Subgroups — #31912680

Order 18176.

442 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597111	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985807	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985805	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985774	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586435	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556150	C₂²	1	1	yes	characteristic
9	#17986467	—	2	1	no	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17986093	C₄	1	2	yes	—
12	#17986086	—	2	2	no	—
13	#17986081	C₄	1	2	yes	—
14	C₂×#14556064	C₂²	1	1	yes	characteristic
15	#17985993	—	2	1	no	—
16	M₁₆×Dic₇₁	—	2	1	no	—
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985807	C₄	1	4	yes	complemented
19	#17985805	C₄	1	4	yes	complemented
20	#17985774	—	2	2	no	—
21	#17985763	—	2	1	no	—
22	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
24	#427×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#11765865	—	2	1	no	—
26	#14556150	—	2	1	no	—
27	#14556150	—	2	1	no	—
28	#14556150	C₂×C₄	1	2	yes	—
29	#14556150	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
31	C₂×#11765854	—	2	1	no	—
32	C₂×D₅₆₈	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765849	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
35	#14556093	—	2	2	no	—
36	#14556090	—	2	2	no	—
37	(C₂²⋊D₇₁)⋊C₄	—	4	2	no	—
38	#14556085	—	4	1	no	—
39	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
40	C₄²×D₇₁	—	4	1	no	—
41	C₂×C₇₁⋊C₁₆	—	2	1	no	—
42	C₂×C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
44	#14556064	—	2	1	no	—
45	#14556064	C₂×C₄	1	2	yes	complemented
46	#14556051	—	2	2	no	—
47	#14556048	—	2	2	no	—
48	#14556038	—	4	1	no	—
49	C₈×Dic₇₁	—	4	1	no	—
50	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
51	C₈×Dic₇₁	D₄	1	2	yes	—
52	C₂×M₁₆×C₇₁	—	2	1	no	—
53	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	#117×C₇₁	—	2	1	no	—
55	C₂×C₁₆×C₇₁	—	2	1	no	—
56	C₂×C₁₆×C₇₁	—	2	1	no	—
57	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
58	#108×C₇₁	—	2	1	no	—
59	#108×C₇₁	C₂×C₄	1	2	yes	complemented
60	#107×C₇₁	C₂×C₄	1	2	yes	—
61	#106×C₇₁	—	2	1	no	—
62	#11765865	—	2	2	no	—
63	#11765865	—	2	2	no	—
64	#11765865	—	4	1	no	—
65	#11765865	—	4	1	no	—
66	C₂×Dic₁₄₂	—	2	1	no	—
67	C₂×D₂₈₄	—	2	1	no	—
68	C₂×C₂²⋊D₇₁	—	4	1	no	—
69	C₂²×Dic₇₁	SD₁₆	1	2	yes	—
70	C₂×C₄×D₇₁	—	4	1	no	—
71	C₂×C₄×D₇₁	—	4	2	no	—
72	Dic₂₈₄	—	2	1	no	—
73	Dic₂₈₄	C₄²	1	2	yes	—
74	#11765854	—	2	2	no	—
75	#11765854	—	2	2	no	—
76	D₅₆₈	—	2	1	no	—
77	D₅₆₈	C₄²	1	2	yes	—
78	#11765849	—	2	2	no	—
79	#11765849	—	4	1	no	—
80	C₂×C₇₁⋊C₈	—	2	1	no	—
81	C₂×C₇₁⋊C₈	—	2	1	no	—
82	C₂×C₇₁⋊C₈	—	4	1	no	—
83	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
84	Dic₇₁⋊C₄	—	4	2	no	—
85	(C₂×D₇₁)⋊C₄	—	4	2	no	—
86	C₄×Dic₇₁	—	4	2	no	—
87	C₄×Dic₇₁	C₂²⋊C₄	1	2	yes	—
88	C₄×Dic₇₁	D₈	1	1	yes	characteristic
89	C₄×Dic₇₁	D₈	1	1	yes	characteristic
90	C₇₁⋊C₁₆	—	2	2	no	—
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	M₁₆×C₇₁	—	2	2	no	—
93	M₁₆×C₇₁	—	4	1	no	—
94	C₂×C₈×C₇₁	—	4	1	no	—
95	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
100	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	—	2	2	no	—
102	C₂²⋊C₄×C₇₁	—	4	1	no	—
103	C₄²×C₇₁	—	4	1	no	—
104	C₁₆×C₇₁	—	2	2	no	—
