Subgroups — #32150467

Order 18176.

636 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601442	C₂	1	1	yes	maximal, characteristic
2	#22596948	C₂	1	1	yes	maximal, characteristic
3	#22596932	C₂	1	1	yes	maximal, characteristic
4	#22590112	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590111	C₂	1	4	yes	maximal, complemented
6	C₂×#17985823	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17985822	C₂	1	1	yes	maximal, characteristic, complemented
8	#365×D₇₁	C₂	1	4	yes	maximal, complemented
9	C₇₁×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986765	—	2	1	no	—
11	C₂×#14556134	C₂²	1	1	yes	characteristic
12	#17986479	—	2	1	no	—
13	#17986417	C₄	1	4	yes	complemented
14	#17986415	C₄	1	4	yes	complemented
15	C₂×#14556068	C₂²	1	1	yes	characteristic
16	M₃₂×D₇₁	C₂²	1	4	yes	—
17	#17986047	—	2	1	no	—
18	#17986046	—	2	2	no	—
19	C₂×#14556064	C₂²	1	1	yes	characteristic
20	#108×D₇₁	C₂²	1	4	yes	—
21	#17986042	—	2	1	no	—
22	#17986040	—	2	2	no	—
23	#17986034	C₂²	1	2	yes	—
24	C₂×#14556060	C₂²	1	1	yes	characteristic
25	#106×D₇₁	C₂²	1	2	yes	—
26	#17985823	C₂²	1	4	yes	—
27	#17985822	C₂²	1	4	yes	—
28	#17985821	C₂²	1	2	yes	—
29	#17985821	C₂²	1	2	yes	—
30	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
32	#427×C₇₁	C₂²	1	1	yes	characteristic
33	#365×C₇₁	C₂²	1	2	yes	—
34	#365×C₇₁	C₂²	1	2	yes	—
35	C₂×#11765865	D₄	1	1	yes	characteristic
36	#14556136	—	4	2	no	—
37	#14556134	—	4	2	no	—
38	#14556134	C₂×C₄	1	2	yes	—
39	#14556134	C₂×C₄	1	2	yes	—
40	#14556134	Q₈	1	2	yes	—
41	#14556134	Q₈	1	2	yes	—
42	C₂×#11765849	—	2	1	no	—
43	C₂×#11765849	D₄	1	1	yes	characteristic
44	C₂×#11765848	D₄	1	2	yes	—
45	M₁₆×D₇₁	—	2	4	no	—
46	C₂×C₈×D₇₁	C₂³	1	2	yes	—
47	C₄×Dic₁₄₂	—	4	1	no	—
48	C₄×D₂₈₄	—	4	1	no	—
49	(C₂²⋊D₇₁)⋊C₄	—	4	2	no	—
50	#14556086	—	2	2	no	—
51	C₂×C₄⋊Dic₇₁	—	2	1	no	—
52	C₄⋊C₄×D₇₁	—	2	2	no	—
53	#14556068	C₂×C₄	1	2	yes	complemented
54	#14556068	C₂³	1	2	yes	—
55	#14556067	C₂×C₄	1	8	yes	complemented
56	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
57	C₁₆×D₇₁	C₂×C₄	1	8	yes	complemented
58	#14556064	—	2	1	no	—
59	#14556064	C₂³	1	2	yes	—
60	#14556063	—	2	4	no	—
61	#14556061	—	2	2	no	—
62	#14556060	—	2	1	no	—
63	#14556060	C₂³	1	2	yes	—
64	#14556059	—	2	2	no	—
65	C₂×M₁₆×C₇₁	—	2	1	no	—
66	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
67	#117×C₇₁	—	2	1	no	—
68	M₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
69	M₃₂×C₇₁	C₂³	1	2	yes	—
70	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
71	#108×C₇₁	—	2	1	no	—
72	#108×C₇₁	C₂³	1	2	yes	—
73	#107×C₇₁	—	2	1	no	—
74	#106×C₇₁	C₂³	1	1	yes	characteristic
75	#106×C₇₁	C₂³	1	1	yes	characteristic
76	#11765865	C₄⋊C₄	1	2	yes	—
77	#11765865	C₄⋊C₄	1	2	yes	—
78	#11765865	SD₁₆	1	2	yes	—
79	#11765865	SD₁₆	1	2	yes	—
80	C₂×Dic₁₄₂	SD₁₆	1	1	yes	characteristic
81	C₂×D₂₈₄	SD₁₆	1	1	yes	characteristic
82	C₂×C₂²⋊D₇₁	SD₁₆	1	2	yes	—
83	C₂²×Dic₇₁	—	4	1	no	—
84	C₂×C₄×D₇₁	—	4	2	no	—
85	C₂×C₄×D₇₁	C₂×D₄	1	2	yes	—
86	#11765849	—	2	2	no	—
87	#11765849	—	4	1	no	—
88	#11765849	C₄⋊C₄	1	2	yes	—
89	#11765849	C₄⋊C₄	1	2	yes	—
90	#11765848	—	4	4	no	—
91	#11765848	C₄⋊C₄	1	4	yes	—
92	#11765848	C₄⋊C₄	1	4	yes	—
93	C₂×C₇₁⋊C₈	—	4	1	no	—
94	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
95	C₈×D₇₁	—	4	4	no	—
96	C₈×D₇₁	C₂²×C₄	1	4	yes	—
97	C₈×D₇₁	C₂×Q₈	1	4	yes	—
98	C₄⋊Dic₇₁	—	2	2	no	—
99	C₄⋊Dic₇₁	—	4	1	no	—
100	Dic₇₁⋊C₄	—	4	2	no	—
101	(C₂×D₇₁)⋊C₄	—	4	2	no	—
102	C₄×Dic₇₁	—	4	2	no	—
103	C₇₁⋊C₁₆	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
