Subgroups — C₆₇×#20405 #30699656

Order 17152.

1562 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1759	C₂	1	4	yes	maximal, complemented
7	C₄×#104×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1733	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	#520×C₆₇	—	2	2	no	—
14	#520×C₆₇	C₄	1	4	yes	complemented
15	#519×C₆₇	C₄	1	2	yes	complemented
16	#518×C₆₇	—	2	2	no	—
17	#518×C₆₇	C₄	1	2	yes	complemented
18	C₄×#41×C₆₇	—	2	2	no	—
19	C₄×#41×C₆₇	C₂²	1	4	yes	—
20	C₂×#117×C₆₇	—	2	2	no	—
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#421×C₆₇	C₄	1	8	yes	complemented
26	C₂×#104×C₆₇	C₄	1	8	yes	complemented
27	C₂×#104×C₆₇	C₂²	1	2	yes	—
28	C₂×#104×C₆₇	C₂²	1	4	yes	—
29	C₂×#104×C₆₇	C₂²	1	2	yes	—
30	#405×C₆₇	—	2	2	no	—
31	#405×C₆₇	—	2	2	no	—
32	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
33	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
35	#376×C₆₇	C₂²	1	2	yes	complemented
36	C₄³×C₆₇	C₂²	1	2	yes	complemented
37	#339×C₆₇	C₂²	1	2	yes	complemented
38	#339×C₆₇	C₂²	1	2	yes	complemented
39	#339×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×#41×C₆₇	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
60	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	—	2	1	no	—
64	C₄×Q₈×C₆₇	—	2	4	no	—
65	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
66	C₄×D₄×C₆₇	—	2	8	no	—
67	C₄×D₄×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	8	no	—
69	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
70	#117×C₆₇	—	2	1	no	—
71	#117×C₆₇	—	2	1	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	4	no	—
74	#117×C₆₇	—	2	4	no	—
75	#117×C₆₇	—	2	4	no	—
76	#117×C₆₇	C₂×C₄	1	2	yes	—
77	#117×C₆₇	C₂×C₄	1	2	yes	—
78	#117×C₆₇	C₂×C₄	1	2	yes	—
79	#117×C₆₇	C₂×C₄	1	4	yes	complemented
80	#117×C₆₇	D₄	1	2	yes	—
81	#117×C₆₇	D₄	1	2	yes	—
82	#117×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	—	2	1	no	—
93	C₂×C₄²×C₆₇	—	2	2	no	—
94	C₂×C₄²×C₆₇	—	2	4	no	—
95	C₂×C₄²×C₆₇	—	2	4	no	—
96	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	complemented
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
101	#104×C₆₇	C₂×C₄	1	8	yes	complemented
102	#104×C₆₇	C₂×C₄	1	8	yes	complemented
103	#104×C₆₇	C₂×C₄	1	16	yes	complemented
104	#97×C₆₇	—	2	4	no	—
105	C₄×C₈×C₆₇	—	2	4	no	—
106	#95×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	1	no	—
108	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
109	#41×C₆₇	—	2	4	no	—
110	#41×C₆₇	—	2	4	no	—
111	#41×C₆₇	—	2	4	no	—
112	#41×C₆₇	—	2	4	no	—
113	#41×C₆₇	—	2	4	no	—
114	#41×C₆₇	—	2	2	no	—
115	#41×C₆₇	—	2	2	no	—
116	#41×C₆₇	C₄²	1	8	yes	complemented
117	#41×C₆₇	C₂²×C₄	1	4	yes	—
118	#41×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	C₄²	1	4	yes	complemented
122	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	C₄²	1	4	yes	complemented
130	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	8	no	—
142	C₂²×C₄×C₆₇	—	4	4	no	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	#41	1	2	yes	—
152	C₂²×C₄×C₆₇	#41	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	M₁₆×C₆₇	—	2	4	no	—
155	M₁₆×C₆₇	—	2	4	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	M₁₆×C₆₇	—	2	2	no	—
158	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
159	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	—	2	4	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	8	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
168	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
170	C₂²⋊C₄×C₆₇	—	2	2	no	—
171	C₂²⋊C₄×C₆₇	—	2	4	no	—
172	C₂²⋊C₄×C₆₇	—	2	2	no	—
173	C₂²⋊C₄×C₆₇	—	2	8	no	—
174	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
175	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
176	C₄²×C₆₇	—	2	8	no	—
177	C₄²×C₆₇	—	2	2	no	—
178	C₄²×C₆₇	—	2	4	no	—
179	C₄²×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	2	no	—
183	C₄²×C₆₇	—	4	8	no	—
184	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
