Subgroups — #30826508

Order 17152.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570792	C₂	1	1	yes	maximal, characteristic
2	#22569300	C₂	1	2	yes	maximal
3	#22559118	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559117	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559103	C₂	1	2	yes	maximal, complemented
6	#22559029	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#11761618	C₂	1	2	yes	maximal, complemented
8	#22558969	C₂	1	1	yes	maximal, characteristic
9	#22558680	C₂	1	2	yes	maximal
10	#22558494	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863950	C₂²	1	2	yes	—
13	#16863800	C₄	1	1	yes	characteristic
14	#16863774	C₄	1	1	yes	characteristic
15	#16863745	C₄	1	2	yes	—
16	#16863638	C₄	1	1	yes	characteristic
17	#16863636	C₄	1	2	yes	—
18	#16863634	C₄	1	1	yes	characteristic
19	#16863527	C₂²	1	2	yes	—
20	#16863526	—	2	2	no	—
21	#16863525	—	2	2	no	—
22	#16863523	C₂²	1	2	yes	—
23	C₈×Dic₁₃₄	C₂²	1	2	yes	—
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	C₄×#11761618	C₂²	1	2	yes	—
26	#16863456	—	2	2	no	—
27	#16863455	—	2	2	no	—
28	#16863452	C₂²	1	2	yes	—
29	#16863447	C₂²	1	1	yes	characteristic
30	#16863447	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863147	C₂²	1	1	yes	characteristic
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863117	C₂²	1	1	yes	characteristic
38	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761618	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
49	#14524042	C₂×C₄	1	2	yes	—
50	#14524041	C₂×C₄	1	2	yes	—
51	#14524033	—	2	2	no	—
52	#14524023	C₂×C₄	1	2	yes	—
53	#14524021	—	2	2	no	—
54	#14524019	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃₄	—	2	2	no	—
56	C₄×Dic₁₃₄	C₂³	1	2	yes	—
57	#14524000	—	2	2	no	—
58	C₄²×D₆₇	C₂³	1	1	yes	characteristic
59	#14523977	C₂×C₄	1	1	yes	characteristic
60	#14523976	C₂×C₄	1	2	yes	—
61	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
63	Dic₆₇⋊C₈	—	2	2	no	—
64	Dic₆₇⋊C₈	—	2	1	no	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	D₄	1	2	yes	—
67	#14523968	C₂×C₄	1	2	yes	—
68	#14523967	C₂×C₄	1	2	yes	—
69	#14523966	C₂×C₄	1	2	yes	—
70	(C₂×D₆₇)⋊C₈	—	2	2	no	—
71	(C₂×D₆₇)⋊C₈	—	2	1	no	—
72	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
73	#14523954	—	2	2	no	—
74	#14523953	—	2	2	no	—
75	#14523953	—	2	2	no	—
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₆₇	—	2	2	no	—
80	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
84	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₆₇	—	2	2	no	—
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₃₄	—	2	2	no	—
93	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
97	Dic₂₆₈	—	2	2	no	—
98	#11761618	—	2	4	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
104	C₈×D₆₇	—	2	2	no	—
105	C₈×D₆₇	—	2	2	no	—
106	C₄⋊Dic₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
108	Dic₆₇⋊C₄	—	2	2	no	—
109	Dic₆₇⋊C₄	—	2	2	no	—
110	Dic₆₇⋊C₄	—	2	2	no	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	(C₂×D₆₇)⋊C₄	—	2	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	C₄×Dic₆₇	—	2	2	no	—
116	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₆₇	#41	1	1	yes	characteristic
