Subgroups — #31914401

Order 18176.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985803	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#22586431	C₂	1	2	yes	maximal, complemented
5	C₂×#467×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₇₁	—	2	1	no	—
7	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
8	C₂×#14556062	—	2	4	no	—
9	C₂×#14556062	—	2	1	no	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	#17985803	—	2	4	no	—
12	C₂²⋊(C₇₁⋊C₁₆)	—	2	2	no	—
13	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₇₁	—	2	1	no	—
16	#467×C₇₁	—	2	2	no	—
17	#467×C₇₁	C₄	1	4	yes	complemented
18	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#103×C₇₁	—	2	1	no	—
20	C₂³×Dic₇₁	—	4	1	no	—
21	C₂×C₄⋊Dic₇₁	—	2	4	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×C₄⋊Dic₇₁	—	4	2	no	—
24	C₂×C₄⋊Dic₇₁	—	4	1	no	—
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
26	C₂×C₇₁⋊C₁₆	—	2	2	no	—
27	C₂×C₇₁⋊C₁₆	—	2	2	no	—
28	#14556062	—	2	4	no	—
29	#14556062	—	2	2	no	—
30	#14556062	D₄	1	4	yes	—
31	C₂²×Q₈×C₇₁	—	2	1	no	—
32	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
33	C₂×Q₁₆×C₇₁	—	2	2	no	—
34	C₂×Q₁₆×C₇₁	—	2	1	no	—
35	C₂×Q₁₆×C₇₁	—	2	2	no	—
36	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
37	C₂²×C₈×C₇₁	—	2	2	no	—
38	C₂²×C₈×C₇₁	—	2	2	no	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	D₄	1	2	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₇₁	—	2	4	no	—
44	C₂²⋊Q₈×C₇₁	—	4	2	no	—
45	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
47	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
48	#107×C₇₁	—	2	1	no	—
49	#107×C₇₁	C₂×C₄	1	2	yes	complemented
50	#103×C₇₁	—	2	4	no	—
51	C₂²×Dic₇₁	—	4	4	no	—
52	C₂²×Dic₇₁	—	4	2	no	—
53	C₂²×Dic₇₁	—	8	2	no	—
54	C₂²×Dic₇₁	—	8	2	no	—
55	C₄⋊Dic₇₁	—	2	4	no	—
56	C₄⋊Dic₇₁	—	4	4	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₇₁⋊C₁₆	—	4	2	no	—
59	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
60	C₂×Q₈×C₇₁	—	2	2	no	—
61	C₂×Q₈×C₇₁	—	4	1	no	—
62	C₂×Q₈×C₇₁	—	4	1	no	—
63	C₂×Q₈×C₇₁	—	4	1	no	—
64	C₂×Q₈×C₇₁	—	4	1	no	—
65	C₂×Q₈×C₇₁	—	4	1	no	—
66	C₂²×C₄×C₇₁	—	2	1	no	—
67	C₂²×C₄×C₇₁	—	2	1	no	—
68	C₂²×C₄×C₇₁	—	2	1	no	—
69	C₂²×C₄×C₇₁	—	2	1	no	—
70	C₂²×C₄×C₇₁	—	4	1	no	—
71	C₂²×C₄×C₇₁	—	4	1	no	—
72	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₇₁	—	2	4	no	—
77	Q₁₆×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	—	2	1	no	—
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	4	no	—
81	C₂×C₈×C₇₁	—	2	4	no	—
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	#41	1	2	yes	—
87	C₄⋊C₄×C₇₁	—	2	2	no	—
88	C₄⋊C₄×C₇₁	—	4	4	no	—
89	C₄⋊C₄×C₇₁	—	4	1	no	—
90	C₂²⋊C₄×C₇₁	—	4	4	no	—
91	C₂³×C₇₁	—	2	1	no	—
92	C₂³×C₇₁	—	2	1	no	—
93	C₂³×C₇₁	—	2	1	no	—
94	C₂³×C₇₁	—	2	1	no	—
95	C₂³×C₇₁	#102	1	2	yes	—
96	C₂³×C₇₁	SD₃₂	1	2	yes	—
97	C₂³×C₇₁	Q₃₂	1	2	yes	—
98	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₇₁	—	4	2	no	—
