Group #22570549 — subgroups

Subgroups — C₂×#16863765 #22570549

Order 8576.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₂×#14524035	C₂	1	1	yes	maximal, characteristic
3	#16863765	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14523968	C₂	1	4	yes	maximal
5	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	#14524043	—	2	2	no	—
8	#14524043	C₂²	1	4	yes	—
9	#14524035	—	2	2	no	—
10	#14524035	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₇	—	2	2	no	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₄	—	2	2	no	—
14	#14523968	C₂²	1	16	yes	—
15	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#97×C₆₇	C₂²	1	4	yes	—
18	C₂×Dic₁₃₄	—	2	8	no	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	C₂³	1	4	yes	—
21	C₂²×Dic₆₇	—	2	2	no	—
22	C₂²×Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	8	no	—
24	C₄⋊Dic₆₇	—	2	2	no	—
25	C₄⋊Dic₆₇	C₂³	1	4	yes	—
26	Dic₆₇⋊C₄	—	2	8	no	—
27	C₂²×C₄×C₆₇	D₄	1	2	yes	—
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₆₇	—	2	4	no	—
30	C₂×C₈×C₆₇	C₂³	1	4	yes	—
31	C₄²×C₆₇	D₄	1	2	yes	—
32	C₄²×C₆₇	C₂³	1	2	yes	—
33	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	#41	1	4	yes	—
38	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₆₇	—	2	4	no	—
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	4	4	no	—
42	C₂×Dic₆₇	—	4	4	no	—
43	Dic₁₃₄	—	2	8	no	—
44	Dic₁₃₄	—	4	4	no	—
45	C₈×C₆₇	—	2	4	no	—
46	C₂²×C₆₇	#124	1	2	yes	—
47	C₂²×C₆₇	#137	1	4	yes	—
48	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₆₇	—	2	4	no	—
50	C₄×C₆₇	#124	1	2	yes	—
51	C₄×C₆₇	C₂×#41	1	2	yes	—
52	Dic₆₇	—	4	4	no	—
53	Dic₆₇	—	4	4	no	—
54	C₂×C₆₇	#491	1	4	yes	complemented
55	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
56	C₂×C₆₇	C₂×#137	1	2	yes	—
57	C₂×#491	—	67	1	no	maximal, Sylow
58	C₆₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	67	1	no	—
60	C₂×#125	—	67	1	no	—
61	#491	—	67	8	no	—
62	C₂×#103	—	67	4	no	—
63	C₂×#97	D₆₇	1	1	yes	characteristic
64	C₂²×Q₈	—	67	2	no	—
65	C₂²×C₈	C₂×D₆₇	1	2	yes	—
66	C₄⋊Q₈	—	67	4	no	—
67	C₄⋊Q₈	—	134	2	no	—
68	#125	—	67	4	no	—
69	#125	—	134	2	no	—
70	C₂×C₄⋊C₄	—	67	2	no	—
71	C₂×C₄⋊C₄	—	134	2	no	—
72	C₂×C₄⋊C₄	—	134	2	no	—
73	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
74	#103	—	67	16	no	—
75	#97	C₂×D₆₇	1	4	yes	—
76	C₂×Q₈	—	67	4	no	—
77	C₂×Q₈	—	134	8	no	—
78	C₂×Q₈	—	134	2	no	—
79	C₂²×C₄	—	134	2	no	—
80	C₂²×C₄	—	134	2	no	—
81	C₂²×C₄	D₂₆₈	1	2	yes	—
82	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂²×D₆₇	1	4	yes	—
85	C₄⋊C₄	—	67	4	no	—
86	C₄⋊C₄	—	134	2	no	—
87	C₄⋊C₄	—	134	8	no	—
88	C₄⋊C₄	—	134	8	no	—
89	C₄²	D₂₆₈	1	2	yes	—
90	C₄²	C₂²×D₆₇	1	2	yes	—
91	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
92	Q₈	—	268	4	no	—
93	Q₈	—	134	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	268	4	no	—
96	C₂×C₄	—	268	4	no	—
97	C₂×C₄	—	134	4	no	—
98	C₂×C₄	—	134	4	no	—
99	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
100	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
101	C₂×C₄	#11761628	1	4	yes	—
102	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#14524031	1	2	yes	—
105	C₂²	#14524076	1	4	yes	—
106	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	268	4	no	—
109	C₄	—	268	4	no	—
110	C₄	#14524031	1	2	yes	—
111	C₄	C₂×#11761628	1	2	yes	—
112	C₂	#16863765	1	4	yes	direct product factor
113	C₂	C₂×#14524031	1	1	yes	characteristic
114	C₂	C₂×#14524076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

C₂×#14524035

C₂

yes

maximal, characteristic

#16863765

C₂

yes

maximal, direct product factor

C₂×#14523968

C₂

yes

maximal

C₂×#97×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

—

#14524043

C₂²

yes

—

#14524035

—

#14524035

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

#14523968

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

#124

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#124

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×#491

—

maximal, Sylow

C₆₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

—

#491

—

C₂×#103

—

C₂×#97

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

#125

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#103

—

#97

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

D₂₆₈

yes

—

C₄²

C₂²×D₆₇

yes

—

C₂³

C₂×D₂₆₈

yes

characteristic

Q₈

—

268

—

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×D₂₆₈

yes

—

100

C₂×C₄

C₂×D₂₆₈

yes

characteristic

101

C₂×C₄

#11761628

yes

—

102

C₂×C₄

C₂³×D₆₇

yes

characteristic

103

C₈

—

104

C₂²

#14524031

yes

—

105

C₂²

#14524076

yes

—

106

C₂²

C₂²×D₂₆₈

yes

characteristic

107

C₄

—

108

C₄

—

268

—

109

C₄

—

268

—

110

C₄

#14524031

yes

—

111

C₄

C₂×#11761628

yes

—

112

C₂

#16863765

yes

direct product factor

113

C₂

C₂×#14524031

yes

characteristic

114

C₂

C₂×#14524076

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	C₂×#491
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#97×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×#97

D₆₇

hypocenter

C₆₇

C₂×#491

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #16863765	C₂	direct (N × H)
54. C₂×C₆₇	#491	semidirect
58. C₆₇	C₂×#491	semidirect
112. C₂	#16863765	direct (N × H)

H ≅ G/N

type

3. #16863765

C₂

direct (N × H)

54. C₂×C₆₇

#491

semidirect

58. C₆₇

C₂×#491

semidirect

112. C₂

#16863765

direct (N × H)

Subgroups — C₂×#16863765 #22570549

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863765 #22570549

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863765 #22570549