Subgroups — C₂×#22586557 #31918501

Order 18176.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#22586557	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₆₈	—	2	1	no	—
6	Dic₁₁₃₆	—	2	1	no	—
7	Dic₁₁₃₆	C₂²	1	2	yes	—
8	C₇₁⋊C₆₄	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₇₁	—	2	1	no	—
10	Q₆₄×C₇₁	—	2	1	no	—
11	Q₆₄×C₇₁	C₂²	1	2	yes	—
12	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₈₄	—	4	1	no	—
14	Dic₅₆₈	—	2	2	no	—
15	Dic₅₆₈	—	4	1	no	—
16	C₂×Q₁₆×C₇₁	—	4	1	no	—
17	Q₃₂×C₇₁	—	2	2	no	—
18	Q₃₂×C₇₁	—	4	1	no	—
19	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
20	C₃₂×C₇₁	D₄	1	1	yes	characteristic
21	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₂	—	8	1	no	—
23	Dic₂₈₄	—	4	2	no	—
24	Dic₂₈₄	—	8	1	no	—
25	C₂×Q₈×C₇₁	—	8	1	no	—
26	Q₁₆×C₇₁	—	4	2	no	—
27	Q₁₆×C₇₁	—	8	1	no	—
28	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
29	C₁₆×C₇₁	D₈	1	1	yes	characteristic
30	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₇₁	—	8	2	no	—
32	Q₈×C₇₁	—	16	1	no	—
33	C₂×C₄×C₇₁	—	16	1	no	—
34	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
35	C₂×Dic₇₁	—	16	1	no	—
36	Dic₁₄₂	—	8	2	no	—
37	Dic₁₄₂	—	16	1	no	—
38	C₈×C₇₁	D₁₆	1	1	yes	characteristic
39	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
40	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
41	C₄×C₇₁	—	16	2	no	—
42	C₄×C₇₁	D₃₂	1	1	yes	characteristic
43	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
44	Dic₇₁	—	16	2	no	—
45	C₂×Q₁₂₈	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	Q₁₂₈	1	2	yes	complemented
47	C₂×C₇₁	C₂×D₃₂	1	1	yes	characteristic
48	C₂×Q₆₄	—	71	1	no	—
49	C₂×Q₆₄	D₇₁	1	1	yes	characteristic
50	Q₁₂₈	—	71	4	no	—
51	C₂×C₆₄	—	71	1	no	—
52	C₇₁	C₂×Q₁₂₈	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₃₂	—	2	1	no	—
54	C₂×Q₃₂	—	142	1	no	—
55	Q₆₄	—	2	1	no	—
56	Q₆₄	—	71	2	no	—
57	Q₆₄	—	142	1	no	—
58	Q₆₄	C₂×D₇₁	1	2	yes	—
59	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
60	C₆₄	—	71	2	no	—
61	C₂×Q₁₆	—	4	1	no	—
62	C₂×Q₁₆	—	284	1	no	—
63	Q₃₂	—	2	2	no	—
64	Q₃₂	—	4	1	no	—
65	Q₃₂	—	284	1	no	—
66	Q₃₂	—	142	2	no	—
67	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
68	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
69	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
70	C₂×Q₈	—	8	1	no	—
71	C₂×Q₈	—	568	1	no	—
72	Q₁₆	—	4	2	no	—
73	Q₁₆	—	8	1	no	—
74	Q₁₆	—	284	2	no	—
75	Q₁₆	—	568	1	no	—
76	C₂×C₈	#11765850	1	1	yes	characteristic
77	C₁₆	#11765850	1	1	yes	characteristic
78	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
79	Q₈	—	8	2	no	—
80	Q₈	—	16	1	no	—
81	Q₈	—	568	2	no	—
82	Q₈	—	1136	1	no	—
83	C₂×C₄	—	16	1	no	—
84	C₂×C₄	—	1136	1	no	—
85	C₂×C₄	#14556069	1	1	yes	characteristic
86	C₈	#14556069	1	1	yes	characteristic
87	C₈	C₂×#11765850	1	1	yes	characteristic
88	C₂²	#17985834	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	1136	2	no	—
91	C₄	#17985834	1	1	yes	characteristic
92	C₄	C₂×#14556069	1	1	yes	characteristic
93	C₂	#22586557	1	2	yes	direct product factor
94	C₂	C₂×#17985834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985834
commutator, cosocle	C₃₂×C₇₁	C₂³
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₂×Q₆₄×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	D₃₂
hypercenter	C₂×Q₆₄	D₇₁
hypocenter	C₇₁	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
2. #22586557	C₂	direct (N × H)
46. C₂×C₇₁	Q₁₂₈	semidirect
52. C₇₁	C₂×Q₁₂₈	semidirect
93. C₂	#22586557	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₃₆	2
#22586557	2
C₂×C₇₁⋊C₆₄	2
C₂×Q₆₄×C₇₁	2
C₂×Q₁₂₈	71

Subgroups — C2×#22586557 #31918501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586557 #31918501