Subgroups — #31896135

Order 18176.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589099	C₂	1	2	yes	maximal
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	#22588207	C₂	1	1	yes	maximal, characteristic
4	#22586224	C₂	1	2	yes	maximal
5	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	2	no	—
7	C₂×#14556062	—	2	1	no	—
8	C₂×#14556062	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14556038	—	2	1	no	—
11	#17985869	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	#17985762	—	2	2	no	—
14	#17985753	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#334×C₇₁	C₄	1	4	yes	complemented
18	C₂²×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₁	—	2	1	no	—
25	#14556062	—	4	2	no	—
26	#14556062	—	4	4	no	—
27	#14556062	—	4	2	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
29	C₄⋊(C₇₁⋊C₈)	Q₈	1	2	yes	—
30	#14556038	—	4	2	no	—
31	C₇₁⋊#95	—	2	1	no	—
32	C₂²×C₈×C₇₁	—	2	2	no	—
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₁	—	2	2	no	—
37	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	Q₈	1	2	yes	—
40	C₂²×Dic₇₁	—	2	1	no	—
41	C₂²×Dic₇₁	—	2	1	no	—
42	C₂²×Dic₇₁	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	4	1	no	—
45	C₂×C₇₁⋊C₈	—	4	1	no	—
46	C₄⋊Dic₇₁	—	2	2	no	—
47	C₄⋊Dic₇₁	—	2	2	no	—
48	C₄⋊Dic₇₁	—	4	2	no	—
49	C₄⋊Dic₇₁	—	4	2	no	—
50	C₄⋊Dic₇₁	—	4	2	no	—
51	C₄⋊Dic₇₁	—	4	2	no	—
52	C₄⋊Dic₇₁	—	8	2	no	—
53	C₄×Dic₇₁	—	4	1	no	—
54	C₄×Dic₇₁	—	4	1	no	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	—	2	1	no	—
60	C₂×C₈×C₇₁	—	2	4	no	—
61	C₂×C₈×C₇₁	—	4	2	no	—
62	C₂×C₈×C₇₁	—	4	2	no	—
63	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₁	D₈	1	2	yes	—
66	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
67	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₇₁	#41	1	1	yes	characteristic
71	C₂³×C₇₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₁	—	2	1	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#137	1	2	yes	—
86	C₂×Dic₇₁	—	4	1	no	—
87	C₂×Dic₇₁	—	4	1	no	—
88	C₂×Dic₇₁	—	4	1	no	—
89	C₂×Dic₇₁	—	4	1	no	—
90	C₂×Dic₇₁	—	4	1	no	—
91	C₂×Dic₇₁	—	4	1	no	—
92	C₂×Dic₇₁	—	8	2	no	—
93	C₂×Dic₇₁	—	8	2	no	—
94	C₂×Dic₇₁	—	8	2	no	—
95	C₈×C₇₁	—	2	2	no	—
96	C₈×C₇₁	—	4	4	no	—
97	C₈×C₇₁	#107	1	4	yes	complemented
98	C₇₁⋊C₈	—	4	2	no	—
99	C₂²×C₇₁	#389	1	1	yes	characteristic
100	C₂²×C₇₁	#459	1	1	yes	characteristic
101	C₂²×C₇₁	#462	1	1	yes	characteristic
102	C₂²×C₇₁	#469	1	1	yes	characteristic
103	C₂²×C₇₁	#470	1	1	yes	characteristic
104	C₂²×C₇₁	#472	1	1	yes	characteristic
105	C₂²×C₇₁	#473	1	1	yes	characteristic
106	C₄×C₇₁	—	2	1	no	—
107	C₄×C₇₁	—	2	1	no	—
108	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
109	C₄×C₇₁	#428	1	1	yes	characteristic
110	C₄×C₇₁	#440	1	2	yes	—
111	C₄×C₇₁	#474	1	1	yes	characteristic
112	C₄×C₇₁	#477	1	1	yes	characteristic
113	C₄×C₇₁	#501	1	2	yes	—
114	Dic₇₁	—	8	1	no	—
115	Dic₇₁	—	8	1	no	—
116	Dic₇₁	—	16	2	no	—
117	#9620	—	71	1	no	maximal, Sylow
118	C₂×C₇₁	#1687	1	1	yes	characteristic
119	C₂×C₇₁	#1689	1	1	yes	characteristic
120	C₂×C₇₁	#1694	1	1	yes	characteristic
121	C₂×C₇₁	#1696	1	1	yes	characteristic
122	C₂×C₇₁	#1944	1	1	yes	characteristic
123	C₂×C₇₁	#1947	1	1	yes	characteristic
124	C₂×C₇₁	#1952	1	1	yes	characteristic
125	#1947	—	71	2	no	—
126	#1845	—	71	1	no	—
127	#1770	—	71	1	no	—
128	C₂×#334	D₇₁	1	1	yes	characteristic
129	#1296	—	71	2	no	—
130	C₇₁	#9620	1	1	yes	characteristic, complemented, Sylow
131	C₂×#107	—	142	2	no	—
132	C₂×#107	—	142	1	no	—
133	C₂×#107	—	142	1	no	—
134	C₂×C₄⋊C₈	—	71	1	no	—
135	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
136	C₂×#97	—	142	1	no	—
137	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
138	#384	—	71	2	no	—
139	C₄×C₄⋊C₄	—	71	1	no	—
140	#340	—	142	2	no	—
141	#336	—	142	1	no	—
142	#334	Dic₇₁	1	4	yes	complemented
