Group #30692161 — subgroups

Subgroups — C₆₇×#12910 #30692161

Order 17152.

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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×#368×C₆₇	C₂	1	4	yes	maximal
2	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1848	C₂	1	2	yes	maximal
4	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
5	C₂×#108×C₆₇	—	2	2	no	—
6	C₂×#108×C₆₇	C₂²	1	4	yes	—
7	#423×C₆₇	—	2	2	no	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#368×C₆₇	—	2	8	no	—
10	#346×C₆₇	C₄	1	4	yes	—
11	C₂×M₁₆×C₆₇	—	2	4	no	—
12	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
16	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
17	C₂×C₁₆×C₆₇	Q₈	1	4	yes	—
18	#108×C₆₇	—	2	8	no	—
19	#108×C₆₇	—	4	4	no	—
20	C₄⋊C₈×C₆₇	—	4	4	no	—
21	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
24	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
25	M₁₆×C₆₇	—	4	4	no	—
26	M₁₆×C₆₇	—	4	4	no	—
27	C₂×C₈×C₆₇	—	4	4	no	—
28	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₆₇	D₈	1	2	yes	—
30	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
31	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
33	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₄²×C₆₇	D₈	1	2	yes	—
37	C₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
38	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	4	2	no	—
40	C₂×C₄×C₆₇	#107	1	4	yes	—
41	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₆₇	—	8	4	no	—
48	C₈×C₆₇	#107	1	4	yes	—
49	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
53	C₂²×C₆₇	#494	1	1	yes	characteristic
54	C₂²×C₆₇	#500	1	1	yes	characteristic
55	C₄×C₆₇	—	2	4	no	—
56	C₄×C₆₇	#384	1	1	yes	characteristic
57	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
58	C₄×C₆₇	#493	1	1	yes	characteristic
59	C₄×C₆₇	#500	1	1	yes	characteristic
60	#12910	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₆₇	—	2	1	no	—
62	C₂×C₆₇	—	2	1	no	—
63	C₂×C₆₇	#1845	1	1	yes	characteristic
64	C₂×C₆₇	#2161	1	2	yes	—
65	C₂×#368	C₂×C₆₇	1	4	yes	—
66	C₂×#346	C₂×C₆₇	1	1	yes	characteristic
67	#1848	C₂×C₆₇	1	2	yes	—
68	C₆₇	#12910	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
70	C₂×#108	—	2	2	no	—
71	C₂×#108	C₂²×C₆₇	1	4	yes	—
72	#423	—	2	2	no	—
73	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
74	#368	—	2	8	no	—
75	#346	C₄×C₆₇	1	4	yes	—
76	C₂×M₁₆	—	2	4	no	—
77	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
78	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
79	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
80	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
81	C₂×C₁₆	D₄×C₆₇	1	4	yes	—
82	C₂×C₁₆	Q₈×C₆₇	1	4	yes	—
83	#108	—	2	8	no	—
84	#108	—	4	4	no	—
85	C₄⋊C₈	—	4	4	no	—
86	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
87	C₄×C₈	Q₈×C₆₇	1	2	yes	—
88	C₂²×C₄	Q₁₆×C₆₇	1	2	yes	—
89	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
90	M₁₆	—	4	4	no	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
94	C₂×C₈	D₈×C₆₇	1	2	yes	—
95	C₂×C₈	Q₁₆×C₆₇	1	2	yes	—
96	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
98	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
99	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
100	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
101	C₄²	D₈×C₆₇	1	2	yes	—
102	C₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
103	C₂³	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#107×C₆₇	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
107	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
108	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
110	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
111	C₂×C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
112	C₈	—	8	4	no	—
113	C₈	#107×C₆₇	1	4	yes	—
114	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
115	C₈	C₄⋊Q₈×C₆₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#107×C₆₇	1	1	yes	characteristic
118	C₂²	#494×C₆₇	1	1	yes	characteristic
119	C₂²	#500×C₆₇	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#384×C₆₇	1	1	yes	characteristic
122	C₄	C₂×#107×C₆₇	1	1	yes	characteristic
123	C₄	#493×C₆₇	1	1	yes	characteristic
124	C₄	#500×C₆₇	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₆₇×#1845	1	1	yes	characteristic
128	C₂	C₆₇×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#368×C₆₇

