Subgroups — C₆₇×#17500 #30696751

Order 17152.

1324 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1492	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1490	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1484	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₆₇	—	2	1	no	—
8	C₂²×M₁₆×C₆₇	—	2	1	no	—
9	#532×C₆₇	—	2	1	no	—
10	#532×C₆₇	—	2	1	no	—
11	#532×C₆₇	—	2	1	no	—
12	#532×C₆₇	—	2	1	no	—
13	#532×C₆₇	C₄	1	2	yes	—
14	#532×C₆₇	C₄	1	2	yes	—
15	#532×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#127×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#124×C₆₇	—	2	1	no	—
20	C₂×#121×C₆₇	—	2	1	no	—
21	C₂×#121×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#432×C₆₇	—	2	1	no	—
26	#432×C₆₇	C₂²	1	2	yes	—
27	#431×C₆₇	—	2	1	no	—
28	#431×C₆₇	C₂²	1	2	yes	complemented
29	#423×C₆₇	C₂²	1	2	yes	complemented
30	#407×C₆₇	—	2	1	no	—
31	C₄×M₁₆×C₆₇	—	2	1	no	—
32	C₄×M₁₆×C₆₇	—	2	1	no	—
33	C₄×M₁₆×C₆₇	—	2	1	no	—
34	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
35	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#97×C₆₇	—	2	1	no	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
38	D₄⋊C₈×C₆₇	C₂²	1	16	yes	complemented
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	2	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	4	8	no	—
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₆₇	—	2	2	no	—
50	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₆₇	—	2	8	no	—
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
62	#127×C₆₇	—	2	1	no	—
63	#127×C₆₇	—	2	1	no	—
64	#127×C₆₇	C₂×C₄	1	2	yes	—
65	#127×C₆₇	C₂×C₄	1	2	yes	—
66	#124×C₆₇	—	2	2	no	—
67	#124×C₆₇	—	2	2	no	—
68	#124×C₆₇	—	2	2	no	—
69	#124×C₆₇	—	2	2	no	—
70	#121×C₆₇	—	2	2	no	—
71	#121×C₆₇	—	2	2	no	—
72	#121×C₆₇	—	2	2	no	—
73	#121×C₆₇	—	2	2	no	—
74	#121×C₆₇	—	2	2	no	—
75	#121×C₆₇	—	2	2	no	—
76	#121×C₆₇	C₂×C₄	1	4	yes	—
77	#121×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×D₄×C₆₇	—	2	2	no	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₄×D₄×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
85	C₄×D₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
104	C₂²⋊C₈×C₆₇	—	2	2	no	—
105	C₂²⋊C₈×C₆₇	—	2	2	no	—
106	#97×C₆₇	—	2	2	no	—
107	#97×C₆₇	—	2	2	no	—
108	#97×C₆₇	—	2	2	no	—
109	C₄×C₈×C₆₇	—	2	2	no	—
110	C₄×C₈×C₆₇	—	2	2	no	—
111	C₄×C₈×C₆₇	C₂³	1	4	yes	—
112	C₂⁴×C₆₇	—	2	2	no	—
113	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
114	#41×C₆₇	—	4	8	no	—
115	#41×C₆₇	—	4	8	no	—
116	#41×C₆₇	—	4	8	no	—
117	#41×C₆₇	—	4	8	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	4	4	no	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	4	4	no	—
129	C₂×D₄×C₆₇	—	4	4	no	—
130	C₂×D₄×C₆₇	—	4	8	no	—
131	C₂×D₄×C₆₇	—	4	4	no	—
132	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	4	2	no	—
139	C₂²×C₄×C₆₇	—	4	8	no	—
140	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	M₁₆×C₆₇	—	2	8	no	—
159	M₁₆×C₆₇	—	2	8	no	—
160	M₁₆×C₆₇	—	4	4	no	—
161	C₂×C₈×C₆₇	—	2	8	no	—
162	C₂×C₈×C₆₇	—	2	4	no	—
163	C₂×C₈×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	—	2	4	no	—
172	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	—	4	4	no	—
194	C₂³×C₆₇	—	4	4	no	—
195	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	Q₈×C₆₇	—	4	4	no	—
199	Q₈×C₆₇	—	4	4	no	—
200	D₄×C₆₇	—	2	8	no	—
201	D₄×C₆₇	—	4	4	no	—
202	D₄×C₆₇	—	4	4	no	—
203	D₄×C₆₇	—	4	8	no	—
204	D₄×C₆₇	—	4	8	no	—
205	D₄×C₆₇	—	4	4	no	—