185	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
186	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
187	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
188	C₄²×C₆₇	C₂×D₄	1	4	yes	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	—	8	2	no	—
193	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂³×C₆₇	C₄×D₄	1	2	yes	—
196	C₂³×C₆₇	C₄×D₄	1	2	yes	—
197	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₆₇	—	2	2	no	—
199	Q₈×C₆₇	—	2	4	no	—
200	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
201	D₄×C₆₇	—	2	4	no	—
202	D₄×C₆₇	—	2	4	no	—
203	D₄×C₆₇	—	2	8	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	C₂×C₄²	1	4	yes	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	8	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	4	4	no	—
218	C₂×C₄×C₆₇	—	4	8	no	—
219	C₂×C₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	8	8	no	—
222	C₂×C₄×C₆₇	—	8	2	no	—
223	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
224	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
225	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
226	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	#117	1	2	yes	—
229	C₂×C₄×C₆₇	#117	1	2	yes	—
230	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
233	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
242	C₈×C₆₇	—	2	4	no	—
243	C₈×C₆₇	—	2	4	no	—
244	C₂²×C₆₇	—	2	2	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	4	no	—
248	C₂²×C₆₇	—	4	2	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	8	4	no	—
252	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
253	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
254	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
255	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
257	C₄×C₆₇	—	2	4	no	—
258	C₄×C₆₇	—	2	8	no	—
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
264	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
265	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
266	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
267	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
268	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
269	#20405	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	2	1	no	—
271	C₂×C₆₇	—	2	1	no	—
272	C₂×C₆₇	—	2	2	no	—
273	C₂×C₆₇	—	2	2	no	—
274	C₂×C₆₇	—	4	4	no	—
275	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₆₇	#2901	1	2	yes	—
277	#3428	C₂×C₆₇	1	1	yes	characteristic
278	C₂×#421	C₂×C₆₇	1	1	yes	characteristic
279	C₂²×#104	C₂×C₆₇	1	1	yes	characteristic
280	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1759	C₂×C₆₇	1	4	yes	complemented
283	C₄×#104	C₂×C₆₇	1	4	yes	complemented
284	#1733	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#339	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₆₇	#20405	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
288	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
290	#520	—	2	2	no	—
291	#520	C₄×C₆₇	1	4	yes	complemented
292	#519	C₄×C₆₇	1	2	yes	complemented
293	#518	—	2	2	no	—
294	#518	C₄×C₆₇	1	2	yes	complemented
295	C₄×#41	—	2	2	no	—
296	C₄×#41	C₂²×C₆₇	1	4	yes	—
297	C₂×#117	—	2	2	no	—
298	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
301	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
302	#421	C₄×C₆₇	1	8	yes	complemented
303	C₂×#104	C₄×C₆₇	1	8	yes	complemented
304	C₂×#104	C₂²×C₆₇	1	2	yes	—
305	C₂×#104	C₂²×C₆₇	1	2	yes	—
306	C₂×#104	C₂²×C₆₇	1	4	yes	—
307	#405	—	2	2	no	—
308	#405	—	2	2	no	—
309	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
310	C₄×C₄⋊C₄	—	2	2	no	—
311	C₄×C₂²⋊C₄	—	2	2	no	—
312	#376	C₂²×C₆₇	1	2	yes	complemented
313	C₄³	C₂²×C₆₇	1	2	yes	complemented
314	#339	C₂²×C₆₇	1	2	yes	complemented
315	#339	C₂²×C₆₇	1	2	yes	complemented
316	#339	C₂²×C₆₇	1	4	yes	complemented
317	C₂×#41	—	2	1	no	—
318	C₂×#41	—	2	1	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
