Subgroups — #31912450

Order 18176.

414 conjugacy classes of subgroups.

The subgroup lattice has 162 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	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586417	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₂²	1	2	yes	—
9	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
10	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₁	C₄	1	4	yes	complemented
14	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
16	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
17	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₇₁⋊C₁₆	—	2	4	no	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₇₁⋊C₈	C₈	1	4	yes	—
21	C₈×Dic₇₁	—	2	4	no	—
22	C₈×Dic₇₁	—	2	4	no	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	—	2	4	no	—
27	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×Dic₇₁	C₂×C₈	1	4	yes	—
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₂×C₇₁⋊C₈	—	2	2	no	—
33	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
34	C₄×Dic₇₁	—	2	4	no	—
35	C₄×Dic₇₁	—	2	2	no	—
36	C₄×Dic₇₁	—	2	2	no	—
37	C₄×Dic₇₁	C₁₆	1	8	yes	complemented
38	C₇₁⋊C₁₆	—	2	4	no	—
39	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄²	1	2	yes	—
43	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
44	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	#41	1	4	yes	—
47	C₄²×C₇₁	C₄²	1	2	yes	—
48	C₄²×C₇₁	C₂×C₈	1	2	yes	—
49	C₁₆×C₇₁	—	2	4	no	—
50	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₇₁	C₂×C₁₆	1	4	yes	—
54	C₂×C₄×C₇₁	M₃₂	1	4	yes	—
55	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#117	1	2	yes	—
57	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₇₁	—	2	4	no	—
60	C₂×Dic₇₁	—	2	4	no	—
61	C₂×Dic₇₁	C₂×C₁₆	1	8	yes	—
62	C₈×C₇₁	#97	1	4	yes	—
63	C₈×C₇₁	#117	1	2	yes	—
64	C₈×C₇₁	#117	1	2	yes	—
65	C₇₁⋊C₈	—	2	4	no	—
66	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂²×C₇₁	#431	1	2	yes	—
68	C₂²×C₇₁	C₂²×C₁₆	1	2	yes	—
69	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
70	C₄×C₇₁	C₄×C₁₆	1	4	yes	—
71	C₄×C₇₁	#346	1	4	yes	—
72	C₄×C₇₁	#376	1	1	yes	characteristic
73	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
74	C₄×C₇₁	#431	1	2	yes	—
75	Dic₇₁	—	2	8	no	—
76	#12590	—	71	1	no	maximal, Sylow
77	C₂×C₇₁	#1744	1	1	yes	characteristic
78	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
80	C₂×C₇₁	#2162	1	4	yes	—
81	C₂×C₄×C₁₆	—	71	1	no	—
82	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
83	C₄²×C₈	—	71	1	no	—
84	#1374	—	71	4	no	—
85	C₇₁	#12590	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₁₆	—	71	2	no	—
87	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
88	C₂×C₄×C₈	—	71	1	no	—
89	C₂×C₄×C₈	—	71	2	no	—
90	C₂×C₄×C₈	—	71	2	no	—
91	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄³	—	71	1	no	—
93	C₄×C₁₆	—	71	4	no	—
94	C₄×C₁₆	Dic₇₁	1	4	yes	complemented
95	C₂²×C₈	—	71	2	no	—
96	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	C₂×C₄²	—	71	2	no	—
99	C₂×C₄²	—	71	2	no	—
100	C₂×C₄²	—	71	2	no	—
101	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	71	4	no	—
104	C₂×C₁₆	—	142	4	no	—
105	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
106	C₄×C₈	—	71	4	no	—
107	C₄×C₈	—	142	4	no	—
108	C₄×C₈	—	142	4	no	—
109	C₄×C₈	C₄×D₇₁	1	2	yes	—
110	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
111	C₂²×C₄	—	71	4	no	—
112	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
115	C₂×C₈	—	71	4	no	—
116	C₂×C₈	—	142	2	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
119	C₂×C₈	#11765848	1	4	yes	—
120	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
122	C₂×C₈	#11765865	1	4	yes	—
123	C₄²	—	71	8	no	—
124	C₄²	—	142	4	no	—
125	C₄²	—	142	2	no	—
126	C₄²	—	142	2	no	—
127	C₄²	C₄×Dic₇₁	1	2	yes	—
128	C₄²	C₈×D₇₁	1	2	yes	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	142	4	no	—
131	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
132	C₂×C₄	—	71	8	no	—
133	C₂×C₄	—	142	4	no	—
134	C₂×C₄	—	142	4	no	—
135	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
136	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
137	C₂×C₄	C₁₆×D₇₁	1	4	yes	—
138	C₂×C₄	#14556067	1	4	yes	—
139	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
140	C₂×C₄	#14556085	1	2	yes	—
141	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×#11765848	1	2	yes	—
143	C₈	—	142	4	no	—
144	C₈	#14556038	1	4	yes	—
145	C₈	#14556085	1	2	yes	—
146	C₈	#14556087	1	2	yes	—
147	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
148	C₂²	#17986050	1	2	yes	—
149	C₂²	C₂×C₁₆×D₇₁	1	2	yes	—
150	C₂²	C₂×#14556067	1	2	yes	—
151	C₄	—	142	8	no	—
152	C₄	C₁₆×Dic₇₁	1	4	yes	—
153	C₄	#17985774	1	4	yes	—
154	C₄	#17985845	1	1	yes	characteristic
155	C₄	C₂×#14556038	1	1	yes	characteristic
156	C₄	#17986050	1	2	yes	—
157	C₂	#22588085	1	1	yes	characteristic
158	C₂	C₂×C₁₆×Dic₇₁	1	1	yes	characteristic
159	C₂	C₂×#17985774	1	1	yes	characteristic
160	C₂	#22590137	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12590
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₇₁⋊C₁₆	C₄	semidirect
13. C₄×C₁₆×C₇₁	C₄	semidirect
37. C₄×Dic₇₁	C₁₆	semidirect
85. C₇₁	#12590	semidirect
94. C₄×C₁₆	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
C₄×C₈×Dic₇₁	2
#22586417	2
C₂×C₄×C₁₆×C₇₁	2
#12590	71