Subgroups — #30838474

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570720	C₂	1	1	yes	maximal, characteristic
2	#22570250	C₂	1	2	yes	maximal
3	#446×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22559734	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559728	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559727	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559718	C₂	1	2	yes	maximal, complemented
8	#22559717	C₂	1	2	yes	maximal, complemented
9	#22558954	C₂	1	1	yes	maximal, characteristic
10	#22558863	C₂	1	2	yes	maximal
11	C₆₇×#1696	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863951	C₂²	1	2	yes	—
13	C₄×Q₈×D₆₇	C₂²	1	1	yes	characteristic
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863797	—	2	1	no	—
16	#16863797	—	2	1	no	—
17	#16863792	C₂²	1	2	yes	—
18	#16863733	—	2	2	no	—
19	#16863688	C₂²	1	2	yes	—
20	#16863607	C₂²	1	2	yes	—
21	#16863606	C₂²	1	2	yes	—
22	#16863553	—	2	2	no	—
23	#16863527	C₂²	1	1	yes	characteristic
24	#16863526	C₂²	1	1	yes	characteristic
25	#16863525	—	2	1	no	—
26	#16863524	—	2	1	no	—
27	#16863523	C₂²	1	1	yes	characteristic
28	#16863490	—	2	2	no	—
29	#16863488	—	2	2	no	—
30	#16863448	C₂²	1	2	yes	—
31	#16863433	—	2	1	no	—
32	#16863433	—	2	1	no	—
33	C₄⋊C₈×D₆₇	—	2	1	no	—
34	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
35	#97×D₆₇	—	2	1	no	—
36	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
37	#16863148	C₂²	1	2	yes	—
38	#16863147	C₂²	1	1	yes	characteristic
39	#16863145	C₂²	1	2	yes	—
40	#16863143	C₂²	1	2	yes	—
41	#16863141	C₂²	1	2	yes	—
42	#500×C₆₇	C₂²	1	1	yes	characteristic
43	#477×C₆₇	C₂²	1	2	yes	—
44	#446×C₆₇	C₂²	1	1	yes	characteristic
45	#334×C₆₇	C₂²	1	1	yes	characteristic
46	#333×C₆₇	C₂²	1	2	yes	—
47	C₂×Q₈×D₆₇	D₄	1	1	yes	characteristic
48	C₂×Dic₂₆₈	—	2	2	no	—
49	C₂×#11761618	—	2	2	no	—
50	C₂×C₈×D₆₇	—	2	1	no	—
51	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
52	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
53	#14524043	C₂³	1	2	yes	—
54	#14524034	—	2	2	no	—
55	#14524023	—	2	2	no	—
56	#14524019	—	2	2	no	—
57	Q₈×Dic₆₇	—	2	2	no	—
58	Q₈×Dic₆₇	D₄	1	1	yes	characteristic
59	C₄×Dic₁₃₄	—	2	1	no	—
60	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
61	#14524008	—	2	1	no	—
62	#14524008	—	2	2	no	—
63	#14524008	D₄	1	1	yes	characteristic
64	#14524001	—	2	2	no	—
65	C₄⋊C₄×D₆₇	—	2	1	no	—
66	C₄⋊C₄×D₆₇	D₄	1	1	yes	characteristic
67	C₄²×D₆₇	—	2	1	no	—
68	C₄²×D₆₇	C₂³	1	1	yes	characteristic
69	#14523977	—	2	1	no	—
70	#14523977	—	2	1	no	—
71	#14523976	—	2	2	no	—
72	#14523974	—	2	2	no	—
73	#14523974	—	2	2	no	—
74	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
77	Dic₆₇⋊C₈	—	2	1	no	—
78	Dic₆₇⋊C₈	—	2	1	no	—
79	Dic₆₇⋊C₈	—	2	1	no	—
80	#14523968	—	2	2	no	—
81	#14523966	—	2	2	no	—
82	#14523966	—	2	2	no	—
83	(C₂×D₆₇)⋊C₈	—	2	2	no	—
84	#14523954	—	2	1	no	—
85	#14523953	—	2	1	no	—
