Subgroups — #30826510

Order 17152.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 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	#22569305	C₂	1	2	yes	maximal
3	#22559116	C₂	1	2	yes	maximal, complemented
4	#22559107	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559025	C₂	1	2	yes	maximal, complemented
6	Q₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559015	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558971	C₂	1	1	yes	maximal, characteristic
9	#22558680	C₂	1	2	yes	maximal
10	#22558491	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1591	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	#16863526	C₂²	1	2	yes	—
20	#16863524	—	2	2	no	—
21	#16863523	—	2	2	no	—
22	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
23	#16863520	C₂²	1	2	yes	—
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	C₄×#11761618	C₂²	1	2	yes	—
26	#16863458	—	2	2	no	—
27	#16863456	—	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	#16863149	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863118	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×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	1	yes	characteristic
67	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
68	#14523968	C₂×C₄	1	2	yes	—
69	#14523967	C₂×C₄	1	2	yes	—
70	#14523966	C₂×C₄	1	2	yes	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	—	2	1	no	—
73	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
75	#14523953	—	2	2	no	—
76	#14523953	—	2	2	no	—
77	C₄×C₆₇⋊C₈	—	2	2	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₇	—	2	2	no	—
82	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
86	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	—	2	2	no	—
88	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
89	#103×C₆₇	C₂×C₄	1	2	yes	—
90	#97×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₃₄	—	2	2	no	—
95	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
99	Dic₂₆₈	—	2	2	no	—
100	#11761618	—	2	4	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₈×D₆₇	—	2	2	no	—
108	C₈×D₆₇	—	2	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	Dic₆₇⋊C₄	—	2	2	no	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	2	2	no	—
118	C₄×Dic₆₇	—	2	2	no	—
119	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₆₇	#41	1	1	yes	characteristic
122	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₆₇	—	2	2	no	—
143	C₂×Dic₆₇	—	2	2	no	—
144	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
147	C₄×D₆₇	—	2	1	no	—
148	C₄×D₆₇	—	2	1	no	—
149	C₄×D₆₇	—	2	1	no	—
150	C₄×D₆₇	—	2	1	no	—
151	C₄×D₆₇	—	4	1	no	—
152	Dic₁₃₄	—	2	2	no	—
153	Dic₁₃₄	—	2	2	no	—
154	Dic₁₃₄	—	8	2	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	—	2	1	no	—
158	C₈×C₆₇	—	4	1	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	#131	1	2	yes	—
161	C₆₇⋊C₈	—	2	1	no	—
162	C₆₇⋊C₈	—	2	1	no	—
163	C₆₇⋊C₈	—	2	1	no	—
164	C₆₇⋊C₈	—	4	1	no	—
165	C₆₇⋊C₈	—	4	2	no	—
166	C₆₇⋊C₈	#131	1	2	yes	—
167	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₆₇	—	2	1	no	—
169	C₂×D₆₇	—	2	1	no	—
170	C₂×D₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
175	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
176	C₄×C₆₇	#578	1	1	yes	characteristic
177	C₄×C₆₇	#578	1	1	yes	characteristic
178	Dic₆₇	—	2	1	no	—
179	Dic₆₇	—	2	1	no	—
180	Dic₆₇	—	2	1	no	—
181	Dic₆₇	—	4	2	no	—
182	Dic₆₇	—	4	2	no	—
183	#17972	—	67	1	no	maximal, Sylow
184	C₂×C₆₇	#2931	1	1	yes	characteristic
185	C₂×C₆₇	#2950	1	1	yes	characteristic
186	C₂×C₆₇	#2957	1	1	yes	characteristic
187	D₆₇	—	4	1	no	—
188	#3170	—	67	1	no	—
189	#2985	—	67	2	no	—
190	#1593	—	67	2	no	—
191	#1591	—	67	1	no	—
192	#1591	D₆₇	1	1	yes	characteristic, complemented
193	#1579	—	67	2	no	—
194	C₈×Q₁₆	—	67	1	no	—
195	C₈×Q₁₆	—	67	1	no	—
196	#1568	—	67	1	no	—
197	#1497	—	67	2	no	—
198	#1450	—	67	1	no	—
199	C₆₇	#17972	1	1	yes	characteristic, complemented, Sylow
200	#533	—	67	2	no	—
201	#500	—	67	1	no	—
202	#494	—	67	1	no	—
203	#487	—	67	2	no	—
204	#467	—	67	1	no	—
205	#467	—	67	1	no	—
206	#467	—	67	2	no	—
207	#446	—	67	2	no	—
208	#446	—	134	2	no	—
209	#446	—	134	2	no	—
210	#446	C₂×D₆₇	1	2	yes	—
211	C₈×Q₈	—	67	2	no	—
212	C₈×Q₈	—	67	2	no	—
213	C₄×Q₁₆	—	67	2	no	—
214	C₄×Q₁₆	—	67	1	no	—
215	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
