Subgroups — C₆₇×#17983 #30697234

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3171	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1586	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1582	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1581	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1496	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₆₇	—	2	1	no	—
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	#529×C₆₇	C₂²	1	1	yes	characteristic
15	#501×C₆₇	C₄	1	1	yes	characteristic
16	#496×C₆₇	C₄	1	1	yes	characteristic
17	#490×C₆₇	C₄	1	1	yes	characteristic
18	#489×C₆₇	C₄	1	1	yes	characteristic
19	#469×C₆₇	C₄	1	2	yes	—
20	#468×C₆₇	C₄	1	2	yes	—
21	#446×C₆₇	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
24	C₄×D₈×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	—	2	1	no	—
26	#436×C₆₇	C₂²	1	1	yes	characteristic
27	#436×C₆₇	C₂²	1	2	yes	—
28	#435×C₆₇	—	2	1	no	—
29	C₈×D₄×C₆₇	—	2	1	no	—
30	C₈×D₄×C₆₇	—	2	1	no	—
31	C₈×D₄×C₆₇	C₂²	1	2	yes	—
32	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	#433×C₆₇	—	2	1	no	—
34	#432×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	#423×C₆₇	C₂²	1	1	yes	characteristic
37	#408×C₆₇	—	2	1	no	—
38	#405×C₆₇	—	2	1	no	—
39	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	2	yes	—
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂×#41×C₆₇	—	2	1	no	—
50	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
52	#131×C₆₇	—	4	4	no	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
63	#126×C₆₇	—	2	1	no	—
64	#125×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#124×C₆₇	—	2	1	no	—
66	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#123×C₆₇	—	2	1	no	—
68	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	#121×C₆₇	—	2	1	no	—
70	#121×C₆₇	C₂×C₄	1	2	yes	—
71	#121×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₆₇	—	2	1	no	—
74	C₄×D₄×C₆₇	—	2	1	no	—
75	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₄×D₄×C₆₇	C₂³	1	2	yes	—
77	#117×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂×C₄	1	2	yes	—
80	#106×C₆₇	C₂×C₄	1	2	yes	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	—	2	1	no	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	#102×C₆₇	C₂×C₄	1	2	yes	—
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₂²⋊C₈×C₆₇	—	2	1	no	—
94	C₂²⋊C₈×C₆₇	—	2	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	2	no	—
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	C₂²⋊C₈×C₆₇	—	2	1	no	—
98	#97×C₆₇	—	2	1	no	—
99	#97×C₆₇	—	2	1	no	—
100	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄×C₈×C₆₇	—	2	1	no	—
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	#41×C₆₇	—	4	2	no	—
110	#41×C₆₇	—	4	2	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×D₄×C₆₇	—	2	1	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
124	SD₁₆×C₆₇	—	4	2	no	—
125	D₈×C₆₇	—	4	2	no	—
126	M₁₆×C₆₇	—	4	2	no	—
127	M₁₆×C₆₇	—	4	4	no	—
128	M₁₆×C₆₇	—	4	2	no	—
129	M₁₆×C₆₇	—	4	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	4	1	no	—
136	C₂×C₈×C₆₇	—	4	2	no	—
137	C₂×C₈×C₆₇	—	4	4	no	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₆₇	—	2	1	no	—
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	2	1	no	—
155	C₂²⋊C₄×C₆₇	—	2	2	no	—
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₆₇	#41	1	1	yes	characteristic
163	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₂³×C₆₇	—	2	1	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
167	Q₈×C₆₇	—	2	2	no	—
168	Q₈×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	2	2	no	—
170	D₄×C₆₇	—	2	2	no	—
171	D₄×C₆₇	—	2	2	no	—
172	D₄×C₆₇	—	4	2	no	—