86	#14523953	—	2	1	no	—
87	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₆₇	—	2	2	no	—
89	C₈×Dic₆₇	D₄	1	1	yes	characteristic
90	C₈×Dic₆₇	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
92	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
93	#107×C₆₇	—	2	1	no	—
94	#107×C₆₇	—	2	1	no	—
95	#107×C₆₇	—	2	2	no	—
96	C₄⋊C₈×C₆₇	—	2	1	no	—
97	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
98	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
99	#103×C₆₇	—	2	2	no	—
100	#97×C₆₇	—	2	1	no	—
101	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₃₄	—	2	1	no	—
103	C₂×Dic₁₃₄	—	2	2	no	—
104	C₂×Dic₁₃₄	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₆₇	—	4	2	no	—
106	C₂×C₄×D₆₇	—	2	1	no	—
107	C₂×C₄×D₆₇	—	2	1	no	—
108	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
111	Dic₂₆₈	—	4	4	no	—
112	#11761618	—	4	4	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₈×D₆₇	—	2	2	no	—
118	C₈×D₆₇	—	4	2	no	—
119	C₈×D₆₇	Q₁₆	1	4	yes	complemented
120	C₄⋊Dic₆₇	—	2	1	no	—
121	C₄⋊Dic₆₇	—	2	2	no	—
122	C₄⋊Dic₆₇	—	2	2	no	—
123	C₄⋊Dic₆₇	—	4	2	no	—
124	C₄⋊Dic₆₇	C₂×D₄	1	1	yes	characteristic
125	Dic₆₇⋊C₄	—	2	1	no	—
126	Dic₆₇⋊C₄	—	2	1	no	—
127	Dic₆₇⋊C₄	—	2	1	no	—
128	Dic₆₇⋊C₄	—	4	2	no	—
129	(C₂×D₆₇)⋊C₄	—	4	2	no	—
130	C₄×Dic₆₇	—	2	1	no	—
131	C₄×Dic₆₇	—	2	1	no	—
132	C₄×Dic₆₇	—	2	1	no	—
133	C₄×Dic₆₇	—	4	2	no	—
134	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₆₇	#41	1	1	yes	characteristic
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	1	no	—
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×D₆₇	#121	1	1	yes	characteristic
150	Q₈×C₆₇	—	2	2	no	—
151	Q₈×C₆₇	—	8	2	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₆₇	—	2	1	no	—
159	C₂×Dic₆₇	—	2	1	no	—
160	C₂×Dic₆₇	—	4	2	no	—
161	C₂×Dic₆₇	#121	1	1	yes	characteristic
162	C₂×Dic₆₇	#121	1	1	yes	characteristic
163	C₂×Dic₆₇	#121	1	1	yes	characteristic
164	C₄×D₆₇	—	2	2	no	—
165	C₄×D₆₇	—	4	1	no	—
166	C₄×D₆₇	—	4	2	no	—
167	C₄×D₆₇	—	4	2	no	—
168	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
169	C₄×D₆₇	#137	1	2	yes	—
170	Dic₁₃₄	—	2	2	no	—
171	Dic₁₃₄	—	4	2	no	—
172	Dic₁₃₄	—	4	2	no	—
173	Dic₁₃₄	—	4	2	no	—
174	C₈×C₆₇	—	2	1	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
178	C₆₇⋊C₈	—	2	1	no	—
179	C₆₇⋊C₈	—	4	1	no	—
180	C₆₇⋊C₈	—	4	2	no	—
181	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
182	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
183	C₂×D₆₇	—	2	1	no	—
184	C₂×D₆₇	—	2	1	no	—
185	C₂×D₆₇	#462	1	2	yes	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
192	C₄×C₆₇	#575	1	2	yes	—
193	Dic₆₇	—	2	1	no	—
194	Dic₆₇	—	2	1	no	—
195	Dic₆₇	—	4	1	no	—
196	Dic₆₇	—	4	1	no	—
197	Dic₆₇	—	8	2	no	—
198	Dic₆₇	#462	1	2	yes	—
199	#19150	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
201	C₂×C₆₇	#3054	1	1	yes	characteristic
202	C₂×C₆₇	#3067	1	1	yes	characteristic
