Subgroups — #31011363

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570783	C₂	1	2	yes	maximal, complemented
2	#22562483	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562457	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562431	C₂	1	2	yes	maximal, complemented
5	#22562295	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562293	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562055	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562053	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863941	—	2	2	no	—
13	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
14	C₂×#14523984	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863771	C₂²	1	2	yes	—
18	#16863741	—	2	2	no	—
19	#16863632	C₂²	1	2	yes	—
20	#16863630	—	2	2	no	—
21	D₈×Dic₆₇	C₂²	1	2	yes	—
22	#16863478	C₂²	1	2	yes	—
23	#16863476	—	2	2	no	—
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863222	C₂²	1	1	yes	characteristic
26	#16863221	C₂²	1	2	yes	—
27	#16863198	C₂²	1	2	yes	—
28	#16863197	C₂²	1	2	yes	—
29	#16863196	C₂²	1	2	yes	—
30	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#466×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761627	—	4	2	no	—
40	C₂×D₅₃₆	C₂³	1	2	yes	—
41	C₂×#11761613	—	2	2	no	—
42	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
43	#14524042	—	2	2	no	—
44	#14524039	—	2	2	no	—
45	#14524029	—	4	2	no	—
46	#14524017	—	2	2	no	—
47	#14524015	—	4	2	no	—
48	D₄×Dic₆₇	—	2	2	no	—
49	#14524005	—	2	2	no	—
50	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
51	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
52	D₁₀₇₂	—	2	2	no	—
53	#14523984	—	2	4	no	—
54	C₂×C₆₇⋊C₁₆	—	2	2	no	—
55	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14523977	C₂³	1	2	yes	—
58	#14523976	—	2	2	no	—
59	#14523965	—	2	2	no	—
60	#14523964	—	2	2	no	—
61	#14523963	—	2	2	no	—
62	C₈×Dic₆₇	D₄	1	1	yes	characteristic
63	C₈×Dic₆₇	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#121×C₆₇	—	2	2	no	—
69	D₁₆×C₆₇	—	2	2	no	—
70	C₂×C₁₆×C₆₇	—	2	2	no	—
71	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	2	yes	—
74	#102×C₆₇	—	2	2	no	—
75	#11761627	—	8	8	no	—
76	C₂×Dic₁₃₄	—	4	2	no	—
77	C₂×D₂₆₈	—	2	2	no	—
78	C₂×C₂²⋊D₆₇	—	4	2	no	—
79	C₂×C₂²⋊D₆₇	—	4	2	no	—
80	C₂×C₂²⋊D₆₇	—	4	2	no	—
81	C₂²×Dic₆₇	—	4	2	no	—
82	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
83	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
84	C₂×C₄×D₆₇	—	4	2	no	—
85	D₅₃₆	—	2	2	no	—
86	D₅₃₆	—	2	2	no	—
87	#11761613	—	4	4	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₄⋊Dic₆₇	—	2	2	no	—
92	Dic₆₇⋊C₄	—	4	2	no	—
93	C₂²⋊Dic₆₇	—	4	2	no	—
94	(C₂×D₆₇)⋊C₄	—	4	2	no	—
95	(C₂×D₆₇)⋊C₄	—	4	2	no	—
96	C₄×Dic₆₇	D₈	1	1	yes	characteristic
97	C₄×Dic₆₇	D₈	1	1	yes	characteristic
98	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
100	C₆₇⋊C₁₆	—	2	1	no	—
101	C₆₇⋊C₁₆	#41	1	2	yes	complemented
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	D₈×C₆₇	—	2	2	no	—
106	D₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₁₆×C₆₇	—	2	1	no	—
