Subgroups — #30827609

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 382 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	#22569157	C₂	1	2	yes	maximal
3	#22559105	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559057	C₂	1	2	yes	maximal, complemented
5	#22559056	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#335×D₆₇	C₂	1	1	yes	maximal, characteristic
9	#22558610	C₂	1	2	yes	maximal
10	C₈×#11761611	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	#16863524	—	2	2	no	—
20	#16863523	—	2	2	no	—
21	#16863523	C₂²	1	2	yes	—
22	Q₈×C₆₇⋊C₈	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	#16863447	C₂²	1	1	yes	characteristic
27	#16863421	—	2	2	no	—
28	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
29	#16863353	—	2	2	no	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic
33	#16863148	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	#16863117	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×#11761611	D₄	1	1	yes	characteristic
48	C₂×#11761611	D₄	1	1	yes	characteristic
49	C₂×C₈×D₆₇	—	2	2	no	—
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
52	#14524042	C₂×C₄	1	2	yes	—
53	#14524041	C₂×C₄	1	2	yes	—
54	#14524033	—	2	2	no	—
55	#14524023	C₂×C₄	1	2	yes	—
56	#14524021	—	2	2	no	—
57	#14524019	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₃₄	—	2	2	no	—
59	C₄×Dic₁₃₄	C₂³	1	2	yes	—
60	#14524000	—	2	2	no	—
61	C₄²×D₆₇	C₂³	1	1	yes	characteristic
62	#14523977	C₂×C₄	1	1	yes	characteristic
63	#14523976	C₂×C₄	1	2	yes	—
64	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	Dic₆₇⋊C₈	—	2	1	no	—
69	Dic₆₇⋊C₈	—	2	2	no	—
70	#14523968	C₂×C₄	1	2	yes	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523966	C₂×C₄	1	2	yes	—
73	(C₂×D₆₇)⋊C₈	—	2	1	no	—
74	#14523953	—	2	2	no	—
75	#14523953	D₄	1	1	yes	characteristic
76	#14523953	D₄	1	1	yes	characteristic
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	—	4	2	no	—
101	#11761611	—	2	2	no	—
102	#11761611	Q₁₆	1	4	yes	complemented
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	1	no	—
105	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₈×D₆₇	—	2	2	no	—
110	C₈×D₆₇	—	4	1	no	—
111	C₈×D₆₇	—	4	4	no	—
112	C₄⋊Dic₆₇	—	2	2	no	—
113	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	Dic₆₇⋊C₄	—	2	2	no	—
117	Dic₆₇⋊C₄	—	2	2	no	—
118	Dic₆₇⋊C₄	—	2	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	2	no	—
120	(C₂×D₆₇)⋊C₄	—	2	2	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₆₇	#41	1	1	yes	characteristic
125	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
126	Q₁₆×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	2	no	—
139	Q₈×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
150	C₄×D₆₇	—	2	1	no	—
151	C₄×D₆₇	—	2	1	no	—
152	C₄×D₆₇	—	4	1	no	—
153	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
154	C₄×D₆₇	#135	1	2	yes	—
155	Dic₁₃₄	—	2	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	8	2	no	—
158	C₈×C₆₇	—	2	1	no	—
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	—	4	1	no	—
161	C₈×C₆₇	—	4	2	no	—
162	C₈×C₆₇	#131	1	2	yes	—
163	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