173	D₄×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	2	2	no	—
196	C₈×C₆₇	—	2	1	no	—
197	C₈×C₆₇	—	2	1	no	—
198	C₈×C₆₇	—	2	1	no	—
199	C₈×C₆₇	—	4	2	no	—
200	C₈×C₆₇	—	4	2	no	—
201	C₈×C₆₇	—	4	1	no	—
202	C₈×C₆₇	—	4	1	no	—
203	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
204	C₂²×C₆₇	—	2	1	no	—
205	C₂²×C₆₇	—	2	1	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	4	1	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄×C₆₇	—	2	1	no	—
215	C₄×C₆₇	—	2	1	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	4	1	no	—
219	C₄×C₆₇	—	4	1	no	—
220	C₄×C₆₇	—	4	2	no	—
221	C₄×C₆₇	—	4	1	no	—
222	C₄×C₆₇	#568	1	2	yes	—
223	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₄×C₆₇	#575	1	1	yes	characteristic
225	#17983	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂×C₆₇	—	4	1	no	—
227	C₂×C₆₇	—	8	2	no	—
228	C₂×C₆₇	—	8	1	no	—
229	C₂×C₆₇	#2932	1	1	yes	characteristic
230	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂×C₆₇	#2955	1	1	yes	characteristic
232	#3171	C₂×C₆₇	1	1	yes	characteristic
233	#2966	C₂×C₆₇	1	1	yes	characteristic, complemented
234	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1588	C₂×C₆₇	1	2	yes	complemented
236	#1586	C₂×C₆₇	1	2	yes	complemented
237	#1582	C₂×C₆₇	1	2	yes	complemented
238	#1581	C₂×C₆₇	1	2	yes	complemented
239	#1569	C₂×C₆₇	1	1	yes	characteristic, complemented
240	#1496	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#1495	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₆₇	#17983	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#131	—	2	1	no	—
245	#532	C₂²×C₆₇	1	1	yes	characteristic
246	#529	C₂²×C₆₇	1	1	yes	characteristic
247	#501	C₄×C₆₇	1	1	yes	characteristic
248	#496	C₄×C₆₇	1	1	yes	characteristic
249	#490	C₄×C₆₇	1	1	yes	characteristic
250	#489	C₄×C₆₇	1	1	yes	characteristic
251	#469	C₄×C₆₇	1	2	yes	—
252	#468	C₄×C₆₇	1	2	yes	—
253	#446	C₂²×C₆₇	1	1	yes	characteristic
254	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
256	C₄×D₈	C₂²×C₆₇	1	2	yes	—
257	#436	—	2	1	no	—
258	#436	C₂²×C₆₇	1	1	yes	characteristic
259	#436	C₂²×C₆₇	1	2	yes	—
260	#435	—	2	1	no	—
261	C₈×D₄	—	2	1	no	—
262	C₈×D₄	—	2	1	no	—
263	C₈×D₄	C₂²×C₆₇	1	2	yes	—
264	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
265	#433	—	2	1	no	—
266	#432	C₂²×C₆₇	1	1	yes	characteristic
267	#431	C₂²×C₆₇	1	1	yes	characteristic
268	#423	C₂²×C₆₇	1	1	yes	characteristic
269	#408	—	2	1	no	—
270	#405	—	2	1	no	—
271	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
273	#335	C₂²×C₆₇	1	2	yes	—
274	#334	C₂²×C₆₇	1	2	yes	—
275	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
276	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
277	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
278	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
279	#322	C₂²×C₆₇	1	2	yes	—
280	#322	C₂²×C₆₇	1	2	yes	—
281	C₂×#41	—	2	1	no	—
282	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
283	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
284	#131	—	4	4	no	—
285	C₂×M₁₆	—	2	1	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	1	no	—
288	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
293	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
294	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
295	#126	—	2	1	no	—
296	#125	C₂×C₄×C₆₇	1	1	yes	characteristic
297	#124	—	2	1	no	—
298	#124	C₂×C₄×C₆₇	1	1	yes	characteristic
299	#123	—	2	1	no	—
300	C₂²⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
301	#121	—	2	1	no	—
302	#121	C₂×C₄×C₆₇	1	1	yes	characteristic
303	#121	C₂×C₄×C₆₇	1	2	yes	—