203	D₆₇	—	2	2	no	—
204	#3160	—	67	1	no	—
205	#3104	—	67	2	no	—
206	C₂×#446	—	67	1	no	—
207	#1696	—	67	1	no	—
208	#1696	D₆₇	1	1	yes	characteristic, complemented
209	#1694	—	67	1	no	—
210	#1694	—	67	1	no	—
211	#1692	—	67	2	no	—
212	#1692	—	67	2	no	—
213	#1564	—	67	1	no	—
214	#1542	—	67	2	no	—
215	C₆₇	#19150	1	1	yes	characteristic, complemented, Sylow
216	#533	—	67	2	no	—
217	C₂×C₄×Q₈	—	67	1	no	—
218	#500	—	67	1	no	—
219	#500	C₂×D₆₇	1	1	yes	characteristic
220	#499	—	134	1	no	—
221	#499	—	134	1	no	—
222	#498	—	67	2	no	—
223	#484	—	134	2	no	—
224	#477	—	67	2	no	—
225	#477	C₂×D₆₇	1	2	yes	—
226	#462	—	67	2	no	—
227	#462	—	67	2	no	—
228	#451	—	134	2	no	—
229	#446	—	67	1	no	—
230	#446	—	67	1	no	—
231	#446	—	67	1	no	—
232	#446	—	134	1	no	—
233	#446	—	134	1	no	—
234	#446	C₂×D₆₇	1	1	yes	characteristic
235	C₄×Q₁₆	—	134	2	no	—
236	C₄×Q₁₆	—	134	2	no	—
237	#431	—	67	2	no	—
238	#427	—	134	1	no	—
239	#427	—	134	1	no	—
240	C₂×C₄⋊C₈	—	67	1	no	—
241	C₂×C₄⋊C₈	—	134	1	no	—
242	C₂×#97	—	134	1	no	—
243	C₂×C₄×C₈	—	67	1	no	—
244	#335	—	67	2	no	—
245	#334	—	67	1	no	—
246	#334	C₂×D₆₇	1	1	yes	characteristic
247	#333	—	67	2	no	—
248	#333	C₂×D₆₇	1	2	yes	—
249	#332	—	67	2	no	—
250	#332	—	67	2	no	—
251	C₂²×Q₈	—	67	1	no	—
252	C₂×Q₁₆	—	134	2	no	—
253	C₂×Q₁₆	—	134	2	no	—
254	C₂²×C₈	—	67	1	no	—
255	C₂²×C₈	—	67	1	no	—
256	C₂²×C₈	—	134	1	no	—
257	C₄⋊Q₈	—	67	2	no	—
258	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
259	#125	—	134	2	no	—
260	C₂²⋊Q₈	—	134	2	no	—
261	C₂²⋊Q₈	—	134	2	no	—
262	C₄×Q₈	—	67	1	no	—
263	C₄×Q₈	—	67	1	no	—
264	C₄×Q₈	—	67	1	no	—
265	C₄×Q₈	—	134	1	no	—
266	C₄×Q₈	—	134	2	no	—
267	C₄×Q₈	—	134	2	no	—
268	C₄×Q₈	—	134	1	no	—
269	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
270	#117	—	134	2	no	—
271	C₂×C₄⋊C₄	—	67	1	no	—
272	C₂×C₄⋊C₄	—	134	1	no	—
273	C₂×C₄²	—	67	1	no	—
274	C₂×C₄²	—	134	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	2	2	no	—
278	#107	—	134	2	no	—
279	#107	—	134	1	no	—
280	#107	—	134	1	no	—
281	#106	—	134	2	no	—
282	#106	—	134	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	67	1	no	—
285	C₄⋊C₈	—	67	2	no	—
286	C₄⋊C₈	—	134	1	no	—
287	C₄⋊C₈	—	134	1	no	—
288	C₄⋊C₈	—	134	1	no	—
289	C₄⋊C₈	—	134	1	no	—
290	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	134	2	no	—
294	#103	—	134	2	no	—
295	#103	—	134	2	no	—
296	C₂²⋊C₈	—	134	2	no	—
297	#97	—	2	1	no	—
298	#97	—	134	1	no	—
299	#97	—	134	1	no	—
300	#97	—	134	1	no	—
301	C₄×C₈	—	67	1	no	—
302	C₄×C₈	—	67	1	no	—
303	C₄×C₈	—	67	1	no	—
304	C₄×C₈	—	134	2	no	—
305	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	268	2	no	—
308	C₂×Q₈	—	67	1	no	—
309	C₂×Q₈	—	134	1	no	—
310	C₂×Q₈	—	134	2	no	—
311	C₂×Q₈	D₄×D₆₇	1	1	yes	characteristic
312	C₂²×C₄	—	67	1	no	—
313	C₂²×C₄	—	67	1	no	—
314	C₂²×C₄	—	67	1	no	—
315	C₂²×C₄	—	134	1	no	—
316	C₂²×C₄	—	134	1	no	—
317	Q₁₆	—	268	4	no	—