164	C₆₇⋊C₈	—	2	1	no	—
165	C₆₇⋊C₈	—	2	1	no	—
166	C₆₇⋊C₈	—	2	1	no	—
167	C₆₇⋊C₈	—	4	1	no	—
168	C₆₇⋊C₈	—	4	2	no	—
169	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
170	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₇	—	2	1	no	—
172	C₂×D₆₇	—	2	1	no	—
173	C₂×D₆₇	C₄×Q₁₆	1	2	yes	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	4	2	no	—
176	C₄×C₆₇	—	4	2	no	—
177	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
178	C₄×C₆₇	#568	1	1	yes	characteristic
179	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
181	Dic₆₇	—	2	1	no	—
182	Dic₆₇	—	2	1	no	—
183	Dic₆₇	—	4	2	no	—
184	Dic₆₇	—	4	2	no	—
185	Dic₆₇	C₄×Q₁₆	1	2	yes	—
186	#18068	—	67	1	no	maximal, Sylow
187	C₂×C₆₇	#2935	1	1	yes	characteristic
188	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
189	C₂×C₆₇	#2957	1	1	yes	characteristic
190	D₆₇	—	2	2	no	—
191	#3170	—	67	1	no	—
192	#2962	—	67	2	no	—
193	#1591	—	67	1	no	—
194	#1591	D₆₇	1	1	yes	characteristic, complemented
195	#1584	—	67	2	no	—
196	#1584	—	67	2	no	—
197	C₈×Q₁₆	—	67	1	no	—
198	C₈×Q₁₆	—	67	1	no	—
199	C₂×#335	—	67	1	no	—
200	#1481	—	67	2	no	—
201	C₈×M₁₆	—	67	1	no	—
202	C₆₇	#18068	1	1	yes	characteristic, complemented, Sylow
203	#533	—	67	2	no	—
204	#500	—	67	1	no	—
205	#494	—	67	1	no	—
206	#487	—	67	2	no	—
207	#467	—	67	1	no	—
208	#467	—	67	1	no	—
209	#467	—	67	2	no	—
210	#446	—	67	2	no	—
211	#446	—	134	2	no	—
212	#446	—	134	2	no	—
213	#446	C₂×D₆₇	1	2	yes	—
214	C₈×Q₈	—	67	2	no	—
215	C₈×Q₈	—	67	2	no	—
216	C₄×Q₁₆	—	67	2	no	—
217	C₄×Q₁₆	—	67	1	no	—
218	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
219	#431	—	67	1	no	—
220	#424	—	134	2	no	—
221	C₂×C₄⋊C₈	—	67	2	no	—
222	#405	—	134	2	no	—
223	C₄×M₁₆	—	67	1	no	—
224	C₂×C₄×C₈	—	67	1	no	—
225	#335	—	67	2	no	—
226	#335	—	67	1	no	—
227	#335	C₂×D₆₇	1	1	yes	characteristic
228	Q₈⋊C₈	—	67	2	no	—
229	Q₈⋊C₈	—	67	2	no	—
230	Q₈⋊C₈	—	67	2	no	—
231	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
232	#322	—	67	1	no	—
233	#322	—	67	1	no	—
234	C₈²	—	67	1	no	—
235	C₈²	C₂×D₆₇	1	1	yes	characteristic
236	C₂×Q₁₆	—	67	2	no	—
237	C₂×Q₁₆	—	67	1	no	—
238	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
239	C₂×M₁₆	—	67	1	no	—
240	C₂×M₁₆	—	67	1	no	—
241	C₂²×C₈	—	67	1	no	—
242	C₂²×C₈	—	67	1	no	—
243	C₂²×C₈	—	134	2	no	—
244	C₄⋊Q₈	—	67	2	no	—
245	C₄⋊Q₈	—	67	2	no	—
246	#125	—	134	2	no	—
247	C₂²⋊Q₈	—	67	2	no	—
248	C₂²⋊Q₈	—	67	2	no	—
249	C₂²⋊Q₈	—	134	2	no	—
250	C₄×Q₈	—	67	2	no	—
251	C₄×Q₈	—	134	2	no	—
252	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
253	#117	—	134	2	no	—
254	C₂×C₄²	—	67	1	no	—
255	#107	—	67	1	no	—
256	#107	—	67	2	no	—
257	#107	C₄×D₆₇	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	67	2	no	—
260	C₄⋊C₈	—	134	2	no	—
261	C₄⋊C₈	—	134	2	no	—
262	C₄⋊C₈	—	134	1	no	—
263	C₄⋊C₈	—	134	2	no	—
264	C₄⋊C₈	—	134	2	no	—
265	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
266	#103	—	67	2	no	—
267	#103	—	67	2	no	—
268	#103	—	67	2	no	—
269	#103	C₄×D₆₇	1	2	yes	—
270	C₂²⋊C₈	—	134	1	no	—
271	#97	—	2	2	no	—
272	#97	—	67	1	no	—
273	#97	—	67	1	no	—
274	#97	—	134	2	no	—
275	C₄×C₈	—	67	1	no	—
276	C₄×C₈	—	67	1	no	—
277	C₄×C₈	—	67	1	no	—
