Subgroups — #31884639

Order 18176.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×#14556062	—	2	1	no	—
6	C₂×#14556062	C₄	1	2	yes	—
7	#17985869	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#335×C₇₁	—	2	1	no	—
13	#335×C₇₁	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₄⋊Dic₇₁	—	4	1	no	—
16	C₂×C₄⋊Dic₇₁	—	4	1	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	4	1	no	—
19	C₂×C₇₁⋊C₁₆	—	4	1	no	—
20	#14556062	—	4	2	no	—
21	#14556062	—	4	1	no	—
22	#14556062	—	4	1	no	—
23	C₂²×C₈×C₇₁	—	4	1	no	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₄⋊C₈×C₇₁	—	4	1	no	—
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₁	—	4	2	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	—	8	2	no	—
37	C₄⋊Dic₇₁	—	8	1	no	—
38	C₄⋊Dic₇₁	—	8	1	no	—
39	C₇₁⋊C₁₆	—	4	2	no	—
40	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	4	2	no	—
46	C₂×C₈×C₇₁	—	8	1	no	—
47	C₂×C₈×C₇₁	—	8	1	no	—
48	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
56	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
59	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
60	C₂³×C₇₁	#102	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	2	1	no	—
62	C₂×C₄×C₇₁	—	2	1	no	—
63	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#100	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#101	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
73	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₇₁	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₇₁	—	8	2	no	—
78	C₂×Dic₇₁	—	8	2	no	—
79	C₂×Dic₇₁	—	8	2	no	—
80	C₈×C₇₁	—	2	1	no	—
81	C₈×C₇₁	—	2	1	no	—
82	C₈×C₇₁	—	8	2	no	—
83	C₈×C₇₁	#102	1	1	yes	characteristic
84	C₈×C₇₁	#103	1	1	yes	characteristic
85	C₈×C₇₁	#106	1	1	yes	characteristic
86	C₈×C₇₁	#107	1	1	yes	characteristic
87	C₂²×C₇₁	#331	1	1	yes	characteristic
88	C₂²×C₇₁	#332	1	1	yes	characteristic
89	C₂²×C₇₁	#340	1	1	yes	characteristic
90	C₂²×C₇₁	#357	1	1	yes	characteristic
91	C₂²×C₇₁	#358	1	1	yes	characteristic
92	C₂²×C₇₁	#361	1	1	yes	characteristic
93	C₂²×C₇₁	#362	1	1	yes	characteristic
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	—	2	1	no	—
96	C₄×C₇₁	#332	1	1	yes	characteristic
97	C₄×C₇₁	#333	1	1	yes	characteristic
98	C₄×C₇₁	#340	1	1	yes	characteristic
99	C₄×C₇₁	#343	1	1	yes	characteristic
100	C₄×C₇₁	#357	1	1	yes	characteristic
101	C₄×C₇₁	#358	1	1	yes	characteristic
102	C₄×C₇₁	#366	1	1	yes	characteristic
103	C₄×C₇₁	#367	1	1	yes	characteristic
104	Dic₇₁	—	16	2	no	—
105	#7274	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	#1296	1	1	yes	characteristic
107	C₂×C₇₁	#1361	1	1	yes	characteristic
108	C₂×C₇₁	#1362	1	1	yes	characteristic
109	C₂×C₇₁	#1364	1	1	yes	characteristic
110	C₂×C₇₁	#1365	1	1	yes	characteristic
111	C₂×C₇₁	#1380	1	1	yes	characteristic
112	C₂×C₇₁	#1387	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	71	1	no	—
114	#1845	—	71	1	no	—
115	C₂×#335	D₇₁	1	1	yes	characteristic
116	C₇₁	#7274	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₁₆	—	142	1	no	—
118	C₂×#107	—	71	2	no	—
119	C₂×#107	—	142	1	no	—
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
122	#384	—	142	1	no	—
123	C₄⋊C₁₆	—	71	2	no	—
124	C₄⋊C₁₆	—	142	1	no	—
125	#335	—	2	1	no	—
126	#335	Dic₇₁	1	2	yes	complemented
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
129	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	284	1	no	—
131	C₂×C₄⋊C₄	—	284	1	no	—
132	C₂×C₄⋊C₄	—	142	2	no	—
