Group #22568672 — subgroups

Subgroups — #22568672

Order 8576.

716 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 249 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×#14524091	C₂	1	1	yes	maximal, characteristic
2	#16863874	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863369	C₂	1	12	yes	maximal, complemented
4	#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#14524091	—	2	4	no	—
6	#14524091	—	2	4	no	—
7	C₂×#11761629	C₄	1	4	yes	complemented
8	C₂×#11761628	C₂²	1	6	yes	—
9	C₂×Q₈×D₆₇	C₄	1	4	yes	complemented
10	Q₈×Dic₆₇	—	2	4	no	—
11	#14524002	C₂²	1	6	yes	—
12	#14523957	C₂²	1	16	yes	—
13	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₆₇	—	2	4	no	—
15	#117×C₆₇	C₂²	1	6	yes	—
16	#11761629	—	2	16	no	—
17	#11761628	—	2	12	no	—
18	#11761628	—	2	12	no	—
19	C₂×Dic₁₃₄	C₂×C₄	1	6	yes	—
20	Q₈×D₆₇	—	2	16	no	—
21	C₂×D₂₆₈	C₂×C₄	1	6	yes	—
22	C₂×C₂²⋊D₆₇	C₂³	1	4	yes	—
23	C₂×C₄×D₆₇	C₂×C₄	1	12	yes	—
24	C₄⋊Dic₆₇	—	2	6	no	—
25	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
26	C₄×Dic₆₇	—	2	6	no	—
27	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×Q₈×C₆₇	D₄	1	4	yes	—
29	C₂×Q₈×C₆₇	D₄	1	4	yes	—
30	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
31	C₄⋊C₄×C₆₇	—	2	6	no	—
32	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
33	C₄²×C₆₇	—	2	6	no	—
34	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂²×D₆₇	C₂²×C₄	1	4	yes	—
36	Q₈×C₆₇	C₂²⋊C₄	1	16	yes	—
37	C₂×C₄×C₆₇	—	2	4	no	—
38	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
39	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
40	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
41	C₂²⋊D₆₇	—	2	4	no	—
42	C₂²⋊D₆₇	—	2	4	no	—
43	C₂×Dic₆₇	—	2	4	no	—
44	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
45	D₂₆₈	—	2	12	no	—
46	C₄×D₆₇	—	2	24	no	—
47	Dic₁₃₄	—	2	12	no	—
48	C₂²×C₆₇	—	4	1	no	—
49	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₆₇	—	2	4	no	—
53	C₂×D₆₇	—	4	4	no	—
54	C₄×C₆₇	—	4	4	no	—
55	C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
56	Dic₆₇	—	2	4	no	—
57	Dic₆₇	—	4	4	no	—
58	C₂×C₆₇	—	2	1	no	—
59	C₂×C₆₇	—	2	1	no	—
60	C₂×C₆₇	—	2	1	no	—
61	C₂×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	D₆₇	—	4	4	no	—
63	#2884	—	67	1	no	maximal, Sylow
64	C₆₇	#2884	1	1	yes	characteristic, complemented, Sylow
65	C₂×2₋¹⁺⁴	—	67	1	no	—
66	#519	—	67	1	no	—
67	#519	D₆₇	1	1	yes	characteristic, complemented
68	#410	—	67	12	no	—
69	2₋¹⁺⁴	—	134	4	no	—
70	2₋¹⁺⁴	—	134	4	no	—
71	C₂×#41	—	67	4	no	—
72	C₂×#41	—	67	6	no	—
73	C₂²×Q₈	—	67	4	no	—
74	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
75	C₄×Q₈	—	2	4	no	—
76	C₄×Q₈	—	134	4	no	—
77	#117	—	67	6	no	—
78	#117	C₂×D₆₇	1	6	yes	—
79	#99	—	67	16	no	—
80	#41	—	134	16	no	—
81	#41	—	134	12	no	—
82	#41	—	134	12	no	—
83	C₂×Q₈	—	67	6	no	—
84	C₂×Q₈	—	134	16	no	—
85	C₂×Q₈	C₄×D₆₇	1	4	yes	—
86	C₂×Q₈	D₂₆₈	1	4	yes	—
87	C₂×Q₈	C₂²⋊D₆₇	1	4	yes	—
88	C₂×D₄	—	67	4	no	—
89	C₂×D₄	—	67	6	no	—
90	C₂²×C₄	—	67	12	no	—
91	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
92	C₄⋊C₄	—	2	6	no	—
93	C₄⋊C₄	—	134	6	no	—
94	C₂²⋊C₄	—	67	4	no	—
95	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
96	C₄²	—	2	6	no	—
97	C₄²	—	134	6	no	—
98	C₂³	—	67	4	no	—
99	C₂³	C₂³×D₆₇	1	1	yes	characteristic
100	Q₈	—	134	12	no	—
101	Q₈	(C₂×D₆₇)⋊C₄	1	16	yes	—
102	D₄	—	134	4	no	—
103	D₄	—	134	4	no	—
104	D₄	—	134	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	67	4	no	—
107	C₂×C₄	—	134	4	no	—
108	C₂×C₄	—	134	24	no	—
109	C₂×C₄	C₂×C₄×D₆₇	1	6	yes	—
110	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
111	C₂×C₄	C₂×D₂₆₈	1	6	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	—	268	4	no	—
114	C₂²	—	134	4	no	—
115	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
116	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
117	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	268	4	no	—
120	C₄	—	134	4	no	—
121	C₄	C₂×(C₂×D₆₇)⋊C₄	1	12	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	268	4	no	—
126	C₂	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524091

