Group #30772251 — subgroups

Subgroups — #30772251

Order 17152.

342 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16863217	C₂	1	4	yes	maximal
2	#22561965	C₂	1	2	yes	maximal
3	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	C₂²	1	4	yes	—
5	#16863217	—	2	8	no	—
6	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
7	C₂×#97×C₆₇	C₄	1	2	yes	—
8	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#376×C₆₇	C₄	1	1	yes	characteristic
12	#14523983	—	2	8	no	—
13	C₂×C₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	#97×C₆₇	—	2	4	no	—
20	#97×C₆₇	C₈	1	4	yes	—
21	C₄×C₈×C₆₇	D₄	1	2	yes	—
22	C₄×C₈×C₆₇	D₄	1	4	yes	—
23	C₄×C₈×C₆₇	Q₈	1	4	yes	—
24	C₄×C₈×C₆₇	Q₈	1	2	yes	—
25	#95×C₆₇	—	2	2	no	—
26	C₆₇⋊C₁₆	—	4	4	no	—
27	C₂²×C₄×C₆₇	—	2	2	no	—
28	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
35	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
36	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₆₇	—	2	2	no	—
40	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₆₇	C₂×C₈	1	2	yes	—
43	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	—	4	4	no	—
47	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
52	C₈×C₆₇	—	2	2	no	—
53	C₈×C₆₇	C₄⋊C₈	1	8	yes	—
54	C₈×C₆₇	#127	1	1	yes	characteristic
55	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
56	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₆₇	—	2	2	no	—
58	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₆₇	#423	1	2	yes	—
60	C₄×C₆₇	—	2	4	no	—
61	C₄×C₆₇	—	4	4	no	—
62	C₄×C₆₇	#384	1	1	yes	characteristic
63	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
64	C₄×C₆₇	#423	1	1	yes	characteristic
65	#12901	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	#1842	1	1	yes	characteristic
68	C₂×C₆₇	#2153	1	2	yes	—
69	C₂×#364	—	67	4	no	—
70	#2150	—	67	2	no	—
71	#1726	D₆₇	1	1	yes	characteristic
72	C₆₇	#12901	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	67	4	no	—
74	C₂×#97	Dic₆₇	1	2	yes	—
75	C₂×#97	Dic₆₇	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
78	#376	Dic₆₇	1	1	yes	characteristic
79	#364	—	134	8	no	—
80	C₄⋊C₁₆	—	134	4	no	—
81	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
83	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
85	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
86	M₃₂	—	134	8	no	—
87	C₂×C₁₆	—	134	4	no	—
88	#97	—	2	4	no	—
89	#97	C₆₇⋊C₈	1	4	yes	—
90	C₄×C₈	Dic₁₃₄	1	4	yes	—
91	C₄×C₈	Dic₁₃₄	1	2	yes	—
92	C₄×C₈	D₂₆₈	1	4	yes	—
93	C₄×C₈	D₂₆₈	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11761612	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
100	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
101	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
102	C₂×C₈	#11761612	1	4	yes	—
103	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
104	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
105	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
106	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊Dic₆₇	1	4	yes	—
109	C₄²	C₄⋊Dic₆₇	1	2	yes	—
110	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
111	C₁₆	—	268	4	no	—
112	C₂³	C₂×#11761612	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
120	C₂×C₄	C₂×#11761612	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊(C₆₇⋊C₈)	1	8	yes	—
123	C₈	#14524040	1	1	yes	characteristic
124	C₈	#14524043	1	2	yes	—
125	C₈	#14524043	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
128	C₂²	#16863420	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#16863266	1	1	yes	characteristic
132	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
133	C₄	#16863420	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#22560361	1	1	yes	characteristic
136	C₂	#22561975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863217

C₂

yes

maximal

#22561965

C₂

yes

maximal

C₆₇×#1726

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

#16863217

—

C₄⋊(C₆₇⋊C₁₆)

—

C₂×#97×C₆₇

C₄

yes

—

C₂×#97×C₆₇

C₄

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₄

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

#97×C₆₇

—

#97×C₆₇

C₈

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#95×C₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₄⋊C₈

yes

—

C₈×C₆₇

#127

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#423

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#384

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₈

yes

—

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1842

yes

characteristic

C₂×C₆₇

#2153

yes

—

C₂×#364

—

#2150

—

#1726

D₆₇

yes

characteristic

C₆₇

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

Dic₆₇

yes

—

C₂×#97

Dic₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

Dic₆₇

yes

characteristic

#364

—

134

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

M₃₂

—

134

—

C₂×C₁₆

—

134

—

#97

—

#97

C₆₇⋊C₈

yes

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

D₂₆₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11761612

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

100

C₂×C₈

C₄⋊Dic₆₇

yes

—

101

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

102

C₂×C₈

#11761612

yes

—

103

C₂×C₈

C₂×D₂₆₈

yes

—

104

C₂×C₈

C₂×D₂₆₈

yes

characteristic

105

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

106

C₂×C₈

C₂×Dic₁₃₄

yes

—

107

C₄²

—

108

C₄²

C₄⋊Dic₆₇

yes

—

109

C₄²

C₄⋊Dic₆₇

yes

—

110

C₄²

C₂×C₆₇⋊C₈

yes

—

111

C₁₆

—

268

—

112

C₂³

C₂×#11761612

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄⋊(C₆₇⋊C₈)

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

119

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

120

C₂×C₄

C₂×#11761612

yes

—

121

C₈

—

122

C₈

C₄⋊(C₆₇⋊C₈)

yes

—

123

C₈

#14524040

yes

characteristic

124

C₈

#14524043

yes

—

125

C₈

#14524043

yes

characteristic

126

C₂²

—

127

C₂²

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

128

C₂²

#16863420

yes

—

129

C₄

—

130

C₄

—

131

C₄

#16863266

yes

characteristic

132

C₄

C₂×C₄⋊(C₆₇⋊C₈)

yes

—

133

C₄

#16863420

yes

characteristic

134

C₂

—

135

C₂

#22560361

yes

characteristic

136

C₂

#22561975

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#12901
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1726

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
72. C₆₇	#12901	semidirect

H ≅ G/N

type

72. C₆₇

#12901

semidirect

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

subgroup	index
C₂×#16863217	2
#22561965	2
C₆₇×#1726	2
#12901	67

subgroup

index

C₂×#16863217

#22561965

C₆₇×#1726

#12901

Subgroups — #30772251

Conjugacy classes of subgroups

Complements

Maximal subgroups