304	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₄×D₄	C₂³×C₆₇	1	2	yes	—
309	#117	—	2	1	no	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	#107	C₂×C₄×C₆₇	1	2	yes	—
312	#106	C₂×C₄×C₆₇	1	2	yes	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
320	#103	C₂×C₄×C₆₇	1	2	yes	—
321	#102	C₂×C₄×C₆₇	1	2	yes	—
322	#102	C₂×C₄×C₆₇	1	2	yes	—
323	#102	C₂×C₄×C₆₇	1	2	yes	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	2	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	2	1	no	—
330	#97	—	2	1	no	—
331	#97	—	2	1	no	—
332	#97	C₂×C₄×C₆₇	1	1	yes	characteristic
333	#97	C₂×C₄×C₆₇	1	1	yes	characteristic
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
336	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₆₇	1	2	yes	—
338	C₄×C₈	C₂³×C₆₇	1	2	yes	—
339	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
340	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
350	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
356	SD₁₆	—	4	2	no	—
357	D₈	—	4	2	no	—
358	M₁₆	—	4	2	no	—
359	M₁₆	—	4	4	no	—
360	M₁₆	—	4	2	no	—
361	M₁₆	—	4	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	1	no	—
368	C₂×C₈	—	4	2	no	—
369	C₂×C₈	—	4	4	no	—
370	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
372	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
373	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
382	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
383	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
393	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₄²	#41×C₆₇	1	1	yes	characteristic
395	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
396	C₂³	—	2	1	no	—
397	C₂³	—	2	2	no	—
398	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	2	2	no	—
403	D₄	—	2	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	4	no	—
407	D₄	—	4	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
421	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
422	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
424	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	2	2	no	—
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	2	1	no	—
431	C₈	—	4	2	no	—
432	C₈	—	4	2	no	—
433	C₈	—	4	1	no	—
434	C₈	—	4	1	no	—
435	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	#568×C₆₇	1	2	yes	—
455	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
456	C₄	#575×C₆₇	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	8	2	no	—
459	C₂	—	8	1	no	—
460	C₂	C₆₇×#2932	1	1	yes	characteristic
461	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
462	C₂	C₆₇×#2955	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	C₂×C₄×D₄
prime radical	C₆₇×#3171	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2966	C₂	semidirect
3. C₆₇×#2931	C₂	semidirect
4. C₆₇×#1588	C₂	semidirect
5. C₆₇×#1586	C₂	semidirect
6. C₆₇×#1582	C₂	semidirect
7. C₆₇×#1581	C₂	semidirect
8. C₆₇×#1569	C₂	semidirect
9. C₆₇×#1496	C₂	semidirect
10. C₆₇×#1495	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
225. #17983	C₆₇	direct (N × H)
233. #2966	C₂×C₆₇	semidirect
234. #2931	C₂×C₆₇	semidirect
235. #1588	C₂×C₆₇	semidirect
236. #1586	C₂×C₆₇	semidirect
237. #1582	C₂×C₆₇	semidirect
238. #1581	C₂×C₆₇	semidirect
239. #1569	C₂×C₆₇	semidirect
240. #1496	C₂×C₆₇	semidirect
241. #1495	C₂×C₆₇	semidirect
242. #1451	C₂×C₆₇	semidirect
243. C₆₇	#17983	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3171	2
C₆₇×#2966	2
C₆₇×#2931	2
C₆₇×#1588	2
C₆₇×#1586	2
C₆₇×#1582	2
C₆₇×#1581	2
C₆₇×#1569	2
C₆₇×#1496	2
C₆₇×#1495	2
C₆₇×#1451	2
#17983	67

Subgroups — C67×#17983 #30697234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17983 #30697234