105	M₁₆×C₇₁	—	2	2	no	—
106	M₁₆×C₇₁	—	4	1	no	—
107	C₂×C₈×C₇₁	—	4	1	no	—
108	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₁	—	2	1	no	—
115	C₄⋊C₄×C₇₁	—	2	1	no	—
116	C₂²⋊C₄×C₇₁	—	4	1	no	—
117	C₄²×C₇₁	—	4	1	no	—
118	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
119	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₇₁	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	—	4	1	no	—
123	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₇₁	#106	1	4	yes	—
130	C₂²⋊D₇₁	#106	1	4	yes	—
131	C₂×Dic₇₁	—	4	2	no	—
132	C₂×Dic₇₁	—	16	1	no	—
133	C₂×Dic₇₁	C₂×SD₁₆	1	2	yes	—
134	D₂₈₄	#106	1	2	yes	—
135	D₂₈₄	#106	1	2	yes	—
136	C₄×D₇₁	—	16	2	no	—
137	C₄×D₇₁	C₂×C₄⋊C₄	1	4	yes	—
138	C₄×D₇₁	C₂×SD₁₆	1	4	yes	—
139	Dic₁₄₂	#106	1	2	yes	—
140	Dic₁₄₂	#106	1	2	yes	—
141	C₈×C₇₁	—	4	2	no	—
142	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
146	C₇₁⋊C₈	—	4	2	no	—
147	C₇₁⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
148	C₂²×C₇₁	—	4	1	no	—
149	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
150	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
151	C₂²×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂×D₇₁	—	4	2	no	—
153	C₂×D₇₁	C₂×#106	1	4	yes	—
154	C₄×C₇₁	—	8	1	no	—
155	C₄×C₇₁	—	8	1	no	—
156	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
157	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
158	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₇₁	—	8	2	no	—
161	Dic₇₁	C₂×#106	1	4	yes	—
162	#33404	—	71	1	no	maximal, Sylow
163	C₂×C₇₁	—	2	1	no	—
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	C₂²×#106	1	1	yes	characteristic
167	D₇₁	—	2	4	no	—
168	#3407	—	71	1	no	—
169	#2916	—	71	1	no	—
170	#2912	—	71	1	no	—
171	#2155	—	71	1	no	—
172	#2155	—	71	4	no	—
173	#2155	D₇₁	1	1	yes	characteristic, complemented
174	C₂×#365	—	71	1	no	—
175	C₂×#365	—	71	1	no	—
176	C₂×#365	—	71	4	no	—
177	C₇₁	#33404	1	1	yes	characteristic, complemented, Sylow
178	#568	—	142	1	no	—
179	C₂×#131	—	71	1	no	—
180	#520	—	142	1	no	—
181	#504	—	71	4	no	—
182	#504	—	71	4	no	—
183	C₂×M₃₂	—	71	1	no	—
184	C₂×M₃₂	—	71	4	no	—
185	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
186	#430	—	142	2	no	—
187	#430	—	142	1	no	—
188	C₂×#108	—	71	1	no	—
189	C₂×#108	—	71	4	no	—
190	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
191	#428	—	142	2	no	—
192	#428	—	142	1	no	—
193	#427	—	71	2	no	—
194	#427	C₂×D₇₁	1	1	yes	characteristic
195	C₂×#106	—	71	1	no	—
196	C₂×#106	—	71	2	no	—
197	#365	—	71	4	no	—
198	#365	—	71	4	no	—
199	#365	—	71	2	no	—
200	#365	—	71	2	no	—
201	#365	C₂×D₇₁	1	2	yes	—
202	#365	C₂×D₇₁	1	2	yes	—
203	C₂×#41	—	71	1	no	—
204	#131	—	284	2	no	—
205	#131	—	284	2	no	—
206	#131	—	71	2	no	—
207	#131	—	71	2	no	—
208	#131	—	71	2	no	—
209	#131	—	71	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	71	1	no	—
212	C₂×M₁₆	—	71	2	no	—
213	C₂×M₁₆	—	142	4	no	—
214	C₂×M₁₆	—	142	1	no	—
215	C₂²×C₈	—	71	2	no	—
216	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
217	C₄×Q₈	—	284	1	no	—
218	C₄×D₄	—	284	1	no	—
219	C₄×D₄	—	284	2	no	—
220	#117	—	2	1	no	—
221	#117	—	142	2	no	—
222	C₂×C₄⋊C₄	—	142	2	no	—
223	C₂×C₄⋊C₄	—	142	1	no	—
224	M₃₂	—	71	2	no	—
225	M₃₂	—	71	2	no	—
226	M₃₂	—	71	8	no	—
227	M₃₂	C₄×D₇₁	1	2	yes	complemented
228	M₃₂	C₂²×D₇₁	1	2	yes	—
229	C₂×C₁₆	—	71	2	no	—
230	C₂×C₁₆	—	71	8	no	—
231	C₂×C₁₆	C₄×D₇₁	1	2	yes	complemented
232	#108	—	2	1	no	—
233	#108	—	71	2	no	—
234	#108	—	142	4	no	—
235	#108	—	142	1	no	—