105	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂²×D₇₁	—	4	1	no	—
107	C₂×C₄×C₇₁	—	4	2	no	—
108	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
114	C₂²⋊D₇₁	—	4	2	no	—
115	C₂²⋊D₇₁	—	4	2	no	—
116	C₂×Dic₇₁	—	4	2	no	—
117	C₂×Dic₇₁	—	4	1	no	—
118	C₂×Dic₇₁	#102	1	2	yes	—
119	C₂×Dic₇₁	#102	1	2	yes	—
120	D₂₈₄	—	2	2	no	—
121	D₂₈₄	—	4	1	no	—
122	C₄×D₇₁	—	4	2	no	—
123	C₄×D₇₁	—	4	2	no	—
124	C₄×D₇₁	—	8	4	no	—
125	Dic₁₄₂	—	2	2	no	—
126	Dic₁₄₂	—	4	1	no	—
127	C₈×C₇₁	—	4	2	no	—
128	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₇₁	#117	1	1	yes	characteristic
130	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
131	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
132	C₇₁⋊C₈	—	2	2	no	—
133	C₇₁⋊C₈	—	4	2	no	—
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
136	C₂²×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
137	C₂²×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
138	C₂×D₇₁	—	4	2	no	—
139	C₂×D₇₁	—	8	2	no	—
140	C₄×C₇₁	—	8	2	no	—
141	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₇₁	#416	1	1	yes	characteristic
143	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
144	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
145	Dic₇₁	—	2	2	no	—
146	Dic₇₁	—	2	2	no	—
147	Dic₇₁	—	4	2	no	—
148	#12640	—	71	1	no	maximal, Sylow
149	C₂×C₇₁	—	2	1	no	—
150	C₂×C₇₁	—	2	1	no	—
151	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
152	C₂×C₇₁	#2143	1	2	yes	—
153	D₇₁	—	8	2	no	—
154	C₄×#135	—	71	1	no	—
155	C₂×#359	—	71	1	no	—
156	C₂×#359	—	71	1	no	—
157	C₂×#346	—	71	1	no	—
158	C₄×#108	—	71	1	no	—
159	#1382	—	71	1	no	—
160	#1382	D₇₁	1	1	yes	characteristic, complemented
161	C₇₁	#12640	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	—	71	1	no	—
163	C₄×#41	—	142	1	no	—
164	C₂²×C₁₆	—	71	1	no	—
165	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
166	C₄×Q₁₆	—	71	2	no	—
167	C₄×SD₁₆	—	142	2	no	—
168	C₄×D₈	—	71	2	no	—
169	C₂×#108	—	71	1	no	—
170	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
171	#427	C₂×D₇₁	1	1	yes	characteristic
172	#416	—	142	1	no	—
173	C₄×M₁₆	—	142	1	no	—
174	C₂×C₄×C₈	—	71	1	no	—
175	#359	—	71	4	no	—
176	#359	—	71	4	no	—
177	#346	—	142	2	no	—
178	#341	—	142	1	no	—
179	C₂×#41	—	142	1	no	—
180	#135	—	71	2	no	—
181	#135	—	71	2	no	—
182	#135	—	142	1	no	—
183	#135	—	142	1	no	—
184	C₂×Q₁₆	—	71	1	no	—
185	C₂×SD₁₆	—	142	1	no	—
186	C₂×D₈	—	71	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	142	1	no	—
189	C₂²×C₈	—	71	1	no	—
190	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
191	C₄×Q₈	—	142	2	no	—
192	C₄×D₄	—	284	2	no	—
193	C₄×D₄	—	142	2	no	—
194	#117	—	2	1	no	—
195	#117	—	284	1	no	—
196	C₂×C₄²	—	284	1	no	—
197	C₂×C₄²	—	71	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	71	2	no	—
201	C₂×C₁₆	—	142	1	no	—
202	C₂×C₁₆	—	142	1	no	—
203	C₂×C₁₆	C₄×D₇₁	1	2	yes	complemented
204	#108	—	2	1	no	—
205	#108	—	71	2	no	—
206	#108	—	142	1	no	—
207	#108	C₄×D₇₁	1	2	yes	complemented
208	#107	C₄×D₇₁	1	2	yes	—
209	#106	—	2	1	no	—
210	#103	—	142	2	no	—
211	#102	—	142	2	no	—
212	#97	—	284	1	no	—
213	C₄×C₈	—	284	1	no	—
214	C₄×C₈	—	71	2	no	—
215	C₄×C₈	—	71	2	no	—
216	#41	—	284	1	no	—
217	#41	—	284	1	no	—
218	#41	—	142	2	no	—
219	#41	—	142	2	no	—
220	C₂×Q₈	—	142	1	no	—
221	C₂×D₄	—	284	1	no	—
