Subgroups — #31917632

Order 18176.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985833	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#22586474	C₂	1	2	yes	maximal, complemented
4	C₄×C₇₁⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
6	#17985833	C₄	1	4	yes	complemented
7	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	complemented
8	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
12	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	#405×C₇₁	C₄	1	1	yes	characteristic
14	#346×C₇₁	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
17	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
18	C₇₁⋊C₃₂	C₂×C₄	1	4	yes	—
19	C₂×M₁₆×C₇₁	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	#117×C₇₁	C₈	1	1	yes	characteristic
22	M₃₂×C₇₁	C₈	1	4	yes	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
28	#97×C₇₁	C₈	1	2	yes	—
29	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₇₁	C₁₆	1	4	yes	—
32	C₂×C₈×C₇₁	C₄²	1	2	yes	—
33	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₁	C₁₆	1	2	yes	—
38	C₂²⋊C₄×C₇₁	C₁₆	1	2	yes	—
39	C₄²×C₇₁	C₂×C₈	1	2	yes	—
40	C₁₆×C₇₁	C₄²	1	2	yes	—
41	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
42	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₁	C₄×C₈	1	2	yes	—
51	C₈×C₇₁	C₂×C₁₆	1	4	yes	—
52	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
53	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
54	C₂²×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	C₄×C₁₆	1	2	yes	—
56	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₇₁	—	2	4	no	—
58	C₄×C₇₁	C₄×C₁₆	1	2	yes	—
59	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
61	#13549	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₇₁	#2258	1	2	yes	—
65	C₂×M₆₄	—	71	1	no	—
66	C₂²×C₃₂	—	71	1	no	—
67	#2108	D₇₁	1	1	yes	characteristic
68	#1397	—	71	2	no	—
69	C₄×C₃₂	—	71	2	no	—
70	C₇₁	#13549	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
72	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	#405	Dic₇₁	1	1	yes	characteristic
74	M₆₄	—	71	4	no	—
75	C₂×C₃₂	—	71	4	no	—
76	C₂×C₃₂	—	71	1	no	—
77	C₂×C₃₂	—	71	2	no	—
78	C₂×C₃₂	—	71	1	no	—
79	#346	Dic₇₁	1	2	yes	—
80	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
81	C₂×M₁₆	C₇₁⋊C₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
83	#117	C₇₁⋊C₈	1	1	yes	characteristic
84	M₃₂	C₇₁⋊C₈	1	4	yes	—
85	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
86	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
88	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
89	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
90	#97	C₇₁⋊C₈	1	2	yes	—
91	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
92	C₃₂	—	71	2	no	—
93	C₃₂	—	71	2	no	—
94	C₃₂	—	71	4	no	—
95	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
96	M₁₆	C₇₁⋊C₁₆	1	4	yes	—
97	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
98	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
99	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
100	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
101	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
102	C₄⋊C₄	C₇₁⋊C₁₆	1	2	yes	—
103	C₂²⋊C₄	C₇₁⋊C₁₆	1	2	yes	—
104	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
105	C₁₆	C₄×Dic₇₁	1	2	yes	—
106	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
107	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₂³	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
110	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
111	C₂×C₄	C₂×C₇₁⋊C₁₆	1	4	yes	—
112	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
113	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
115	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
116	C₈	C₂×C₇₁⋊C₁₆	1	4	yes	—
117	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
118	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₇₁⋊C₁₆	1	2	yes	—
121	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₇₁⋊C₁₆	1	2	yes	—
124	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
125	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₇₁⋊C₁₆	1	1	yes	characteristic
128	C₂	#22590634	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#13549
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #22586474	C₂	semidirect
4. C₄×C₇₁⋊C₃₂	C₂	semidirect
6. #17985833	C₄	semidirect
7. C₂×C₇₁⋊C₃₂	C₄	semidirect
70. C₇₁	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985833	2
C₂²×C₇₁⋊C₃₂	2
#22586474	2
C₄×C₇₁⋊C₃₂	2
C₇₁×#2108	2
#13549	71