206	D₄×C₆₇	—	4	4	no	—
207	D₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	1	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	—	4	4	no	—
227	C₂×C₄×C₆₇	—	4	8	no	—
228	C₂×C₄×C₆₇	#104	1	4	yes	—
229	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₈×C₆₇	—	2	8	no	—
243	C₈×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	2	1	no	—
245	C₂²×C₆₇	—	2	1	no	—
246	C₂²×C₆₇	—	2	1	no	—
247	C₂²×C₆₇	—	2	1	no	—
248	C₂²×C₆₇	—	2	1	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	—	8	8	no	—
256	C₂²×C₆₇	#418	1	2	yes	—
257	C₂²×C₆₇	C₂×#104	1	2	yes	—
258	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
259	C₄×C₆₇	—	2	1	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	2	1	no	—
264	C₄×C₆₇	—	2	1	no	—
265	C₄×C₆₇	—	2	1	no	—
266	C₄×C₆₇	—	4	4	no	—
267	C₄×C₆₇	#418	1	2	yes	—
268	C₄×C₆₇	C₂×#104	1	2	yes	—
269	C₄×C₆₇	C₂×#104	1	2	yes	—
270	C₄×C₆₇	#568	1	2	yes	—
271	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
272	#17500	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₆₇	—	2	2	no	—
274	C₂×C₆₇	—	2	2	no	—
275	C₂×C₆₇	—	4	1	no	—
276	C₂×C₆₇	—	8	4	no	—
277	C₂×C₆₇	#2879	1	1	yes	characteristic
278	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
279	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
280	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
281	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#1492	C₂×C₆₇	1	4	yes	complemented
284	#1490	C₂×C₆₇	1	4	yes	complemented
285	#1484	C₂×C₆₇	1	4	yes	complemented
286	C₆₇	#17500	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×#41	—	2	1	no	—
288	C₂²×M₁₆	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	—	2	1	no	—
293	#532	C₄×C₆₇	1	2	yes	—
294	#532	C₄×C₆₇	1	2	yes	—
295	#532	C₂²×C₆₇	1	2	yes	—
296	#532	C₂²×C₆₇	1	2	yes	—
297	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
298	C₂×#127	C₄×C₆₇	1	1	yes	characteristic
299	C₂×#124	—	2	1	no	—
300	C₂×#121	—	2	1	no	—
301	C₂×#121	C₄×C₆₇	1	2	yes	—
302	C₂×C₄×D₄	—	2	1	no	—
303	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
304	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
305	#432	—	2	1	no	—
306	#432	C₂²×C₆₇	1	2	yes	—
307	#431	—	2	1	no	—
308	#431	C₂²×C₆₇	1	2	yes	complemented
309	#423	C₂²×C₆₇	1	2	yes	complemented
310	#407	—	2	1	no	—
311	C₄×M₁₆	—	2	1	no	—
312	C₄×M₁₆	—	2	1	no	—
313	C₄×M₁₆	—	2	1	no	—
314	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
315	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
316	C₂×#97	—	2	1	no	—
317	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
318	D₄⋊C₈	C₂²×C₆₇	1	16	yes	complemented
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	4	8	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
329	C₂³×C₄	—	2	2	no	—
330	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
332	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
333	C₂×M₁₆	—	2	8	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	2	no	—
336	C₂×M₁₆	—	2	2	no	—
337	C₂²×C₈	—	2	2	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
341	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
342	#127	—	2	1	no	—
343	#127	—	2	1	no	—
344	#127	C₂×C₄×C₆₇	1	2	yes	—
345	#127	C₂×C₄×C₆₇	1	2	yes	—
346	#124	—	2	2	no	—
347	#124	—	2	2	no	—
348	#124	—	2	2	no	—
349	#124	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	C₂×C₄×C₆₇	1	4	yes	—
357	#121	C₂×C₄×C₆₇	1	4	yes	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