114	C₁₆×C₆₇	#41	1	2	yes	complemented
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	8	4	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₆₇	—	8	4	no	—
127	C₂²⋊D₆₇	—	8	4	no	—
128	C₂²⋊D₆₇	—	8	4	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	8	4	no	—
132	C₂×Dic₆₇	D₁₆	1	4	yes	complemented
133	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
137	D₂₆₈	—	4	2	no	—
138	D₂₆₈	—	4	2	no	—
139	C₄×D₆₇	—	8	4	no	—
140	Dic₁₃₄	—	8	4	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	#135	1	2	yes	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
152	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₆₇	—	8	2	no	—
154	C₂×D₆₇	—	8	2	no	—
155	C₂×D₆₇	—	8	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	8	2	no	—
158	C₄×C₆₇	#532	1	1	yes	characteristic
159	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
160	Dic₆₇	—	2	1	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	C₂×D₁₆	1	2	yes	—
163	Dic₆₇	C₂×D₁₆	1	2	yes	—
164	Dic₆₇	#508	1	2	yes	—
165	#33733	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	—	2	2	no	—
167	C₂×C₆₇	—	16	2	no	—
168	C₂×C₆₇	#3169	1	1	yes	characteristic
169	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
170	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
171	D₆₇	—	16	2	no	—
172	#3169	—	67	2	no	—
173	#2252	—	67	1	no	—
174	#2246	—	67	1	no	—
175	#2240	—	67	2	no	—
176	#2215	—	67	1	no	—
177	#2215	—	67	2	no	—
178	#2215	D₆₇	1	1	yes	characteristic, complemented
179	C₄×D₁₆	—	67	1	no	—
180	C₄×D₁₆	—	67	1	no	—
181	C₄×D₁₆	—	67	2	no	—
182	C₂×C₄×C₁₆	—	67	1	no	—
183	C₆₇	#33733	1	1	yes	characteristic, complemented, Sylow
184	#532	—	134	2	no	—
185	C₂×D₁₆	—	67	2	no	—
186	C₂×D₁₆	—	67	1	no	—
187	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
188	C₂²×C₁₆	—	67	1	no	—
189	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
190	#500	—	67	2	no	—
191	#493	—	67	2	no	—
192	#486	—	134	2	no	—
193	#466	—	67	2	no	—
194	#466	—	134	2	no	—
195	#466	C₂×D₆₇	1	2	yes	—
196	C₄×D₈	—	67	2	no	—
197	C₄×D₈	—	67	2	no	—
198	C₄×D₈	—	134	2	no	—
199	C₂×C₄×C₈	—	67	1	no	—
200	#366	—	67	1	no	—
201	#366	—	67	2	no	—
202	#366	C₂×D₆₇	1	1	yes	characteristic
203	#357	—	67	2	no	—
204	#357	—	67	2	no	—
205	#357	—	67	2	no	—
206	#357	C₂×D₆₇	1	2	yes	—
207	C₄×C₁₆	—	67	1	no	—
208	C₄×C₁₆	—	67	1	no	—
209	C₄×C₁₆	—	67	1	no	—
210	C₄×C₁₆	—	67	1	no	—
211	C₂×#41	—	268	2	no	—
212	C₂×D₈	—	67	2	no	—
213	C₂×D₈	—	134	2	no	—
214	C₂×D₈	C₂²×D₆₇	1	2	yes	—
215	C₂²×C₈	—	67	1	no	—
216	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	134	2	no	—
218	#127	—	134	2	no	—
219	#124	—	268	2	no	—
220	#121	—	2	2	no	—
221	#121	—	268	2	no	—
222	#121	—	134	2	no	—
223	C₄×D₄	—	268	2	no	—
224	C₄×D₄	—	134	2	no	—
225	C₄×D₄	—	134	2	no	—
226	C₂×C₄²	—	67	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	134	4	no	—
229	D₁₆	—	134	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	67	1	no	—
232	C₂×C₁₆	—	67	1	no	—
233	C₂×C₁₆	—	134	2	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	#107	—	67	2	no	—
237	#107	—	134	2	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	134	2	no	—
