Subgroups — C₂×#22586474 #31917625

Order 18176.

254 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#1397
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22586474	C₂	direct (N × H)
4. C₂×C₇₁⋊C₃₂	C₄	semidirect
11. C₇₁⋊C₃₂	C₂×C₄	semidirect
54. C₂×C₇₁	#1397	semidirect
60. C₇₁	C₂×#1397	semidirect
110. C₂	#22586474	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇₁⋊C₃₂	2
#22586474	2
C₂×C₄×C₁₆×C₇₁	2
C₂×#1397	71

Subgroups — C2×#22586474 #31917625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586474 #31917625