365	C₄×D₄	C₂³×C₆₇	1	4	yes	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
382	C₄⋊C₈	—	2	2	no	—
383	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	#97	—	2	2	no	—
387	#97	—	2	2	no	—
388	#97	—	2	2	no	—
389	C₄×C₈	—	2	2	no	—
390	C₄×C₈	—	2	2	no	—
391	C₄×C₈	C₂³×C₆₇	1	4	yes	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
394	#41	—	4	8	no	—
395	#41	—	4	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	4	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	8	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	1	yes	characteristic
423	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
438	M₁₆	—	2	8	no	—
439	M₁₆	—	2	8	no	—
440	M₁₆	—	4	4	no	—
441	C₂×C₈	—	2	8	no	—
442	C₂×C₈	—	2	4	no	—
443	C₂×C₈	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
455	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
456	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
457	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
459	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
460	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
467	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	4	no	—
475	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
476	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
477	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
478	Q₈	—	4	4	no	—
479	Q₈	—	4	4	no	—
480	D₄	—	2	8	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	8	no	—
485	D₄	—	4	4	no	—
486	D₄	—	4	4	no	—
487	D₄	—	4	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	#104×C₆₇	1	4	yes	—
509	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
513	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
518	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
519	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
522	C₈	—	2	8	no	—
523	C₈	—	4	4	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	8	8	no	—
536	C₂²	#418×C₆₇	1	2	yes	—
537	C₂²	C₂×#104×C₆₇	1	2	yes	—
538	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	4	4	no	—
547	C₄	#418×C₆₇	1	2	yes	—
548	C₄	C₂×#104×C₆₇	1	2	yes	—
549	C₄	C₂×#104×C₆₇	1	2	yes	—
550	C₄	#568×C₆₇	1	2	yes	—
551	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
552	C₂	—	2	2	no	—
553	C₂	—	2	2	no	—
554	C₂	—	4	1	no	—
555	C₂	—	8	4	no	—
556	C₂	C₆₇×#2879	1	1	yes	characteristic
557	C₂	C₂×#418×C₆₇	1	1	yes	characteristic
558	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2920	C₂	semidirect
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1492	C₂	semidirect
5. C₆₇×#1490	C₂	semidirect
6. C₆₇×#1484	C₂	semidirect
28. #431×C₆₇	C₂²	semidirect
29. #423×C₆₇	C₂²	semidirect
35. C₄×M₁₆×C₆₇	C₂²	semidirect
37. C₂×C₄×C₈×C₆₇	C₂²	semidirect
38. D₄⋊C₈×C₆₇	C₂²	semidirect
272. #17500	C₆₇	direct (N × H)
281. #2920	C₂×C₆₇	semidirect
282. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
283. #1492	C₂×C₆₇	semidirect
284. #1490	C₂×C₆₇	semidirect
285. #1484	C₂×C₆₇	semidirect
286. C₆₇	#17500	direct (N × H)
308. #431	C₂²×C₆₇	semidirect
309. #423	C₂²×C₆₇	semidirect
315. C₄×M₁₆	C₂²×C₆₇	semidirect
317. C₂×C₄×C₈	C₂²×C₆₇	semidirect
318. D₄⋊C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₆₇×#2920	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#1492	2
C₆₇×#1490	2
C₆₇×#1484	2
#17500	67

Subgroups — C67×#17500 #30696751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17500 #30696751