236	#108	C₂²×D₇₁	1	2	yes	—
237	#107	—	2	1	no	—
238	#107	—	142	2	no	—
239	#106	—	71	2	no	—
240	#106	—	142	2	no	—
241	#106	—	142	1	no	—
242	#106	C₂²×D₇₁	1	1	yes	characteristic
243	#106	C₂²×D₇₁	1	1	yes	characteristic
244	#41	—	71	2	no	—
245	#41	—	71	2	no	—
246	#41	—	71	2	no	—
247	#41	—	71	2	no	—
248	C₂×Q₈	—	71	1	no	—
249	C₂×D₄	—	71	2	no	—
250	C₂×D₄	—	71	1	no	—
251	C₂²×C₄	—	284	2	no	—
252	C₂²×C₄	—	284	1	no	—
253	C₂²×C₄	—	71	2	no	—
254	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	284	4	no	—
258	M₁₆	—	284	1	no	—
259	M₁₆	—	71	2	no	—
260	M₁₆	—	71	2	no	—
261	M₁₆	—	71	4	no	—
262	M₁₆	—	71	4	no	—
263	M₁₆	—	142	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	284	4	no	—
266	C₂×C₈	—	284	1	no	—
267	C₂×C₈	—	71	2	no	—
268	C₂×C₈	—	71	4	no	—
269	C₂×C₈	—	71	4	no	—
270	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
272	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
273	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	284	2	no	—
279	C₄⋊C₄	—	284	1	no	—
280	C₄⋊C₄	—	142	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	284	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	284	2	no	—
285	C₁₆	—	71	4	no	—
286	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
287	C₂³	—	71	2	no	—
288	C₂³	SD₁₆×D₇₁	1	1	yes	characteristic
289	Q₈	—	71	2	no	—
290	Q₈	—	71	2	no	—
291	D₄	—	71	4	no	—
292	D₄	—	71	4	no	—
293	D₄	—	71	2	no	—
294	D₄	—	71	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	71	2	no	—
299	C₂×C₄	—	71	4	no	—
300	C₂×C₄	—	71	4	no	—
301	C₂×C₄	—	1136	2	no	—
302	C₂×C₄	—	1136	1	no	—
303	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	284	2	no	—
311	C₈	—	71	4	no	—
312	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
313	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
314	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
315	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
316	C₂²	—	4	1	no	—
317	C₂²	—	284	2	no	—
318	C₂²	—	71	4	no	—
319	C₂²	#106×D₇₁	1	1	yes	characteristic
320	C₂²	#106×D₇₁	1	1	yes	characteristic
321	C₂²	C₂×SD₁₆×D₇₁	1	1	yes	characteristic
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	568	2	no	—
325	C₄	—	71	4	no	—
326	C₄	#106×D₇₁	1	1	yes	characteristic
327	C₄	#106×D₇₁	1	1	yes	characteristic
328	C₄	C₂×C₄⋊C₄×D₇₁	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₇₁	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	142	4	no	—
334	C₂	C₂×#106×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#106×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2155	C₂
socle	C₂×C₇₁	C₂²×#106
prime radical	C₂×#11765865	D₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2155	D₇₁
hypocenter	C₇₁	#33404
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
4. #22590112	C₂	semidirect
5. #22590111	C₂	semidirect
6. C₂×#17985823	C₂	semidirect
7. C₂×#17985822	C₂	semidirect
8. #365×D₇₁	C₂	semidirect
9. C₇₁×#2155	C₂	semidirect
13. #17986417	C₄	semidirect
14. #17986415	C₄	semidirect
53. #14556068	C₂×C₄	semidirect
55. #14556067	C₂×C₄	semidirect
56. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
57. C₁₆×D₇₁	C₂×C₄	semidirect
68. M₃₂×C₇₁	C₂×C₄	semidirect
70. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
173. #2155	D₇₁	semidirect
177. C₇₁	#33404	semidirect
227. M₃₂	C₄×D₇₁	semidirect
231. C₂×C₁₆	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22601442	2
#22596948	2
#22596932	2
#22590112	2
#22590111	2
C₂×#17985823	2
C₂×#17985822	2
#365×D₇₁	2
C₇₁×#2155	2
#33404	71