216	#433	—	134	2	no	—
217	#433	—	134	2	no	—
218	#432	—	67	2	no	—
219	#431	—	67	1	no	—
220	#431	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	#335	—	67	1	no	—
223	#335	C₂×D₆₇	1	1	yes	characteristic
224	#334	—	67	2	no	—
225	Q₈⋊C₈	—	67	2	no	—
226	Q₈⋊C₈	—	67	2	no	—
227	Q₈⋊C₈	—	67	2	no	—
228	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
229	#322	—	67	1	no	—
230	#322	—	67	1	no	—
231	C₈²	—	67	1	no	—
232	C₈²	C₂×D₆₇	1	1	yes	characteristic
233	C₂×Q₁₆	—	67	2	no	—
234	C₂×Q₁₆	—	67	1	no	—
235	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	—	67	1	no	—
238	C₄⋊Q₈	—	67	2	no	—
239	C₄⋊Q₈	—	67	2	no	—
240	#125	—	134	2	no	—
241	C₂²⋊Q₈	—	67	2	no	—
242	C₂²⋊Q₈	—	67	2	no	—
243	C₂²⋊Q₈	—	134	2	no	—
244	C₄×Q₈	—	67	2	no	—
245	C₄×Q₈	—	134	2	no	—
246	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
247	#117	—	134	2	no	—
248	C₂×C₄²	—	67	1	no	—
249	#107	—	67	1	no	—
250	#107	—	67	2	no	—
251	#107	C₄×D₆₇	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	67	1	no	—
254	C₄⋊C₈	—	67	1	no	—
255	C₄⋊C₈	—	67	2	no	—
256	C₄⋊C₈	—	134	2	no	—
257	C₄⋊C₈	—	134	1	no	—
258	C₄⋊C₈	—	134	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
261	#103	—	67	2	no	—
262	#103	—	67	2	no	—
263	#103	—	67	2	no	—
264	#103	C₄×D₆₇	1	2	yes	—
265	C₂²⋊C₈	—	67	1	no	—
266	C₂²⋊C₈	—	67	1	no	—
267	C₂²⋊C₈	—	134	1	no	—
268	C₂²⋊C₈	—	134	2	no	—
269	#97	—	2	2	no	—
270	#97	—	134	2	no	—
271	#97	—	134	2	no	—
272	C₄×C₈	—	67	1	no	—
273	C₄×C₈	—	67	1	no	—
274	C₄×C₈	—	67	1	no	—
275	C₄×C₈	—	67	1	no	—
276	C₄×C₈	—	67	1	no	—
277	C₄×C₈	—	134	2	no	—
278	C₄×C₈	—	134	2	no	—
279	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
282	C₂×Q₈	—	67	2	no	—
283	C₂×Q₈	—	134	2	no	—
284	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
285	C₂²×C₄	—	67	1	no	—
286	C₂²×C₄	—	67	1	no	—
287	C₂²×C₄	—	67	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	134	4	no	—
290	Q₁₆	—	134	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	67	1	no	—
294	C₂×C₈	—	67	1	no	—
295	C₂×C₈	—	67	1	no	—
296	C₂×C₈	—	67	1	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	—	134	1	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
304	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	67	2	no	—
307	C₄⋊C₄	—	134	2	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₄⋊C₄	—	134	2	no	—
311	C₄⋊C₄	—	134	2	no	—
312	C₄⋊C₄	—	134	2	no	—
313	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	67	1	no	—
318	C₄²	—	67	1	no	—
319	C₄²	—	67	1	no	—
320	C₄²	—	134	2	no	—
321	C₄²	C₂³×D₆₇	1	1	yes	characteristic
322	C₂³	—	67	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	536	2	no	—
326	Q₈	—	134	2	no	—
327	Q₈	—	134	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	268	1	no	—
331	C₂×C₄	—	67	1	no	—
332	C₂×C₄	—	67	1	no	—
333	C₂×C₄	—	67	1	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	—	134	2	no	—
336	C₂×C₄	—	134	1	no	—
337	C₂×C₄	—	134	1	no	—
338	C₂×C₄	—	134	1	no	—
339	C₂×C₄	—	134	1	no	—
340	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
342	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	268	1	no	—
349	C₈	—	268	2	no	—
350	C₈	—	67	2	no	—
351	C₈	—	134	1	no	—
352	C₈	—	134	1	no	—
353	C₈	—	134	1	no	—
354	C₈	#14524051	1	2	yes	—
355	C₂²	—	134	1	no	—
356	C₂²	—	134	1	no	—
357	C₂²	—	134	1	no	—
358	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	268	2	no	—
363	C₄	—	268	2	no	—
364	C₄	—	134	1	no	—
365	C₄	—	134	1	no	—
366	C₄	—	134	1	no	—
367	C₄	#131×D₆₇	1	1	yes	characteristic
368	C₄	C₂×#14524051	1	1	yes	characteristic
369	C₄	#16864209	1	1	yes	characteristic
370	C₄	#16864210	1	1	yes	characteristic
371	C₂	—	268	1	no	—
372	C₂	#22568918	1	1	yes	characteristic
373	C₂	#22569072	1	1	yes	characteristic
374	C₂	#22569121	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×#1591	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1591	D₆₇
hypocenter	C₆₇	#17972
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #22559116	C₂	semidirect
4. #22559107	C₂	semidirect
5. #22559025	C₂	semidirect
6. Q₁₆×C₆₇⋊C₈	C₂	semidirect
7. #22559015	C₂	semidirect
11. C₆₇×#1591	C₂	semidirect
192. #1591	D₆₇	semidirect
199. 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
#22569305	2
#22559116	2
#22559107	2
#22559025	2
Q₁₆×C₆₇⋊C₈	2
#22559015	2
#22558971	2
#22558680	2
#22558491	2
C₆₇×#1591	2
#17972	67