Subgroups — #31896624

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589260	C₂	1	2	yes	maximal, complemented
2	C₄×#14556060	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586835	C₂	1	2	yes	maximal, complemented
4	#22586629	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#3169	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	1	yes	characteristic
7	C₂×#14556060	—	2	1	no	—
8	C₂×#14556055	—	2	4	no	—
9	C₄×#11765849	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
11	#17985762	—	2	1	no	—
12	#17985733	—	2	4	no	—
13	#17985721	—	2	1	no	—
14	#17985720	—	2	1	no	—
15	#532×C₇₁	C₂²	1	2	yes	—
16	#500×C₇₁	C₄	1	1	yes	characteristic, complemented
17	#493×C₇₁	C₄	1	1	yes	characteristic, complemented
18	#486×C₇₁	C₄	1	2	yes	complemented
19	#466×C₇₁	—	2	1	no	—
20	#466×C₇₁	—	2	1	no	—
21	C₄×D₈×C₇₁	—	2	1	no	—
22	C₄×D₈×C₇₁	—	2	1	no	—
23	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#11765849	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄×Dic₇₁	—	2	2	no	—
27	#14556060	—	2	4	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
29	#14556055	—	4	8	no	—
30	C₂²⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
31	#14556039	—	2	1	no	—
32	C₄×C₇₁⋊C₈	—	2	1	no	—
33	C₄×C₇₁⋊C₈	—	2	1	no	—
34	C₇₁⋊#95	—	2	2	no	—
35	C₂×#41×C₇₁	—	2	2	no	—
36	C₂×D₈×C₇₁	—	2	1	no	—
37	C₂×D₈×C₇₁	—	2	1	no	—
38	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
41	#127×C₇₁	C₂×C₄	1	2	yes	—
42	#124×C₇₁	—	2	2	no	—
43	#121×C₇₁	—	2	2	no	—
44	#121×C₇₁	—	2	2	no	—
45	C₄×D₄×C₇₁	—	2	2	no	—
46	C₄×D₄×C₇₁	—	2	2	no	—
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	#107×C₇₁	—	2	1	no	—
49	#107×C₇₁	—	2	1	no	—
50	#102×C₇₁	—	2	2	no	—
51	#102×C₇₁	—	2	2	no	—
52	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
56	C₂²×Dic₇₁	—	4	2	no	—
57	#11765849	—	2	4	no	—
58	C₂×C₇₁⋊C₈	—	2	2	no	—
59	C₂×C₇₁⋊C₈	—	2	2	no	—
60	C₄⋊Dic₇₁	—	2	4	no	—
61	C₄⋊Dic₇₁	—	4	2	no	—
62	C₄×Dic₇₁	—	4	4	no	—
63	#41×C₇₁	—	4	4	no	—
64	#41×C₇₁	—	4	4	no	—
65	C₂×Q₈×C₇₁	—	2	2	no	—
66	C₂×D₄×C₇₁	—	2	2	no	—
67	C₂×D₄×C₇₁	—	2	2	no	—
68	C₂×D₄×C₇₁	—	2	2	no	—
69	C₂×D₄×C₇₁	—	4	2	no	—
70	C₂²×C₄×C₇₁	—	4	2	no	—
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	D₈×C₇₁	—	4	2	no	—
75	D₈×C₇₁	—	4	2	no	—
76	C₂×C₈×C₇₁	—	2	2	no	—
77	C₂×C₈×C₇₁	—	4	1	no	—
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₇₁	—	2	2	no	—
83	C₄⋊C₄×C₇₁	—	2	2	no	—
84	C₂²⋊C₄×C₇₁	—	4	2	no	—
85	C₂²⋊C₄×C₇₁	—	4	2	no	—
86	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇₁	#41	1	1	yes	characteristic
89	C₄²×C₇₁	#41	1	1	yes	characteristic
90	C₂³×C₇₁	—	4	2	no	—
91	C₂³×C₇₁	#127	1	1	yes	characteristic
92	Q₈×C₇₁	—	4	4	no	—
93	D₄×C₇₁	—	4	2	no	—
94	D₄×C₇₁	—	4	2	no	—
95	D₄×C₇₁	—	4	4	no	—
96	D₄×C₇₁	—	8	2	no	—
97	D₄×C₇₁	—	8	2	no	—
98	D₄×C₇₁	—	8	2	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	4	1	no	—
102	C₂×C₄×C₇₁	—	4	2	no	—
103	C₂×C₄×C₇₁	—	8	4	no	—
104	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×Dic₇₁	—	4	4	no	—
112	C₂×Dic₇₁	—	8	2	no	—
113	C₂×Dic₇₁	—	8	2	no	—
114	C₈×C₇₁	—	2	1	no	—
115	C₈×C₇₁	—	2	1	no	—
116	C₈×C₇₁	—	2	1	no	—
117	C₈×C₇₁	#104	1	2	yes	complemented
118	C₇₁⋊C₈	—	4	2	no	—
119	C₇₁⋊C₈	—	4	2	no	—
120	C₂²×C₇₁	—	2	1	no	—
121	C₂²×C₇₁	—	2	1	no	—
122	C₂²×C₇₁	—	8	2	no	—
123	C₂²×C₇₁	—	8	2	no	—
124	C₂²×C₇₁	—	8	2	no	—
125	C₂²×C₇₁	#390	1	1	yes	characteristic
126	C₂²×C₇₁	#496	1	2	yes	—
127	C₄×C₇₁	—	2	1	no	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	—	8	2	no	—
131	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
132	C₄×C₇₁	#421	1	1	yes	characteristic
133	C₄×C₇₁	#442	1	2	yes	—
134	C₄×C₇₁	#469	1	2	yes	—
135	Dic₇₁	—	8	4	no	—
136	#9721	—	71	1	no	maximal, Sylow
137	C₂×C₇₁	—	2	2	no	—
138	C₂×C₇₁	—	16	2	no	—
139	C₂×C₇₁	#1972	1	1	yes	characteristic
140	C₂×C₇₁	#1976	1	1	yes	characteristic