133	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
134	C₂×C₁₆	—	284	1	no	—
135	C₂×C₁₆	—	284	1	no	—
136	C₂×C₁₆	—	142	2	no	—
137	#107	—	284	1	no	—
138	#107	—	284	1	no	—
139	#107	—	284	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
143	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂²×C₄	—	284	2	no	—
147	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	#11765850	1	1	yes	characteristic
149	C₂²×C₄	#11765852	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
156	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
158	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
159	C₂×C₈	D₅₆₈	1	1	yes	characteristic
160	C₂×C₈	#11765854	1	1	yes	characteristic
161	C₂×C₈	#11765854	1	1	yes	characteristic
162	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
163	C₄⋊C₄	—	284	2	no	—
164	C₄⋊C₄	—	284	2	no	—
165	C₄⋊C₄	—	568	2	no	—
166	C₄⋊C₄	—	568	1	no	—
167	C₄⋊C₄	—	568	1	no	—
168	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
169	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
170	C₄²	#11765853	1	1	yes	characteristic
171	C₄²	#11765855	1	1	yes	characteristic
172	C₁₆	—	284	2	no	—
173	C₂³	#14556048	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	568	2	no	—
177	C₂×C₄	—	568	2	no	—
178	C₂×C₄	—	568	2	no	—
179	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
180	C₂×C₄	#14556044	1	1	yes	characteristic
181	C₂×C₄	#14556046	1	1	yes	characteristic
182	C₂×C₄	#14556049	1	1	yes	characteristic
183	C₂×C₄	#14556050	1	1	yes	characteristic
184	C₂×C₄	#14556051	1	1	yes	characteristic
185	C₂×C₄	#14556052	1	1	yes	characteristic
186	C₂×C₄	#14556053	1	1	yes	characteristic
187	C₂×C₄	#14556059	1	1	yes	characteristic
188	C₂×C₄	#14556061	1	1	yes	characteristic
189	C₂×C₄	#14556069	1	1	yes	characteristic
190	C₂×C₄	#14556071	1	1	yes	characteristic
191	C₂×C₄	#14556072	1	1	yes	characteristic
192	C₂×C₄	#14556074	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	8	2	no	—
196	C₈	#14556050	1	1	yes	characteristic
197	C₈	#14556053	1	1	yes	characteristic
198	C₈	#14556060	1	1	yes	characteristic
199	C₈	#14556062	1	1	yes	characteristic
200	C₂²	#17985742	1	1	yes	characteristic
201	C₂²	#17985746	1	1	yes	characteristic
202	C₂²	#17985761	1	1	yes	characteristic
203	C₂²	#17985799	1	1	yes	characteristic
204	C₂²	#17985802	1	1	yes	characteristic
205	C₂²	#17985811	1	1	yes	characteristic
206	C₂²	#17985814	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	1136	2	no	—
210	C₄	#17985744	1	1	yes	characteristic
211	C₄	#17985747	1	1	yes	characteristic
212	C₄	#17985762	1	1	yes	characteristic
213	C₄	#17985767	1	1	yes	characteristic
214	C₄	#17985800	1	1	yes	characteristic
215	C₄	#17985803	1	1	yes	characteristic
216	C₄	#17985824	1	1	yes	characteristic
217	C₄	#17985826	1	1	yes	characteristic
218	C₂	#22586224	1	1	yes	characteristic
219	C₂	#22586385	1	1	yes	characteristic
220	C₂	#22586388	1	1	yes	characteristic
221	C₂	#22586396	1	1	yes	characteristic
222	C₂	#22586399	1	1	yes	characteristic
223	C₂	#22586431	1	1	yes	characteristic
224	C₂	#22586450	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556048
commutator	C₂×C₈×C₇₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#102
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#7274
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
13. #335×C₇₁	C₄	semidirect
116. C₇₁	#7274	semidirect
126. #335	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22588500	2
C₂×#335×C₇₁	2
#7274	71