Group #22570706 — subgroups

Subgroups — C₂×#16863792 #22570706

Order 8576.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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	#16863792	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523974	C₂	1	4	yes	maximal
4	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	—	2	1	no	—
7	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#14524043	—	2	2	no	—
9	#14524043	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₆₇	—	2	1	no	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	#14523974	C₂²	1	16	yes	—
13	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₇	—	2	1	no	—
15	C₄⋊Q₈×C₆₇	—	2	2	no	—
16	C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₄	—	2	4	no	—
21	C₂×Dic₁₃₄	—	4	4	no	—
22	C₂²×Dic₆₇	—	2	2	no	—
23	C₂×C₆₇⋊C₈	Q₈	1	8	yes	complemented
24	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
25	C₄⋊Dic₆₇	—	2	4	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₆₇	—	2	4	no	—
29	C₂×Q₈×C₆₇	—	4	4	no	—
30	C₂²×C₄×C₆₇	—	2	2	no	—
31	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
37	C₄²×C₆₇	D₄	1	2	yes	—
38	C₄²×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₆₇	—	4	8	no	—
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	—	4	4	no	—
43	C₂×C₄×C₆₇	SD₁₆	1	4	yes	—
44	C₂×C₄×C₆₇	SD₁₆	1	4	yes	—
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	4	4	no	—
52	Dic₁₃₄	—	4	8	no	—
53	C₆₇⋊C₈	C₂×Q₈	1	8	yes	complemented
54	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
56	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₇	—	4	4	no	—
59	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
61	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
62	C₄×C₆₇	C₂²×Q₈	1	2	yes	—
63	Dic₆₇	—	4	4	no	—
64	C₂×C₆₇	#498	1	4	yes	complemented
65	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×#498	—	67	1	no	maximal, Sylow
69	C₆₇	C₂×#498	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊Q₈	—	67	1	no	—
71	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
72	#498	—	67	8	no	—
73	C₂×#106	—	67	4	no	—
74	C₂×C₄×C₈	—	67	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	134	1	no	—
77	C₂²×C₈	—	67	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	67	4	no	—
80	C₄⋊Q₈	—	134	2	no	—
81	C₄⋊Q₈	C₂×D₆₇	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	67	2	no	—
84	C₂×C₄⋊C₄	—	134	1	no	—
85	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
86	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
87	#106	—	67	16	no	—
88	C₄×C₈	—	67	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	268	4	no	—
92	C₂×Q₈	—	134	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	134	2	no	—
95	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×C₈	—	67	8	no	—
99	C₂×C₈	—	67	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	67	4	no	—
103	C₄⋊C₄	—	134	2	no	—
104	C₄⋊C₄	—	134	4	no	—
105	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
106	C₄²	C₂²⋊D₆₇	1	2	yes	—
107	C₄²	C₂²×D₆₇	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	268	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	268	4	no	—
114	C₂×C₄	—	134	4	no	—
115	C₂×C₄	#11761615	1	4	yes	—
116	C₂×C₄	#11761616	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
120	C₂×C₄	Q₈×D₆₇	1	4	yes	—
121	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
122	C₈	—	67	8	no	—
123	C₂²	#14524041	1	2	yes	—
124	C₂²	C₂×#11761615	1	2	yes	—
125	C₂²	C₂×#11761616	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	268	4	no	—
129	C₄	#14524041	1	2	yes	—
130	C₄	C₂×#11761615	1	4	yes	—
131	C₄	C₂×#11761616	1	4	yes	—
132	C₄	C₂×Q₈×D₆₇	1	2	yes	—
133	C₂	#16863792	1	4	yes	direct product factor
134	C₂	C₂×#14524041	1	1	yes	characteristic
135	C₂	C₂²×#11761615	1	1	yes	characteristic
136	C₂	C₂²×#11761616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

#16863792

C₂

yes

maximal, direct product factor

C₂×#14523974

C₂

yes

maximal

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

#14524043

—

#14524043

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523974

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₆₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

—

Dic₆₇

—

C₂×C₆₇

#498

yes

complemented

C₂×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₆₇

yes

characteristic

#498

—

C₂×#106

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

—

134

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

C₂²×D₆₇

yes

—

106

C₄²

C₂²⋊D₆₇

yes

—

107

C₄²

C₂²×D₆₇

yes

—

108

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

109

Q₈

—

110

Q₈

—

268

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

134

—

115

C₂×C₄

#11761615

yes

—

116

C₂×C₄

#11761616

yes

—

117

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

120

C₂×C₄

Q₈×D₆₇

yes

—

121

C₂×C₄

C₂³×D₆₇

yes

characteristic

122

C₈

—

123

C₂²

#14524041

yes

—

124

C₂²

C₂×#11761615

yes

—

125

C₂²

C₂×#11761616

yes

—

126

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

127

C₄

—

128

C₄

—

268

—

129

C₄

#14524041

yes

—

130

C₄

C₂×#11761615

yes

—

131

C₄

C₂×#11761616

yes

—

132

C₄

C₂×Q₈×D₆₇

yes

—

133

C₂

#16863792

yes

direct product factor

134

C₂

C₂×#14524041

yes

characteristic

135

C₂

C₂²×#11761615

yes

characteristic

136

C₂

C₂²×#11761616

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	C₂×#498
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

C₂×#498

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16863792	C₂	direct (N × H)
23. C₂×C₆₇⋊C₈	Q₈	semidirect
53. C₆₇⋊C₈	C₂×Q₈	semidirect
64. C₂×C₆₇	#498	semidirect
69. C₆₇	C₂×#498	semidirect
133. C₂	#16863792	direct (N × H)

H ≅ G/N

type

2. #16863792

C₂

direct (N × H)

23. C₂×C₆₇⋊C₈

Q₈

semidirect

53. C₆₇⋊C₈

C₂×Q₈

semidirect

64. C₂×C₆₇

#498

semidirect

69. C₆₇

C₂×#498

semidirect

133. C₂

#16863792

direct (N × H)

Subgroups — C₂×#16863792 #22570706

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863792 #22570706

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863792 #22570706