141	C₂×C₇₁	#1980	1	1	yes	characteristic
142	#3169	D₇₁	1	1	yes	characteristic
143	#1980	—	71	2	no	—
144	C₄×#106	—	71	1	no	—
145	#1502	—	71	2	no	—
146	#1451	—	71	1	no	—
147	C₇₁	#9721	1	1	yes	characteristic, complemented, Sylow
148	#532	C₂×D₇₁	1	2	yes	—
149	#500	Dic₇₁	1	1	yes	characteristic, complemented
150	#493	Dic₇₁	1	1	yes	characteristic, complemented
151	#486	Dic₇₁	1	2	yes	complemented
152	#466	—	2	1	no	—
153	#466	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	C₄×D₈	—	2	1	no	—
156	#431	—	71	1	no	—
157	C₂×#106	—	142	1	no	—
158	C₂×#104	—	142	4	no	—
159	C₄×M₁₆	—	71	1	no	—
160	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
161	C₄×C₄⋊C₄	—	71	2	no	—
162	#340	—	142	1	no	—
163	#327	—	142	4	no	—
164	#322	—	142	1	no	—
165	#322	—	142	1	no	—
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×M₁₆	—	71	2	no	—
170	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
173	#127	C₂×Dic₇₁	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	142	2	no	—
180	C₂×C₄²	—	142	2	no	—
181	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#106	—	142	4	no	—
185	C₄⋊C₈	—	142	1	no	—
186	#104	—	284	8	no	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	71	2	no	—
190	#97	—	142	1	no	—
191	C₄×C₈	—	142	1	no	—
192	C₄×C₈	—	142	1	no	—
193	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
194	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
197	#95	—	142	2	no	—
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	284	2	no	—
207	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	142	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	142	2	no	—
216	C₂×C₈	—	142	2	no	—
217	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	284	2	no	—
224	C₄⋊C₄	—	142	4	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	284	4	no	—
228	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
230	C₄²	#11765864	1	1	yes	characteristic
231	C₄²	#11765864	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	#14556124	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	284	4	no	—
247	C₂×C₄	—	568	2	no	—
248	C₂×C₄	—	568	2	no	—
249	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
252	C₂×C₄	#14556099	1	1	yes	characteristic
253	C₂×C₄	#14556099	1	1	yes	characteristic
254	C₂×C₄	#14556115	1	1	yes	characteristic
255	C₂×C₄	#14556152	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	284	2	no	—
260	C₈	—	284	2	no	—
261	C₈	#14556055	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	#17985902	1	1	yes	characteristic
268	C₂²	#17986387	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	2	no	—
273	C₄	—	568	4	no	—
274	C₄	C₂×#14556055	1	1	yes	characteristic
275	C₄	#17986016	1	1	yes	characteristic
276	C₄	#17986111	1	2	yes	—
277	C₄	#17986249	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	#22589217	1	1	yes	characteristic
281	C₂	#22589239	1	1	yes	characteristic
282	C₂	#22589260	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556099
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting, prime radical	C₇₁×#3169	C₂
socle	C₂²×C₇₁	#390
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#3169	D₇₁
hypocenter	C₇₁	#9721
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22589260	C₂	semidirect
2. C₄×#14556060	C₂	semidirect
3. #22586835	C₂	semidirect
4. #22586629	C₂	semidirect
16. #500×C₇₁	C₄	semidirect
17. #493×C₇₁	C₄	semidirect
18. #486×C₇₁	C₄	semidirect
117. C₈×C₇₁	#104	semidirect
147. C₇₁	#9721	semidirect
149. #500	Dic₇₁	semidirect
150. #493	Dic₇₁	semidirect
151. #486	Dic₇₁	semidirect
261. C₈	#14556055	semidirect

Maximal subgroups

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

subgroup	index
#22589260	2
C₄×#14556060	2
#22586835	2
#22586629	2
C₇₁×#3169	2
#9721	71