119	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
120	Q₁₆×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	2	2	no	—
132	Q₈×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₆₇	—	2	2	no	—
139	C₂×Dic₆₇	—	2	2	no	—
140	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
143	C₄×D₆₇	—	2	1	no	—
144	C₄×D₆₇	—	2	1	no	—
145	C₄×D₆₇	—	2	2	no	—
146	C₄×D₆₇	—	4	1	no	—
147	Dic₁₃₄	—	2	2	no	—
148	Dic₁₃₄	—	2	2	no	—
149	Dic₁₃₄	—	8	2	no	—
150	C₈×C₆₇	—	2	1	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	—	4	1	no	—
153	C₈×C₆₇	—	4	2	no	—
154	C₈×C₆₇	#131	1	2	yes	—
155	C₆₇⋊C₈	—	2	1	no	—
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	4	1	no	—
158	C₆₇⋊C₈	—	4	2	no	—
159	C₆₇⋊C₈	#131	1	2	yes	—
160	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×D₆₇	—	2	1	no	—
162	C₂×D₆₇	—	2	1	no	—
163	C₂×D₆₇	—	2	1	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	4	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
168	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
169	C₄×C₆₇	#578	1	2	yes	—
170	Dic₆₇	—	2	1	no	—
171	Dic₆₇	—	2	1	no	—
172	Dic₆₇	—	2	1	no	—
173	Dic₆₇	—	4	2	no	—
174	Dic₆₇	—	4	2	no	—
175	#17972	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	#2931	1	1	yes	characteristic
177	C₂×C₆₇	#2950	1	1	yes	characteristic
178	C₂×C₆₇	#2957	1	1	yes	characteristic
179	D₆₇	—	4	1	no	—
180	#3170	—	67	1	no	—
181	#2985	—	67	2	no	—
182	#1593	—	67	1	no	—
183	#1593	—	67	1	no	—
184	#1591	—	67	2	no	—
185	#1579	—	67	1	no	—
186	#1579	D₆₇	1	1	yes	characteristic, complemented
187	C₈×Q₁₆	—	67	2	no	—
188	#1568	—	67	1	no	—
189	#1497	—	67	2	no	—
190	#1450	—	67	1	no	—
191	C₆₇	#17972	1	1	yes	characteristic, complemented, Sylow
192	#533	—	67	2	no	—
193	#500	—	67	1	no	—
194	#494	—	67	1	no	—
195	#487	—	67	2	no	—
196	#467	—	67	1	no	—
197	#467	—	67	1	no	—
198	#467	—	67	2	no	—
199	#446	—	67	2	no	—
200	#446	—	67	2	no	—
201	#446	—	134	2	no	—
202	#446	—	134	2	no	—
203	C₈×Q₈	—	67	2	no	—
204	C₈×Q₈	C₂×D₆₇	1	2	yes	—
205	C₄×Q₁₆	—	67	2	no	—
206	C₄×Q₁₆	—	67	1	no	—
207	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
208	#433	—	134	2	no	—
209	#433	—	134	2	no	—
210	#432	—	67	2	no	—
211	#431	—	67	1	no	—
212	#431	—	67	1	no	—
213	C₂×C₄×C₈	—	67	1	no	—
214	#335	—	67	2	no	—
215	#334	—	67	1	no	—
216	#334	C₂×D₆₇	1	1	yes	characteristic
217	Q₈⋊C₈	—	67	2	no	—
218	Q₈⋊C₈	—	67	2	no	—
219	Q₈⋊C₈	—	67	2	no	—
220	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
221	#322	—	67	1	no	—
222	#322	C₂×D₆₇	1	1	yes	characteristic
223	C₈²	—	67	1	no	—
224	C₈²	—	67	1	no	—
225	C₂×Q₁₆	—	67	2	no	—
226	C₂×Q₁₆	—	67	1	no	—
227	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₈	—	67	1	no	—
229	C₂²×C₈	—	67	1	no	—
230	C₄⋊Q₈	—	67	2	no	—
231	C₄⋊Q₈	—	67	2	no	—
232	#125	—	134	2	no	—
233	C₂²⋊Q₈	—	67	2	no	—
234	C₂²⋊Q₈	—	67	2	no	—
235	C₂²⋊Q₈	—	134	2	no	—
236	C₄×Q₈	—	67	2	no	—
237	C₄×Q₈	—	134	2	no	—
238	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
239	#117	—	134	2	no	—
240	C₂×C₄²	—	67	1	no	—
241	#107	—	67	1	no	—
242	#107	—	67	2	no	—
243	#107	C₄×D₆₇	1	1	yes	characteristic
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	67	2	no	—
246	C₄⋊C₈	—	67	2	no	—
247	C₄⋊C₈	—	134	2	no	—
248	C₄⋊C₈	—	134	1	no	—
