Group #22561971 — subgroups

Subgroups — C₂×#16863216 #22561971

Order 8576.

168 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863216	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₆₇	C₄	1	1	yes	characteristic
7	#14523983	—	2	1	no	—
8	#14523983	C₂²	1	2	yes	—
9	C₂×C₆₇⋊C₁₆	—	2	1	no	—
10	C₈×Dic₆₇	C₄	1	2	yes	—
11	C₈×Dic₆₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	M₃₂×C₆₇	—	2	1	no	—
14	M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×C₁₆×C₆₇	—	2	1	no	—
16	C₂²×Dic₆₇	—	2	1	no	—
17	C₂×C₆₇⋊C₈	C₈	1	2	yes	—
18	C₂×C₆₇⋊C₈	C₈	1	2	yes	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×Dic₆₇	—	2	1	no	—
21	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
22	C₆₇⋊C₁₆	—	2	2	no	—
23	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₁₆×C₆₇	—	2	2	no	—
31	C₂³×C₆₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₆₇	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₆₇	—	2	4	no	—
39	C₂×Dic₆₇	—	2	2	no	—
40	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₆₇⋊C₈	C₂×C₈	1	4	yes	—
45	C₂²×C₆₇	—	2	2	no	—
46	C₂²×C₆₇	C₄⋊C₈	1	1	yes	characteristic
47	C₂²×C₆₇	C₄⋊C₈	1	1	yes	characteristic
48	C₂²×C₆₇	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
50	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
51	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
53	Dic₆₇	—	2	4	no	—
54	C₂×C₆₇	—	2	1	no	—
55	C₂×C₆₇	—	2	1	no	—
56	C₂×C₆₇	#364	1	2	yes	complemented
57	C₂×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×#364	—	67	1	no	maximal, Sylow
59	C₆₇	C₂×#364	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₃₂	—	67	1	no	—
61	C₂×M₃₂	D₆₇	1	1	yes	characteristic
62	C₂×C₄×C₈	—	67	1	no	—
63	#364	—	67	4	no	—
64	C₂²×C₈	—	67	1	no	—
65	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
66	C₂×C₄²	—	67	1	no	—
67	M₃₂	—	2	1	no	—
68	M₃₂	—	67	2	no	—
69	M₃₂	—	134	1	no	—
70	M₃₂	C₂×D₆₇	1	2	yes	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	134	1	no	—
73	C₄×C₈	—	67	2	no	—
74	C₄×C₈	—	67	2	no	—
75	C₂²×C₄	—	134	1	no	—
76	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
77	C₂×C₈	—	67	2	no	—
78	C₂×C₈	—	67	2	no	—
79	C₂×C₈	—	67	2	no	—
80	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
81	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
82	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
84	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
85	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
86	C₄²	—	67	2	no	—
87	C₄²	—	134	1	no	—
88	C₁₆	—	2	2	no	—
89	C₁₆	—	134	2	no	—
90	C₂³	#11761611	1	1	yes	characteristic
91	C₂×C₄	—	134	4	no	—
92	C₂×C₄	—	134	2	no	—
93	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
96	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
97	C₂×C₄	#11761611	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
99	C₈	—	67	4	no	—
100	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
101	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
102	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
103	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
104	C₂²	—	2	2	no	—
105	C₂²	Dic₆₇⋊C₈	1	1	yes	characteristic
106	C₂²	Dic₆₇⋊C₈	1	1	yes	characteristic
107	C₂²	C₂×#11761611	1	1	yes	characteristic
108	C₄	—	134	4	no	—
109	C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
110	C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
111	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
112	C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#16863216	1	2	yes	direct product factor
116	C₂	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#16863216

C₂

yes

maximal, direct product factor

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₄

yes

characteristic

#14523983

—

#14523983

C₂²

yes

—

C₂×C₆₇⋊C₁₆

—

C₈×Dic₆₇

C₄

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

C₂²

yes

—

C₂×C₁₆×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

C₈

yes

—

C₂×C₆₇⋊C₈

C₈

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₆₇⋊C₈

C₂×C₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊C₈

yes

characteristic

C₂²×C₆₇

C₄⋊C₈

yes

characteristic

C₂²×C₆₇

C₂×M₁₆

yes

characteristic

C₄×C₆₇

C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂²×C₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#364

yes

complemented

C₂×C₆₇

C₂×C₄⋊C₈

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₆₇

yes

characteristic

C₂×C₄×C₈

—

#364

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

134

—

M₃₂

C₂×D₆₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

134

—

C₁₆

—

C₁₆

—

134

—

C₂³

#11761611

yes

characteristic

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

C₈×D₆₇

yes

characteristic

C₂×C₄

C₈×D₆₇

yes

characteristic

C₂×C₄

#11761611

yes

characteristic

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₈

—

100

C₈

Dic₆₇⋊C₄

yes

characteristic

101

C₈

Dic₆₇⋊C₄

yes

characteristic

102

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

103

C₈

C₂×Dic₁₃₄

yes

characteristic

104

C₂²

—

105

C₂²

Dic₆₇⋊C₈

yes

characteristic

106

C₂²

Dic₆₇⋊C₈

yes

characteristic

107

C₂²

C₂×#11761611

yes

characteristic

108

C₄

—

134

—

109

C₄

Dic₆₇⋊C₈

yes

characteristic

110

C₄

Dic₆₇⋊C₈

yes

characteristic

111

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

112

C₄

C₂×C₈×D₆₇

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#16863216

yes

direct product factor

116

C₂

C₂×Dic₆₇⋊C₈

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₄⋊C₈
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#364
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₈

C₂²⋊D₆₇

commutator

C₄×C₆₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂²×C₆₇

C₄⋊C₈

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

C₂×#364

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16863216	C₂	direct (N × H)
56. C₂×C₆₇	#364	semidirect
59. C₆₇	C₂×#364	semidirect
115. C₂	#16863216	direct (N × H)

H ≅ G/N

type

3. #16863216

C₂

direct (N × H)

56. C₂×C₆₇

#364

semidirect

59. C₆₇

C₂×#364

semidirect

115. C₂

#16863216

direct (N × H)

Subgroups — C₂×#16863216 #22561971

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863216 #22561971

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863216 #22561971