241	#102	—	134	2	no	—
242	#102	—	134	2	no	—
243	C₄×C₈	—	67	1	no	—
244	C₄×C₈	—	67	1	no	—
245	C₄×C₈	—	67	1	no	—
246	C₄×C₈	—	67	1	no	—
247	#41	—	536	8	no	—
248	C₂×Q₈	—	268	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	268	2	no	—
252	C₂×D₄	—	268	2	no	—
253	C₂×D₄	—	268	2	no	—
254	C₂×D₄	—	134	2	no	—
255	C₂²×C₄	—	268	2	no	—
256	C₂²×C₄	—	268	2	no	—
257	C₂²×C₄	—	67	1	no	—
258	C₂²×C₄	—	67	1	no	—
259	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	268	4	no	—
263	D₈	—	134	2	no	—
264	D₈	—	134	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	67	1	no	—
268	C₂×C₈	—	67	1	no	—
269	C₂×C₈	—	134	2	no	—
270	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₂²⋊C₄	—	268	2	no	—
279	C₄²	—	67	1	no	—
280	C₄²	—	67	1	no	—
281	C₄²	—	67	1	no	—
282	C₄²	—	67	1	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	67	2	no	—
285	C₁₆	—	134	1	no	—
286	C₁₆	#11761627	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	268	2	no	—
289	C₂³	D₈×D₆₇	1	1	yes	characteristic
290	Q₈	—	536	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	268	2	no	—
295	D₄	—	268	2	no	—
296	D₄	—	536	4	no	—
297	D₄	—	536	4	no	—
298	D₄	—	536	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	536	4	no	—
304	C₂×C₄	—	536	4	no	—
305	C₂×C₄	—	67	4	no	—
306	C₂×C₄	—	67	1	no	—
307	C₂×C₄	—	67	1	no	—
308	C₂×C₄	—	67	1	no	—
309	C₂×C₄	—	67	1	no	—
310	C₂×C₄	—	134	2	no	—
311	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	67	2	no	—
315	C₈	—	134	1	no	—
316	C₈	#41×D₆₇	1	1	yes	characteristic
317	C₈	C₂×#11761627	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	536	2	no	—
324	C₂²	—	536	2	no	—
325	C₂²	—	536	2	no	—
326	C₂²	D₁₆×D₆₇	1	2	yes	—
327	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	536	2	no	—
331	C₄	—	67	2	no	—
332	C₄	—	67	2	no	—
333	C₄	—	67	2	no	—
334	C₄	—	134	1	no	—
335	C₄	#16863941	1	1	yes	characteristic
336	C₄	#135×D₆₇	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	16	2	no	—
339	C₂	—	1072	2	no	—
340	C₂	#22570783	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₆₇	1	1	yes	characteristic
342	C₂	#508×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2215	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#2215	D₆₇
hypocenter	C₆₇	#33733
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #22570783	C₂	semidirect
2. #22562483	C₂	semidirect
3. #22562457	C₂	semidirect
4. #22562431	C₂	semidirect
5. #22562295	C₂	semidirect
6. #22562293	C₂	semidirect
7. D₁₆×Dic₆₇	C₂	semidirect
8. #22562055	C₂	semidirect
9. #22562053	C₂	semidirect
10. C₂×C₁₆×Dic₆₇	C₂	semidirect
11. C₆₇×#2215	C₂	semidirect
101. C₆₇⋊C₁₆	#41	semidirect
114. C₁₆×C₆₇	#41	semidirect
132. C₂×Dic₆₇	D₁₆	semidirect
178. #2215	D₆₇	semidirect
183. C₆₇	#33733	semidirect
286. C₁₆	#11761627	semidirect

Maximal subgroups

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

subgroup	index
#22570783	2
#22562483	2
#22562457	2
#22562431	2
#22562295	2
#22562293	2
D₁₆×Dic₆₇	2
#22562055	2
#22562053	2
C₂×C₁₆×Dic₆₇	2
C₆₇×#2215	2
#33733	67