322	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
323	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×#41	C₂³×C₆₇	1	2	yes	—
325	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
336	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
337	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
338	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	—	2	1	no	—
341	C₄×Q₈	—	2	4	no	—
342	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
343	C₄×D₄	—	2	8	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
347	#117	—	2	1	no	—
348	#117	—	2	1	no	—
349	#117	—	2	2	no	—
350	#117	—	2	4	no	—
351	#117	—	2	4	no	—
352	#117	—	2	4	no	—
353	#117	C₂×C₄×C₆₇	1	2	yes	—
354	#117	C₂×C₄×C₆₇	1	2	yes	—
355	#117	C₂×C₄×C₆₇	1	2	yes	—
356	#117	C₂×C₄×C₆₇	1	4	yes	complemented
357	#117	D₄×C₆₇	1	2	yes	—
358	#117	D₄×C₆₇	1	2	yes	—
359	#117	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	4	no	—
362	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
363	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	1	no	—
366	C₂×C₂²⋊C₄	—	2	4	no	—
367	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	—	2	2	no	—
371	C₂×C₄²	—	2	4	no	—
372	C₂×C₄²	—	2	4	no	—
373	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	complemented
374	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
375	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
378	#104	C₂×C₄×C₆₇	1	8	yes	complemented
379	#104	C₂×C₄×C₆₇	1	8	yes	complemented
380	#104	C₂×C₄×C₆₇	1	16	yes	complemented
381	#97	—	2	4	no	—
382	C₄×C₈	—	2	4	no	—
383	#95	—	2	4	no	—
384	C₂⁴	—	2	1	no	—
385	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
386	#41	—	2	4	no	—
387	#41	—	2	4	no	—
388	#41	—	2	4	no	—
389	#41	—	2	4	no	—
390	#41	—	2	4	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	C₄²×C₆₇	1	8	yes	complemented
394	#41	C₂²×C₄×C₆₇	1	4	yes	—
395	#41	C₂²×C₄×C₆₇	1	4	yes	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	C₄²×C₆₇	1	4	yes	complemented
399	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	C₄²×C₆₇	1	4	yes	complemented
407	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
423	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	#41×C₆₇	1	2	yes	—
429	C₂²×C₄	#41×C₆₇	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
431	M₁₆	—	2	4	no	—
432	M₁₆	—	2	4	no	—
433	M₁₆	—	2	2	no	—
434	M₁₆	—	2	2	no	—
435	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
436	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
437	C₂×C₈	—	2	4	no	—
438	C₂×C₈	—	2	4	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
445	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
446	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
452	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
453	C₄²	—	2	8	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	4	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	4	8	no	—
461	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
462	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
463	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
464	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
465	C₄²	C₂×D₄×C₆₇	1	4	yes	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	8	2	no	—
470	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
471	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
472	C₂³	C₄×D₄×C₆₇	1	2	yes	—
473	C₂³	C₄×D₄×C₆₇	1	2	yes	—
474	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	Q₈	—	2	2	no	—
476	Q₈	—	2	4	no	—
477	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
478	D₄	—	2	4	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	8	no	—
481	D₄	—	2	2	no	—
482	D₄	—	2	4	no	—
483	D₄	C₂×C₄²×C₆₇	1	4	yes	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	8	8	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
501	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