249	C₄⋊C₈	—	134	2	no	—
250	C₄⋊C₈	—	134	2	no	—
251	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
252	#103	—	67	2	no	—
253	#103	—	67	2	no	—
254	#103	—	67	2	no	—
255	#103	C₄×D₆₇	1	2	yes	—
256	C₂²⋊C₈	—	67	2	no	—
257	C₂²⋊C₈	—	134	1	no	—
258	C₂²⋊C₈	—	134	2	no	—
259	#97	—	134	2	no	—
260	#97	—	134	2	no	—
261	#97	—	134	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	1	no	—
265	C₄×C₈	—	67	1	no	—
266	C₄×C₈	—	67	1	no	—
267	C₄×C₈	—	67	1	no	—
268	C₄×C₈	—	134	2	no	—
269	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₂×Q₈	—	67	2	no	—
273	C₂×Q₈	—	134	2	no	—
274	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
275	C₂²×C₄	—	67	1	no	—
276	C₂²×C₄	—	67	1	no	—
277	C₂²×C₄	—	67	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	134	4	no	—
280	Q₁₆	—	134	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	67	1	no	—
284	C₂×C₈	—	67	1	no	—
285	C₂×C₈	—	67	2	no	—
286	C₂×C₈	—	134	2	no	—
287	C₂×C₈	—	134	2	no	—
288	C₂×C₈	—	134	1	no	—
289	C₂×C₈	—	134	2	no	—
290	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	D₄×D₆₇	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	67	2	no	—
295	C₄⋊C₄	—	134	2	no	—
296	C₄⋊C₄	—	134	2	no	—
297	C₄⋊C₄	—	134	2	no	—
298	C₄⋊C₄	—	134	2	no	—
299	C₄⋊C₄	—	134	2	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
302	C₂²⋊C₄	—	134	2	no	—
303	C₂²⋊C₄	—	134	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	67	1	no	—
306	C₄²	—	67	1	no	—
307	C₄²	—	67	1	no	—
308	C₄²	—	134	2	no	—
309	C₄²	C₂³×D₆₇	1	1	yes	characteristic
310	C₂³	—	67	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	536	2	no	—
314	Q₈	—	134	2	no	—
315	Q₈	—	134	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	268	1	no	—
319	C₂×C₄	—	67	1	no	—
320	C₂×C₄	—	67	1	no	—
321	C₂×C₄	—	67	1	no	—
322	C₂×C₄	—	134	2	no	—
323	C₂×C₄	—	134	2	no	—
324	C₂×C₄	—	134	1	no	—
325	C₂×C₄	—	134	1	no	—
326	C₂×C₄	—	134	2	no	—
327	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
329	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	268	1	no	—
335	C₈	—	268	2	no	—
336	C₈	—	67	2	no	—
337	C₈	—	134	1	no	—
338	C₈	—	134	2	no	—
339	C₈	#14524049	1	2	yes	—
340	C₂²	—	134	1	no	—
341	C₂²	—	134	1	no	—
342	C₂²	—	134	1	no	—
343	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	268	2	no	—
348	C₄	—	268	2	no	—
349	C₄	—	134	1	no	—
350	C₄	—	134	1	no	—
351	C₄	—	134	1	no	—
352	C₄	#131×D₆₇	1	1	yes	characteristic
353	C₄	C₂×#14524049	1	1	yes	characteristic
354	C₄	#16864212	1	2	yes	—
355	C₂	—	268	1	no	—
356	C₂	#22568916	1	1	yes	characteristic
357	C₂	#22569074	1	1	yes	characteristic
358	C₂	#22569123	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1579	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1579	D₆₇
hypocenter	C₆₇	#17972
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #22559118	C₂	semidirect
4. #22559117	C₂	semidirect
5. #22559103	C₂	semidirect
6. #22559029	C₂	semidirect
7. C₈×#11761618	C₂	semidirect
11. C₆₇×#1579	C₂	semidirect
186. #1579	D₆₇	semidirect
191. C₆₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#22570792	2
#22569300	2
#22559118	2
#22559117	2
#22559103	2
#22559029	2
C₈×#11761618	2
#22558969	2
#22558680	2
#22558494	2
C₆₇×#1579	2
#17972	67