄²×C₆₇	—	4	2	no	—
234	C₄²×C₆₇	—	4	2	no	—
235	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
236	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
237	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
238	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
239	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
240	C₄²×C₆₇	C₂×D₄	1	2	yes	—
241	C₄²×C₆₇	C₂×D₄	1	2	yes	—
242	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₄²×C₆₇	#41	1	2	yes	complemented
244	C₂³×C₆₇	—	2	2	no	—
245	C₂³×C₆₇	—	2	1	no	—
246	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₆₇	—	2	2	no	—
248	Q₈×C₆₇	—	2	2	no	—
249	Q₈×C₆₇	—	2	1	no	—
250	Q₈×C₆₇	—	2	2	no	—
251	Q₈×C₆₇	—	2	1	no	—
252	Q₈×C₆₇	—	2	2	no	—
253	Q₈×C₆₇	—	4	2	no	—
254	D₄×C₆₇	—	2	1	no	—
255	D₄×C₆₇	—	2	2	no	—
256	D₄×C₆₇	—	2	1	no	—
257	D₄×C₆₇	—	2	2	no	—
258	D₄×C₆₇	—	4	4	no	—
259	D₄×C₆₇	—	4	1	no	—
260	D₄×C₆₇	—	4	2	no	—
261	D₄×C₆₇	—	4	1	no	—
262	D₄×C₆₇	—	4	2	no	—
263	D₄×C₆₇	—	4	2	no	—
264	D₄×C₆₇	—	8	1	no	—
265	D₄×C₆₇	—	8	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	1	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	4	1	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	2	no	—
276	C₂×C₄×C₆₇	—	4	4	no	—
277	C₂×C₄×C₆₇	—	4	1	no	—
278	C₂×C₄×C₆₇	—	4	1	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	1	no	—
281	C₂×C₄×C₆₇	—	4	1	no	—
282	C₂×C₄×C₆₇	—	4	2	no	—
283	C₂×C₄×C₆₇	—	4	1	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	2	no	—
286	C₂×C₄×C₆₇	—	4	2	no	—
287	C₂×C₄×C₆₇	—	4	2	no	—
288	C₂×C₄×C₆₇	—	4	2	no	—
289	C₂×C₄×C₆₇	—	8	1	no	—
290	C₂×C₄×C₆₇	—	8	2	no	—
291	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
293	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
294	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
296	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
297	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
303	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
305	C₈×C₆₇	—	2	2	no	—
306	C₈×C₆₇	—	4	2	no	—
307	C₈×C₆₇	—	4	1	no	—
308	C₂²×C₆₇	—	2	1	no	—
309	C₂²×C₆₇	—	2	1	no	—
310	C₂²×C₆₇	—	4	2	no	—
311	C₂²×C₆₇	—	4	1	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	1	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	1	no	—
316	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
317	C₂²×C₆₇	#548	1	1	yes	characteristic
318	C₂²×C₆₇	#550	1	1	yes	characteristic
319	C₄×C₆₇	—	2	1	no	—
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	2	2	no	—
322	C₄×C₆₇	—	2	1	no	—
323	C₄×C₆₇	—	2	2	no	—
324	C₄×C₆₇	—	2	1	no	—
325	C₄×C₆₇	—	2	2	no	—
326	C₄×C₆₇	—	4	2	no	—
327	C₄×C₆₇	—	4	2	no	—
328	C₄×C₆₇	—	4	2	no	—
329	C₄×C₆₇	—	4	1	no	—
330	C₄×C₆₇	—	4	1	no	—
331	C₄×C₆₇	—	4	1	no	—
332	C₄×C₆₇	—	8	2	no	—
333	C₄×C₆₇	#421	1	2	yes	—
334	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
335	C₄×C₆₇	#520	1	1	yes	characteristic
336	C₄×C₆₇	#548	1	1	yes	characteristic
337	C₄×C₆₇	#550	1	1	yes	characteristic
338	#22551	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
339	C₂×C₆₇	—	2	1	no	—
340	C₂×C₆₇	—	2	1	no	—
341	C₂×C₆₇	—	8	2	no	—
342	C₂×C₆₇	—	8	1	no	—
343	C₂×C₆₇	#2341	1	1	yes	characteristic
344	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
345	C₂×C₆₇	#2901	1	1	yes	characteristic
346	#3480	C₂×C₆₇	1	1	yes	characteristic
347	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
348	#2295	C₂×C₆₇	1	1	yes	characteristic, complemented
349	#1980	C₂×C₆₇	1	1	yes	characteristic, complemented
350	#1980	C₂×C₆₇	1	2	yes	complemented
351	#1963	C₂×C₆₇	1	2	yes	complemented
352	#1962	C₂×C₆₇	1	1	yes	characteristic, complemented
353	#1806	C₂×C₆₇	1	1	yes	characteristic, complemented
354	#1806	C₂×C₆₇	1	2	yes	complemented
355	#1759	C₂×C₆₇	1	2	yes	complemented