100	Q₈×C₇₁	—	4	2	no	—
101	Q₈×C₇₁	—	8	1	no	—
102	Q₈×C₇₁	—	8	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	—	4	2	no	—
111	C₂×C₄×C₇₁	—	8	1	no	—
112	C₂×C₄×C₇₁	—	8	1	no	—
113	C₂×C₄×C₇₁	—	8	1	no	—
114	C₂×C₄×C₇₁	—	8	1	no	—
115	C₂×C₄×C₇₁	#102	1	2	yes	—
116	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
122	C₂×Dic₇₁	—	4	4	no	—
123	C₂×Dic₇₁	—	8	8	no	—
124	C₂×Dic₇₁	—	8	2	no	—
125	C₂×Dic₇₁	—	8	2	no	—
126	C₈×C₇₁	—	2	2	no	—
127	C₈×C₇₁	C₄×D₄	1	2	yes	—
128	C₈×C₇₁	#121	1	2	yes	—
129	C₂²×C₇₁	—	2	2	no	—
130	C₂²×C₇₁	—	2	2	no	—
131	C₂²×C₇₁	—	2	2	no	—
132	C₂²×C₇₁	—	2	2	no	—
133	C₂²×C₇₁	—	2	2	no	—
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	#358	1	4	yes	complemented
136	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
137	C₂²×C₇₁	#447	1	1	yes	characteristic
138	C₂²×C₇₁	#480	1	1	yes	characteristic
139	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
140	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
141	C₂²×C₇₁	#509	1	1	yes	characteristic
142	C₂²×C₇₁	#510	1	1	yes	characteristic
143	C₄×C₇₁	—	2	2	no	—
144	C₄×C₇₁	—	8	2	no	—
145	C₄×C₇₁	—	8	2	no	—
146	C₄×C₇₁	#386	1	1	yes	characteristic
147	C₄×C₇₁	#417	1	1	yes	characteristic
148	C₄×C₇₁	#448	1	1	yes	characteristic
149	C₄×C₇₁	#481	1	1	yes	characteristic
150	Dic₇₁	—	8	4	no	—
151	#12993	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	—	2	4	no	—
153	C₂×C₇₁	#1874	1	1	yes	characteristic
154	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
155	C₂×C₇₁	#2139	1	1	yes	characteristic
156	C₂×C₇₁	#2187	1	1	yes	characteristic
157	C₂×C₇₁	#2188	1	1	yes	characteristic
158	C₂×C₇₁	#2235	1	1	yes	characteristic
159	C₂×C₇₁	#2236	1	1	yes	characteristic
160	C₂×#467	D₇₁	1	1	yes	characteristic
161	C₂²×#107	—	71	1	no	—
162	C₂×#358	—	71	2	no	—
163	C₂×C₂²⋊C₁₆	—	71	1	no	—
164	#1380	—	71	2	no	—
165	C₇₁	#12993	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	—	2	1	no	—
169	C₂²×C₄⋊C₄	—	142	1	no	—
170	C₂²×C₁₆	—	71	2	no	—
171	#467	—	2	2	no	—
172	#467	Dic₇₁	1	4	yes	complemented
173	C₂×#107	—	71	2	no	—
174	C₂×#107	—	142	4	no	—
175	C₂×#107	—	142	1	no	—
176	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
177	C₂×#103	—	2	1	no	—
178	#358	—	142	4	no	—
179	C₂²⋊C₁₆	—	142	2	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂³×C₄	—	284	1	no	—
182	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	complemented
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
192	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
193	C₂²⋊Q₈	—	2	4	no	—
194	C₂²⋊Q₈	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₄⋊C₄	—	284	2	no	—
198	C₂×C₄⋊C₄	—	284	1	no	—
199	C₂×C₄⋊C₄	—	142	4	no	—
200	C₂×C₄⋊C₄	—	142	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₁₆	—	142	2	no	—
203	C₂×C₁₆	—	142	2	no	—
204	C₂×C₁₆	—	142	2	no	—
205	#107	—	2	1	no	—
206	#107	—	71	4	no	—
207	#107	—	142	4	no	—
208	#107	—	142	2	no	—
209	#107	C₂×Dic₇₁	1	2	yes	complemented