278	C₄×C₈	—	67	1	no	—
279	C₄×C₈	—	67	1	no	—
280	C₄×C₈	—	134	2	no	—
281	C₄×C₈	—	134	2	no	—
282	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
285	C₂×Q₈	—	67	2	no	—
286	C₂×Q₈	—	134	2	no	—
287	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
288	C₂²×C₄	—	67	1	no	—
289	C₂²×C₄	—	67	1	no	—
290	C₂²×C₄	—	67	1	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	268	2	no	—
293	Q₁₆	—	134	2	no	—
294	M₁₆	—	67	4	no	—
295	M₁₆	—	134	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	268	1	no	—
299	C₂×C₈	—	268	4	no	—
300	C₂×C₈	—	67	1	no	—
301	C₂×C₈	—	67	1	no	—
302	C₂×C₈	—	67	1	no	—
303	C₂×C₈	—	67	1	no	—
304	C₂×C₈	—	134	2	no	—
305	C₂×C₈	—	134	1	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
308	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
309	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
310	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	67	2	no	—
313	C₄⋊C₄	—	134	2	no	—
314	C₄⋊C₄	—	134	2	no	—
315	C₄⋊C₄	—	134	2	no	—
316	C₄⋊C₄	—	134	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	2	no	—
319	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
320	C₂²⋊C₄	—	134	2	no	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	67	1	no	—
324	C₄²	—	67	1	no	—
325	C₄²	—	67	1	no	—
326	C₄²	—	134	2	no	—
327	C₄²	C₂³×D₆₇	1	1	yes	characteristic
328	C₂³	—	67	1	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	536	2	no	—
332	Q₈	—	134	2	no	—
333	Q₈	—	134	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	268	1	no	—
337	C₂×C₄	—	67	1	no	—
338	C₂×C₄	—	67	2	no	—
339	C₂×C₄	—	67	2	no	—
340	C₂×C₄	—	67	1	no	—
341	C₂×C₄	—	67	1	no	—
342	C₂×C₄	—	134	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	—	134	1	no	—
345	C₂×C₄	—	134	1	no	—
346	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
348	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	268	1	no	—
354	C₈	—	268	2	no	—
355	C₈	—	67	2	no	—
356	C₈	—	134	1	no	—
357	C₈	—	134	1	no	—
358	C₈	—	134	1	no	—
359	C₈	#14524051	1	2	yes	—
360	C₈	Q₁₆×D₆₇	1	2	yes	—
361	C₂²	—	67	2	no	—
362	C₂²	—	134	1	no	—
363	C₂²	—	134	1	no	—
364	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	268	2	no	—
369	C₄	—	268	2	no	—
370	C₄	—	67	2	no	—
371	C₄	—	134	1	no	—
372	C₄	—	134	1	no	—
373	C₄	C₂×#14524051	1	1	yes	characteristic
374	C₄	#16864161	1	1	yes	characteristic
375	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
376	C₄	#135×D₆₇	1	1	yes	characteristic
377	C₂	—	134	2	no	—
378	C₂	#22568954	1	1	yes	characteristic
379	C₂	C₄×Q₁₆×D₆₇	1	1	yes	characteristic
380	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	#18068
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. #22559105	C₂	semidirect
4. #22559057	C₂	semidirect
5. #22559056	C₂	semidirect
6. C₈×Dic₂₆₈	C₂	semidirect
7. Q₁₆×C₆₇⋊C₈	C₂	semidirect
11. C₆₇×#1591	C₂	semidirect
102. #11761611	Q₁₆	semidirect
194. #1591	D₆₇	semidirect
202. C₆₇	#18068	semidirect

Maximal subgroups

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

subgroup	index
#22570792	2
#22569157	2
#22559105	2
#22559057	2
#22559056	2
C₈×Dic₂₆₈	2
Q₁₆×C₆₇⋊C₈	2
#335×D₆₇	2
#22558610	2
C₈×#11761611	2
C₆₇×#1591	2
#18068	67