C₂

yes

maximal

C₂×#346×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×#108×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

—

#423×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#368×C₆₇

—

#346×C₆₇

C₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

Q₈

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

—

C₁₆×C₆₇

C₄⋊C₄

yes

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

#107

yes

—

C₈×C₆₇

C₂×C₄⋊C₄

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1845

yes

characteristic

C₂×C₆₇

#2161

yes

—

C₂×#368

C₂×C₆₇

yes

—

C₂×#346

C₂×C₆₇

yes

characteristic

#1848

C₂×C₆₇

yes

—

C₆₇

#12910

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₆₇

yes

—

C₂×#108

—

C₂×#108

C₂²×C₆₇

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

#368

—

#346

C₄×C₆₇

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₆₇

yes

characteristic

C₂²×C₈

C₂³×C₆₇

yes

characteristic

C₂×C₄²

D₄×C₆₇

yes

characteristic

C₂×C₁₆

C₂×C₄×C₆₇

yes

—

C₂×C₁₆

D₄×C₆₇

yes

—

C₂×C₁₆

Q₈×C₆₇

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₆₇

yes

—

C₄×C₈

Q₈×C₆₇

yes

—

C₂²×C₄

Q₁₆×C₆₇

yes

—

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₆₇

yes

—

C₂×C₈

D₈×C₆₇

yes

—

C₂×C₈

Q₁₆×C₆₇

yes

—

C₂×C₈

C₂²×C₄×C₆₇

yes

characteristic

C₂×C₈

C₂×D₄×C₆₇

yes

—

C₂×C₈

C₂×Q₈×C₆₇

yes

—

C₂×C₈

C₂×Q₈×C₆₇

yes

characteristic

100

C₄²

C₄⋊C₄×C₆₇

yes

—

101

C₄²

D₈×C₆₇

yes

—

102

C₁₆

C₄⋊C₄×C₆₇

yes

—

103

C₂³

C₂×Q₁₆×C₆₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#107×C₆₇

yes

—

106

C₂×C₄

C₂×C₄⋊C₄×C₆₇

yes

characteristic

107

C₂×C₄

#127×C₆₇

yes

characteristic

108

C₂×C₄

C₄⋊Q₈×C₆₇

yes

characteristic

109

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

110

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

111

C₂×C₄

C₂×Q₁₆×C₆₇

yes

characteristic

112

C₈

—

113

C₈

#107×C₆₇

yes

—

114

C₈

C₂×C₄⋊C₄×C₆₇

yes

—

115

C₈

C₄⋊Q₈×C₆₇

yes

—

116

C₂²

—

117

C₂²

C₂×#107×C₆₇

yes

characteristic

118

C₂²

#494×C₆₇

yes

characteristic

119

C₂²

#500×C₆₇

yes

characteristic

120

C₄

—

121

C₄

#384×C₆₇

yes

characteristic

122

C₄

C₂×#107×C₆₇

yes

characteristic

123

C₄

#493×C₆₇

yes

characteristic

124

C₄

#500×C₆₇

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₆₇×#1845

yes

characteristic

128

C₂

C₆₇×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₆₇

Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂³×C₆₇

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
60. #12910	C₆₇	direct (N × H)
68. C₆₇	#12910	direct (N × H)

H ≅ G/N

type

60. #12910

C₆₇

direct (N × H)

68. C₆₇

#12910

direct (N × H)

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

subgroup	index
C₂×#368×C₆₇	2
C₂×#346×C₆₇	2
C₆₇×#1848	2
#12910	67

subgroup

index

C₂×#368×C₆₇

C₂×#346×C₆₇

C₆₇×#1848

#12910

Subgroups — C₆₇×#12910 #30692161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#12910 #30692161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#12910 #30692161