318	Q₁₆	—	268	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	268	2	no	—
322	C₂×C₈	—	67	1	no	—
323	C₂×C₈	—	67	4	no	—
324	C₂×C₈	—	67	1	no	—
325	C₂×C₈	—	134	2	no	—
326	C₂×C₈	—	134	2	no	—
327	C₂×C₈	—	134	1	no	—
328	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
329	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	268	2	no	—
334	C₄⋊C₄	—	268	2	no	—
335	C₄⋊C₄	—	67	1	no	—
336	C₄⋊C₄	—	134	1	no	—
337	C₄⋊C₄	—	134	1	no	—
338	C₄⋊C₄	—	134	1	no	—
339	C₄⋊C₄	—	134	2	no	—
340	C₄⋊C₄	—	134	2	no	—
341	C₄⋊C₄	—	134	1	no	—
342	C₄⋊C₄	D₄×D₆₇	1	1	yes	characteristic
343	C₂²⋊C₄	—	268	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	268	2	no	—
346	C₄²	—	67	1	no	—
347	C₄²	—	67	1	no	—
348	C₄²	—	67	1	no	—
349	C₄²	—	134	1	no	—
350	C₄²	—	134	1	no	—
351	C₄²	—	134	1	no	—
352	C₄²	C₂³×D₆₇	1	1	yes	characteristic
353	C₂³	—	67	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	8	2	no	—
356	Q₈	—	268	2	no	—
357	Q₈	—	268	2	no	—
358	Q₈	—	268	2	no	—
359	Q₈	—	134	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	268	1	no	—
364	C₂×C₄	—	268	2	no	—
365	C₂×C₄	—	268	2	no	—
366	C₂×C₄	—	268	2	no	—
367	C₂×C₄	—	67	1	no	—
368	C₂×C₄	—	67	2	no	—
369	C₂×C₄	—	67	2	no	—
370	C₂×C₄	—	67	1	no	—
371	C₂×C₄	—	67	1	no	—
372	C₂×C₄	—	134	1	no	—
373	C₂×C₄	—	134	1	no	—
374	C₂×C₄	—	134	2	no	—
375	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
377	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₈	—	268	1	no	—
382	C₈	—	268	2	no	—
383	C₈	—	67	2	no	—
384	C₈	—	134	1	no	—
385	C₈	Q₁₆×D₆₇	1	2	yes	—
386	C₂²	—	67	2	no	—
387	C₂²	—	134	1	no	—
388	C₂²	—	134	1	no	—
389	C₂²	#121×D₆₇	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	268	1	no	—
395	C₄	—	268	1	no	—
396	C₄	—	536	2	no	—
397	C₄	—	67	2	no	—
398	C₄	—	134	1	no	—
399	C₄	—	134	1	no	—
400	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
401	C₄	#137×D₆₇	1	1	yes	characteristic
402	C₄	#16864195	1	2	yes	—
403	C₂	—	134	2	no	—
404	C₂	#462×D₆₇	1	1	yes	characteristic
405	C₂	#22569820	1	1	yes	characteristic
406	C₂	#22569927	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₆₇
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1696	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂²×D₆₇	#121
hypercenter	#1696	D₆₇
hypocenter	C₆₇	#19150
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1696

Complements

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

N	H ≅ G/N	type
4. #22559734	C₂	semidirect
5. #22559728	C₂	semidirect
6. #22559727	C₂	semidirect
7. #22559718	C₂	semidirect
8. #22559717	C₂	semidirect
11. C₆₇×#1696	C₂	semidirect
119. C₈×D₆₇	Q₁₆	semidirect
208. #1696	D₆₇	semidirect
215. C₆₇	#19150	semidirect

Maximal subgroups

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

subgroup	index
#22570720	2
#22570250	2
#446×D₆₇	2
#22559734	2
#22559728	2
#22559727	2
#22559718	2
#22559717	2
#22558954	2
#22558863	2
C₆₇×#1696	2
#19150	67