356	C₄×#104	C₂×C₆₇	1	1	yes	characteristic, complemented
357	C₆₇	#22551	1	1	yes	characteristic, direct product factor, Sylow
358	#559	C₄×C₆₇	1	2	yes	complemented
359	#559	C₄×C₆₇	1	1	yes	characteristic, complemented
360	#557	C₄×C₆₇	1	1	yes	characteristic, complemented
361	#556	C₄×C₆₇	1	1	yes	characteristic, complemented
362	#556	C₄×C₆₇	1	2	yes	complemented
363	#553	—	2	2	no	—
364	#553	—	2	1	no	—
365	#550	C₄×C₆₇	1	1	yes	characteristic, complemented
366	#548	—	2	1	no	—
367	#532	C₂²×C₆₇	1	1	yes	characteristic
368	#532	C₂²×C₆₇	1	2	yes	—
369	#529	—	2	1	no	—
370	#529	—	2	2	no	—
371	C₄×#41	—	2	1	no	—
372	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
373	C₄×#41	C₂²×C₆₇	1	2	yes	—
374	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
375	#446	—	2	1	no	—
376	#446	—	2	2	no	—
377	C₈×Q₈	—	2	1	no	—
378	#423	C₂²×C₆₇	1	1	yes	characteristic
379	#423	C₂²×C₆₇	1	2	yes	—
380	C₂×#104	C₂²×C₆₇	1	2	yes	—
381	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
382	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×#104	C₂²×C₆₇	1	2	yes	—
384	C₂×#104	C₂²×C₆₇	1	2	yes	—
385	C₂×#104	C₂²×C₆₇	1	2	yes	—
386	C₂×#104	C₂²×C₆₇	1	2	yes	—
387	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
388	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
389	#391	—	2	1	no	—
390	#389	—	2	1	no	—
391	#387	—	2	2	no	—
392	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
393	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
394	#376	C₂²×C₆₇	1	2	yes	—
395	C₄³	C₂²×C₆₇	1	1	yes	characteristic
396	#339	C₂²×C₆₇	1	1	yes	characteristic
397	#339	C₂²×C₆₇	1	2	yes	—
398	#339	C₂²×C₆₇	1	1	yes	characteristic
399	C₂×#41	—	2	1	no	—
400	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
401	C₂×#41	C₂³×C₆₇	1	2	yes	—
402	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
403	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
404	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
405	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
406	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
407	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
408	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
409	#127	C₂×C₄×C₆₇	1	2	yes	—
410	#126	—	2	2	no	—
411	#126	—	2	2	no	—
412	#126	—	2	2	no	—
413	#125	—	2	2	no	—
414	#125	—	2	1	no	—
415	#124	—	2	1	no	—
416	#124	—	2	2	no	—
417	#124	C₂×C₄×C₆₇	1	2	yes	—
418	#124	C₂×C₄×C₆₇	1	2	yes	—
419	#124	C₂×C₄×C₆₇	1	2	yes	—
420	#123	—	2	2	no	—
421	#123	—	2	2	no	—
422	#123	—	2	2	no	—
423	C₂²⋊Q₈	—	2	1	no	—
424	C₂²⋊Q₈	—	2	2	no	—
425	#121	—	2	1	no	—
426	#121	—	2	2	no	—
427	#121	—	2	1	no	—
428	#121	—	2	2	no	—
429	#121	—	2	2	no	—
430	#121	—	2	1	no	—
431	C₄×Q₈	—	2	1	no	—
432	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
433	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
434	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
435	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
436	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
437	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
438	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
439	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
440	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
441	C₄×Q₈	D₄×C₆₇	1	2	yes	—
442	C₄×D₄	—	2	1	no	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	1	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	1	no	—
448	C₄×D₄	—	2	1	no	—
449	C₄×D₄	—	2	2	no	—
450	C₄×D₄	—	2	2	no	—
451	C₄×D₄	—	2	1	no	—
452	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
453	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
454	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