210	#103	—	2	4	no	—
211	C₂⁴	#11765850	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	284	4	no	—
225	C₂²×C₄	—	284	2	no	—
226	C₂²×C₄	—	568	2	no	—
227	C₂²×C₄	—	568	2	no	—
228	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
229	C₂²×C₄	#11765850	1	1	yes	characteristic
230	C₂²×C₄	#11765853	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
239	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
240	C₂×C₈	D₄×D₇₁	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
242	C₂×C₈	#11765864	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	284	4	no	—
247	C₄⋊C₄	—	284	2	no	—
248	C₄⋊C₄	—	142	4	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₁₆	—	284	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#14556049	1	2	yes	—
256	C₂³	#14556071	1	2	yes	—
257	C₂³	#14556074	1	2	yes	—
258	C₂³	C₂×#11765850	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	284	4	no	—
276	C₂×C₄	—	568	8	no	—
277	C₂×C₄	—	568	2	no	—
278	C₂×C₄	—	568	2	no	—
279	C₂×C₄	#14556049	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
281	C₂×C₄	#14556096	1	1	yes	characteristic
282	C₂×C₄	#14556109	1	1	yes	characteristic
283	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
284	C₂×C₄	#14556153	1	1	yes	characteristic
285	C₂×C₄	#14556158	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	D₄×Dic₇₁	1	2	yes	—
288	C₈	#14556100	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#17985803	1	4	yes	complemented
296	C₂²	C₂×#14556049	1	1	yes	characteristic
297	C₂²	#17986132	1	1	yes	characteristic
298	C₂²	#17986310	1	1	yes	characteristic
299	C₂²	C₂×#14556071	1	1	yes	characteristic
300	C₂²	C₂×#14556074	1	1	yes	characteristic
301	C₂²	#17986434	1	1	yes	characteristic
302	C₂²	#17986439	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	568	4	no	—
307	C₄	#17985879	1	1	yes	characteristic
308	C₄	#17986000	1	1	yes	characteristic
309	C₄	#17986137	1	1	yes	characteristic
310	C₄	#17986315	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#22588658	1	1	yes	characteristic
313	C₂	C₂×#17985803	1	1	yes	characteristic
314	C₂	#22590052	1	1	yes	characteristic
315	C₂	#22590268	1	1	yes	characteristic
316	C₂	#22590273	1	1	yes	characteristic
317	C₂	#22590535	1	1	yes	characteristic
318	C₂	#22590540	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#467×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#467	D₇₁
hypocenter	C₇₁	#12993
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#17985803	C₂	semidirect
4. #22586431	C₂	semidirect
17. #467×C₇₁	C₄	semidirect
36. C₂×Q₁₆×C₇₁	C₂×C₄	semidirect
49. #107×C₇₁	C₂×C₄	semidirect
135. C₂²×C₇₁	#358	semidirect
165. C₇₁	#12993	semidirect
172. #467	Dic₇₁	semidirect
186. C₂×Q₁₆	C₂×Dic₇₁	semidirect
209. #107	C₂×Dic₇₁	semidirect
295. C₂²	#17985803	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556062	2
C₂×#17985803	2
C₂×C₂²⋊(C₇₁⋊C₁₆)	2
#22586431	2
C₂×#467×C₇₁	2
#12993	71