C₂

yes

maximal, characteristic

#16863874

C₂

yes

maximal, characteristic, complemented

#16863369

C₂

yes

maximal, complemented

#519×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524091

—

#14524091

—

C₂×#11761629

C₄

yes

complemented

C₂×#11761628

C₂²

yes

—

C₂×Q₈×D₆₇

C₄

yes

complemented

Q₈×Dic₆₇

—

#14524002

C₂²

yes

—

#14523957

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂²

yes

—

#11761629

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

Q₈×D₆₇

—

C₂×D₂₆₈

C₂×C₄

yes

—

C₂×C₂²⋊D₆₇

C₂³

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

C₂²×C₄

yes

—

Q₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂²×C₄

yes

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×C₂²⋊C₄

yes

characteristic

D₆₇

—

#2884

—

maximal, Sylow

C₆₇

#2884

yes

characteristic, complemented, Sylow

C₂×2₋¹⁺⁴

—

#519

—

#519

D₆₇

yes

characteristic, complemented

#410

—

2₋¹⁺⁴

—

134

—

2₋¹⁺⁴

—

134

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

#117

C₂×D₆₇

yes

—

#99

—

#41

—

134

—

#41

—

134

—

#41

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

C₄×D₆₇

yes

—

C₂×Q₈

D₂₆₈

yes

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₆₇

yes

—

C₄²

—

C₄²

—

134

—

C₂³

—

C₂³

C₂³×D₆₇

yes

characteristic

100

Q₈

—

134

—

101

Q₈

(C₂×D₆₇)⋊C₄

yes

—

102

D₄

—

134

—

103

D₄

—

134

—

104

D₄

—

134

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

134

—

108

C₂×C₄

—

134

—

109

C₂×C₄

C₂×C₄×D₆₇

yes

—

110

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

111

C₂×C₄

C₂×D₂₆₈

yes

—

112

C₂²

—

113

C₂²

—

268

—

114

C₂²

—

134

—

115

C₂²

C₂²×C₄×D₆₇

yes

characteristic

116

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

117

C₂²

C₂²×D₂₆₈

yes

characteristic

118

C₄

—

119

C₄

—

268

—

120

C₄

—

134

—

121

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

268

—

126

C₂

C₂²×(C₂×D₆₇)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₂³×C₄
Frattini	C₂³	C₂³×D₆₇
Fitting	#519×C₆₇	C₂
socle	C₂×C₆₇	C₂²×C₂²⋊C₄
hypercenter	#519	D₆₇
hypocenter	C₆₇	#2884
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×(C₂×D₆₇)⋊C₄

commutator

C₂²×C₆₇

C₂³×C₄

Frattini

C₂³

C₂³×D₆₇

Fitting

#519×C₆₇

C₂

socle

C₂×C₆₇

C₂²×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. #16863874	C₂	semidirect
3. #16863369	C₂	semidirect
4. #519×C₆₇	C₂	semidirect
7. C₂×#11761629	C₄	semidirect
9. C₂×Q₈×D₆₇	C₄	semidirect
64. C₆₇	#2884	semidirect
67. #519	D₆₇	semidirect

H ≅ G/N

type

2. #16863874

C₂

semidirect

3. #16863369

C₂

semidirect

4. #519×C₆₇

C₂

semidirect

7. C₂×#11761629

C₄

semidirect

9. C₂×Q₈×D₆₇

C₄

semidirect

64. C₆₇

#2884

semidirect

67. #519

D₆₇

semidirect

Subgroups — #22568672

Conjugacy classes of subgroups

Complements