455	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
456	#117	—	2	2	no	—
457	#117	—	2	1	no	—
458	#117	—	2	2	no	—
459	#117	—	2	1	no	—
460	#117	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	—	2	2	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
464	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
465	C₂×C₄²	—	2	1	no	—
466	C₂×C₄²	—	2	2	no	—
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	—	2	1	no	—
469	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
470	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
474	C₄⋊C₈	—	2	1	no	—
475	C₄⋊C₈	—	2	2	no	—
476	C₄⋊C₈	—	2	2	no	—
477	C₄⋊C₈	—	2	1	no	—
478	#104	—	2	2	no	—
479	#104	—	2	4	no	—
480	#104	—	2	4	no	—
481	#104	—	2	2	no	—
482	#104	—	4	2	no	—
483	#104	—	4	2	no	—
484	#104	—	4	2	no	—
485	#97	—	2	2	no	—
486	#97	—	2	1	no	—
487	C₄×C₈	—	2	1	no	—
488	C₄×C₈	—	2	2	no	—
489	#95	—	2	2	no	—
490	#95	—	2	2	no	—
491	#95	—	2	1	no	—
492	#95	—	2	2	no	—
493	#41	—	2	1	no	—
494	#41	—	2	2	no	—
495	#41	—	2	2	no	—
496	#41	—	2	1	no	—
497	#41	—	2	2	no	—
498	#41	—	2	1	no	—
499	#41	—	2	1	no	—
500	#41	—	2	2	no	—
501	#41	—	4	4	no	—
502	C₂×Q₈	—	4	2	no	—
503	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
504	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
505	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
506	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
507	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
515	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
527	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
528	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	2	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	2	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	#41×C₆₇	1	2	yes	complemented
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	—	4	4	no	—
549	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
550	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
551	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
552	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
553	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
554	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
555	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
556	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
557	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
558	C₂²⋊C₄	—	2	1	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	2	no	—
566	C₂²⋊C₄	—	2	2	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	2	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₄²	—	2	1	no	—
575	C₄²	—	2	1	no	—
576	C₄²	—	2	2	no	—
577	C₄²	—	2	1	no	—
578	C₄²	—	2	2	no	—
579	C₄²	—	4	2	no	—
580	C₄²	—	4	2	no	—
581	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
582	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
583	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
584	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
585	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
586	C₄²	C₂×D₄×C₆₇	1	2	yes	—
587	C₄²	C₂×D₄×C₆₇	1	2	yes	—
588	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
589	C₄²	#41×C₆₇	1	2	yes	complemented
590	C₂³	—	2	2	no	—
591	C₂³	—	2	1	no	—
592	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
593	Q₈	—	2	2	no	—
594	Q₈	—	2	2	no	—
595	Q₈	—	2	1	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	2	1	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	4	2	no	—
600	D₄	—	2	1	no	—
601	D₄	—	2	2	no	—
602	D₄	—	2	1	no	—
603	D₄	—	2	2	no	—
604	D₄	—	4	4	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	2	no	—
607	D₄	—	4	1	no	—
608	D₄	—	4	2	no	—
609	D₄	—	4	2	no	—
610	D₄	—	8	1	no	—