502	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
503	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
504	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
505	C₂×C₄	#117×C₆₇	1	2	yes	—
506	C₂×C₄	#117×C₆₇	1	2	yes	—
507	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
508	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
509	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
510	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
511	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
513	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
519	C₈	—	2	4	no	—
520	C₈	—	2	4	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	8	4	no	—
529	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
530	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
531	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
532	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
533	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
534	C₄	—	2	4	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	8	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	4	no	—
539	C₄	—	4	4	no	—
540	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
541	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
542	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
543	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
544	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
545	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
546	C₂	—	2	1	no	—
547	C₂	—	2	1	no	—
548	C₂	—	2	2	no	—
549	C₂	—	2	2	no	—
550	C₂	—	4	4	no	—
551	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
552	C₂	C₆₇×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#2873	C₂	semidirect
5. C₄×#117×C₆₇	C₂	semidirect
6. C₆₇×#1759	C₂	semidirect
7. C₄×#104×C₆₇	C₂	semidirect
8. C₆₇×#1733	C₂	semidirect
9. C₂×#339×C₆₇	C₂	semidirect
14. #520×C₆₇	C₄	semidirect
15. #519×C₆₇	C₄	semidirect
17. #518×C₆₇	C₄	semidirect
25. #421×C₆₇	C₄	semidirect
26. C₂×#104×C₆₇	C₄	semidirect
32. C₄×M₁₆×C₆₇	C₂²	semidirect
35. #376×C₆₇	C₂²	semidirect
36. C₄³×C₆₇	C₂²	semidirect
37. #339×C₆₇	C₂²	semidirect
38. #339×C₆₇	C₂²	semidirect
39. #339×C₆₇	C₂²	semidirect
58. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
59. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
65. C₄×Q₈×C₆₇	C₂×C₄	semidirect
69. C₄×D₄×C₆₇	C₂×C₄	semidirect
79. #117×C₆₇	C₂×C₄	semidirect
96. C₂×C₄²×C₆₇	C₂×C₄	semidirect
101. #104×C₆₇	C₂×C₄	semidirect
102. #104×C₆₇	C₂×C₄	semidirect
103. #104×C₆₇	C₂×C₄	semidirect
116. #41×C₆₇	C₄²	semidirect
121. C₂×Q₈×C₆₇	C₄²	semidirect
129. C₂×D₄×C₆₇	C₄²	semidirect
269. #20405	C₆₇	direct (N × H)
280. #2873	C₂×C₆₇	semidirect
281. C₄×#117	C₂×C₆₇	semidirect
282. #1759	C₂×C₆₇	semidirect
283. C₄×#104	C₂×C₆₇	semidirect
284. #1733	C₂×C₆₇	semidirect
285. C₂×#339	C₂×C₆₇	semidirect
286. C₆₇	#20405	direct (N × H)
291. #520	C₄×C₆₇	semidirect
292. #519	C₄×C₆₇	semidirect
294. #518	C₄×C₆₇	semidirect
302. #421	C₄×C₆₇	semidirect
303. C₂×#104	C₄×C₆₇	semidirect
309. C₄×M₁₆	C₂²×C₆₇	semidirect
312. #376	C₂²×C₆₇	semidirect
313. C₄³	C₂²×C₆₇	semidirect
314. #339	C₂²×C₆₇	semidirect
315. #339	C₂²×C₆₇	semidirect
316. #339	C₂²×C₆₇	semidirect
335. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
336. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
342. C₄×Q₈	C₂×C₄×C₆₇	semidirect
346. C₄×D₄	C₂×C₄×C₆₇	semidirect
356. #117	C₂×C₄×C₆₇	semidirect
373. C₂×C₄²	C₂×C₄×C₆₇	semidirect
378. #104	C₂×C₄×C₆₇	semidirect
379. #104	C₂×C₄×C₆₇	semidirect
380. #104	C₂×C₄×C₆₇	semidirect
393. #41	C₄²×C₆₇	semidirect
398. C₂×Q₈	C₄²×C₆₇	semidirect
406. C₂×D₄	C₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3428	2
C₂×#421×C₆₇	2
C₂²×#104×C₆₇	2
C₆₇×#2873	2
C₄×#117×C₆₇	2
C₆₇×#1759	2
C₄×#104×C₆₇	2
C₆₇×#1733	2
C₂×#339×C₆₇	2
#20405	67

Subgroups — C67×#20405 #30699656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20405 #30699656