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	142	1	no	—
145	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
146	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	71	1	no	—
148	C₂×C₄⋊C₄	—	71	1	no	—
149	C₂×C₄⋊C₄	—	142	2	no	—
150	C₂×C₄⋊C₄	—	142	2	no	—
151	C₂×C₄⋊C₄	—	142	2	no	—
152	C₂×C₄²	—	142	1	no	—
153	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
154	#107	—	284	2	no	—
155	#107	—	284	2	no	—
156	#107	—	284	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	71	2	no	—
159	C₄⋊C₈	—	142	1	no	—
160	C₄⋊C₈	C₂×Dic₇₁	1	4	yes	—
161	#97	—	284	2	no	—
162	C₄×C₈	Dic₁₄₂	1	2	yes	—
163	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
164	#95	—	142	1	no	—
165	C₂²×C₄	—	284	2	no	—
166	C₂²×C₄	—	142	1	no	—
167	C₂²×C₄	—	142	1	no	—
168	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
170	C₂²×C₄	#11765866	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	284	1	no	—
177	C₂×C₈	—	284	1	no	—
178	C₂×C₈	—	142	2	no	—
179	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
180	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
181	C₂×C₈	D₅₆₈	1	2	yes	—
182	C₂×C₈	Dic₂₈₄	1	2	yes	—
183	C₄⋊C₄	—	284	2	no	—
184	C₄⋊C₄	—	284	2	no	—
185	C₄⋊C₄	—	284	2	no	—
186	C₄⋊C₄	—	284	2	no	—
187	C₄⋊C₄	—	568	2	no	—
188	C₄⋊C₄	—	142	2	no	—
189	C₄⋊C₄	—	142	2	no	—
190	C₄²	—	284	1	no	—
191	C₄²	—	284	1	no	—
192	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
193	C₄²	Q₈×D₇₁	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
195	C₄²	#11765864	1	1	yes	characteristic
196	C₂³	#14556103	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	284	1	no	—
200	C₂×C₄	—	284	1	no	—
201	C₂×C₄	—	284	1	no	—
202	C₂×C₄	—	284	1	no	—
203	C₂×C₄	—	284	1	no	—
204	C₂×C₄	—	284	1	no	—
205	C₂×C₄	—	568	2	no	—
206	C₂×C₄	—	568	2	no	—
207	C₂×C₄	—	568	2	no	—
208	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
209	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
210	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
211	C₂×C₄	#14556108	1	1	yes	characteristic
212	C₂×C₄	#14556119	1	1	yes	characteristic
213	C₂×C₄	#14556127	1	1	yes	characteristic
214	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
215	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
216	C₂×C₄	#14556155	1	2	yes	—
217	C₂×C₄	#14556156	1	1	yes	characteristic
218	C₂×C₄	#14556157	1	2	yes	—
219	C₂×C₄	#14556161	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	284	2	no	—
223	C₈	#14556062	1	4	yes	complemented
224	C₂²	#17985898	1	1	yes	characteristic
225	C₂²	#17986195	1	1	yes	characteristic
226	C₂²	#17986210	1	1	yes	characteristic
227	C₂²	#17986253	1	1	yes	characteristic
228	C₂²	#17986258	1	1	yes	characteristic
229	C₂²	#17986270	1	1	yes	characteristic
230	C₂²	#17986275	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	568	1	no	—
234	C₄	—	568	1	no	—
235	C₄	—	1136	2	no	—
236	C₄	C₂×#14556062	1	1	yes	characteristic
237	C₄	#17986041	1	1	yes	characteristic
238	C₄	#17986100	1	2	yes	—
239	C₄	#17986280	1	1	yes	characteristic
240	C₄	#17986295	1	1	yes	characteristic
241	C₄	#17986406	1	2	yes	—
242	C₂	#22587820	1	1	yes	characteristic
243	C₂	#22587830	1	1	yes	characteristic
244	C₂	#22587859	1	1	yes	characteristic
245	C₂	#22587869	1	1	yes	characteristic
246	C₂	#22589084	1	1	yes	characteristic
247	C₂	#22589099	1	1	yes	characteristic
248	C₂	#22589126	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556103
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#9620
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. #334×C₇₁	C₄	semidirect
97. C₈×C₇₁	#107	semidirect
130. C₇₁	#9620	semidirect
142. #334	Dic₇₁	semidirect
223. C₈	#14556062	semidirect

Maximal subgroups

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

subgroup	index
#22589099	2
#22588500	2
#22588207	2
#22586224	2
C₂×#334×C₇₁	2
#9620	71