611	D₄	—	8	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	2	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	4	4	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	2	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	2	no	—
631	C₂×C₄	—	4	2	no	—
632	C₂×C₄	—	4	2	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	4	2	no	—
635	C₂×C₄	—	8	1	no	—
636	C₂×C₄	—	8	2	no	—
637	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
638	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
639	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
640	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
641	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
642	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
643	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
644	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
645	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
646	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
647	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
648	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
649	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
650	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
651	C₈	—	2	2	no	—
652	C₈	—	4	1	no	—
653	C₈	—	4	2	no	—
654	C₂²	—	2	1	no	—
655	C₂²	—	2	1	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	4	1	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	1	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	1	no	—
662	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
663	C₂²	#548×C₆₇	1	1	yes	characteristic
664	C₂²	#550×C₆₇	1	1	yes	characteristic
665	C₄	—	2	1	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	1	no	—
669	C₄	—	2	2	no	—
670	C₄	—	2	1	no	—
671	C₄	—	2	2	no	—
672	C₄	—	4	2	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	—	4	1	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	8	2	no	—
679	C₄	#421×C₆₇	1	2	yes	—
680	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
681	C₄	#520×C₆₇	1	1	yes	characteristic
682	C₄	#548×C₆₇	1	1	yes	characteristic
683	C₄	#550×C₆₇	1	1	yes	characteristic
684	C₂	—	2	1	no	—
685	C₂	—	2	1	no	—
686	C₂	—	8	2	no	—
687	C₂	—	8	1	no	—
688	C₂	C₆₇×#2341	1	1	yes	characteristic
689	C₂	C₂×#421×C₆₇	1	1	yes	characteristic
690	C₂	C₆₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2295	C₂	semidirect
4. C₆₇×#1980	C₂	semidirect
5. C₆₇×#1980	C₂	semidirect
6. C₆₇×#1963	C₂	semidirect
7. C₆₇×#1962	C₂	semidirect
8. C₆₇×#1806	C₂	semidirect
9. C₆₇×#1806	C₂	semidirect
10. C₆₇×#1759	C₂	semidirect
11. C₄×#104×C₆₇	C₂	semidirect
12. #559×C₆₇	C₄	semidirect
13. #559×C₆₇	C₄	semidirect
14. #557×C₆₇	C₄	semidirect
15. #556×C₆₇	C₄	semidirect
16. #556×C₆₇	C₄	semidirect
19. #550×C₆₇	C₄	semidirect
189. M₁₆×C₆₇	#41	semidirect
243. C₄²×C₆₇	#41	semidirect
338. #22551	C₆₇	direct (N × H)
347. Q₈×M₁₆	C₂×C₆₇	semidirect
348. #2295	C₂×C₆₇	semidirect
349. #1980	C₂×C₆₇	semidirect
350. #1980	C₂×C₆₇	semidirect
351. #1963	C₂×C₆₇	semidirect
352. #1962	C₂×C₆₇	semidirect
353. #1806	C₂×C₆₇	semidirect
354. #1806	C₂×C₆₇	semidirect
355. #1759	C₂×C₆₇	semidirect
356. C₄×#104	C₂×C₆₇	semidirect
357. C₆₇	#22551	direct (N × H)
358. #559	C₄×C₆₇	semidirect
359. #559	C₄×C₆₇	semidirect
360. #557	C₄×C₆₇	semidirect
361. #556	C₄×C₆₇	semidirect
362. #556	C₄×C₆₇	semidirect
365. #550	C₄×C₆₇	semidirect
535. M₁₆	#41×C₆₇	semidirect
589. C₄²	#41×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3480	2
Q₈×M₁₆×C₆₇	2
C₆₇×#2295	2
C₆₇×#1980	2
C₆₇×#1963	2
C₆₇×#1962	2
C₆₇×#1806	2
C₆₇×#1759	2
C₄×#104×C₆₇	2
#22551	67

Subgroups — C67×#22551 #30701802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22551 #30701802