222	C₂×D₄	—	142	1	no	—
223	C₂²×C₄	—	284	2	no	—
224	C₂²×C₄	—	284	1	no	—
225	C₂²×C₄	—	71	2	no	—
226	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
227	Q₁₆	—	71	2	no	—
228	Q₁₆	—	142	1	no	—
229	SD₁₆	—	142	2	no	—
230	SD₁₆	—	142	2	no	—
231	D₈	—	71	2	no	—
232	D₈	—	142	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	284	1	no	—
236	M₁₆	—	142	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	284	1	no	—
239	C₂×C₈	—	71	2	no	—
240	C₂×C₈	—	142	1	no	—
241	C₂×C₈	—	142	1	no	—
242	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
245	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
246	C₂×C₈	#11765864	1	1	yes	characteristic
247	C₂×C₈	#11765866	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	284	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	284	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	284	2	no	—
254	C₄²	—	71	2	no	—
255	C₄²	—	71	1	no	—
256	C₄²	—	71	1	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	142	2	no	—
259	C₂³	—	284	1	no	—
260	C₂³	SD₁₆×D₇₁	1	1	yes	characteristic
261	Q₈	—	284	1	no	—
262	Q₈	—	142	2	no	—
263	D₄	—	284	2	no	—
264	D₄	—	284	2	no	—
265	D₄	—	284	1	no	—
266	D₄	—	142	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	284	1	no	—
269	C₂×C₄	—	284	2	no	—
270	C₂×C₄	—	284	2	no	—
271	C₂×C₄	—	284	2	no	—
272	C₂×C₄	—	568	4	no	—
273	C₂×C₄	—	71	2	no	—
274	C₂×C₄	—	71	2	no	—
275	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	#14556090	1	1	yes	characteristic
278	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
279	C₂×C₄	#14556149	1	1	yes	characteristic
280	C₂×C₄	#14556151	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	284	2	no	—
283	C₈	—	142	2	no	—
284	C₈	C₄²×D₇₁	1	1	yes	characteristic
285	C₈	#14556086	1	1	yes	characteristic
286	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
287	C₈	#14556090	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	284	2	no	—
290	C₂²	—	568	2	no	—
291	C₂²	#102×D₇₁	1	1	yes	characteristic
292	C₂²	#17986084	1	1	yes	characteristic
293	C₂²	#17986086	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	284	2	no	—
296	C₄	—	142	2	no	—
297	C₄	—	142	2	no	—
298	C₄	#17985846	1	1	yes	characteristic
299	C₄	#17985994	1	1	yes	characteristic
300	C₄	#17986079	1	1	yes	characteristic
301	C₄	#17986081	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	568	2	no	—
305	C₂	#22588154	1	1	yes	characteristic
306	C₂	#22590066	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1382	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂×#14556150	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1382	D₇₁
hypocenter	C₇₁	#12640
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×#17985774	C₂	semidirect
5. #108×Dic₇₁	C₂	semidirect
6. #22586435	C₂	semidirect
7. C₇₁×#1382	C₂	semidirect
18. #17985807	C₄	semidirect
19. #17985805	C₄	semidirect
43. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
45. #14556064	C₂×C₄	semidirect
57. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
59. #108×C₇₁	C₂×C₄	semidirect
160. #1382	D₇₁	semidirect
161. C₇₁	#12640	semidirect
203. C₂×C₁₆	C₄×D₇₁	semidirect
207. #108	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22597111	2
C₂×#17985807	2
C₂×#17985805	2
C₂×#17985774	2
#108×Dic₇₁	2
#22586435	2
C₇₁×#1382	2
#12640	71