Group #30772181 — subgroups

Subgroups — #30772181

Order 17152.

322 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
2	#22558287	C₂	1	4	yes	maximal
3	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
5	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
6	C₂×#97×C₆₇	C₄	1	2	yes	—
7	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	#376×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₆₇⋊C₁₆	—	2	4	no	—
12	C₂×C₆₇⋊C₁₆	—	2	8	no	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
18	#97×C₆₇	—	2	4	no	—
19	#97×C₆₇	—	2	2	no	—
20	C₄×C₈×C₆₇	—	2	4	no	—
21	C₄×C₈×C₆₇	D₄	1	2	yes	—
22	C₄×C₈×C₆₇	Q₈	1	2	yes	—
23	#95×C₆₇	C₈	1	4	yes	—
24	C₆₇⋊C₁₆	—	4	4	no	—
25	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
26	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
33	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₆₇	#41	1	4	yes	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₄²×C₆₇	—	2	2	no	—
38	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₄²×C₆₇	M₁₆	1	2	yes	—
40	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	—	2	8	no	—
44	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#117	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
49	C₈×C₆₇	—	2	2	no	—
50	C₈×C₆₇	—	2	4	no	—
51	C₈×C₆₇	#124	1	2	yes	—
52	C₈×C₆₇	#125	1	2	yes	—
53	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂²×C₆₇	#431	1	2	yes	—
55	C₂²×C₆₇	#504	1	4	yes	—
56	C₄×C₆₇	—	2	4	no	—
57	C₄×C₆₇	—	4	4	no	—
58	C₄×C₆₇	#382	1	1	yes	characteristic
59	C₄×C₆₇	#423	1	1	yes	characteristic
60	C₄×C₆₇	#432	1	2	yes	—
61	#12880	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	#1831	1	1	yes	characteristic
63	C₂×C₆₇	#2151	1	2	yes	—
64	C₂×C₆₇	#2164	1	4	yes	—
65	C₂×C₄⋊C₁₆	—	67	2	no	—
66	#1726	D₆₇	1	1	yes	characteristic
67	#1374	—	67	4	no	—
68	C₆₇	#12880	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	—	67	4	no	—
70	C₂×#97	Dic₆₇	1	2	yes	—
71	C₂×#97	Dic₆₇	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
73	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
74	#376	Dic₆₇	1	1	yes	characteristic
75	C₄⋊C₁₆	—	134	4	no	—
76	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
78	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
80	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
81	C₂×C₁₆	—	134	4	no	—
82	C₂×C₁₆	—	134	8	no	—
83	#97	—	2	4	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₁₃₄	1	2	yes	—
87	C₄×C₈	D₂₆₈	1	2	yes	—
88	#95	C₆₇⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
90	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
96	C₂×C₈	#11761612	1	4	yes	—
97	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
99	C₂×C₈	#11761628	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₄⋊Dic₆₇	1	2	yes	—
103	C₄²	#11761612	1	2	yes	—
104	C₁₆	—	268	4	no	—
105	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
111	C₂×C₄	#14524002	1	2	yes	—
112	C₂×C₄	C₂×#11761612	1	2	yes	—
113	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#14524031	1	2	yes	—
117	C₈	#14524035	1	2	yes	—
118	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
119	C₂²	#16863449	1	2	yes	—
120	C₂²	#16863813	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#16863262	1	1	yes	characteristic
124	C₄	#16863420	1	1	yes	characteristic
125	C₄	#16863453	1	2	yes	—
126	C₂	#22560326	1	1	yes	characteristic
127	C₂	#22561968	1	2	yes	—
128	C₂	#22562016	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal

#22558287

C₂

yes

maximal

C₆₇×#1726

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

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

C₂×C₆₇⋊C₁₆

—

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

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#95×C₆₇

C₈

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#124

yes

—

C₈×C₆₇

#125

yes

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#431

yes

—

C₂²×C₆₇

#504

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2164

yes

—

C₂×C₄⋊C₁₆

—

#1726

D₆₇

yes

characteristic

#1374

—

C₆₇

#12880

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

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

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

C₂×C₁₆

—

134

—

C₂×C₁₆

—

134

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

#95

C₆₇⋊C₈

yes

—

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

#11761612

yes

—

C₂×C₈

C₂×D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

C₂×C₈

#11761628

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊Dic₆₇

yes

—

103

C₄²

#11761612

yes

—

104

C₁₆

—

268

—

105

C₂³

C₂²×C₆₇⋊C₈

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊(C₆₇⋊C₈)

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

111

C₂×C₄

#14524002

yes

—

112

C₂×C₄

C₂×#11761612

yes

—

113

C₂×C₄

C₂×#11761612

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

#14524031

yes

—

117

C₈

#14524035

yes

—

118

C₂²

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

119

C₂²

#16863449

yes

—

120

C₂²

#16863813

yes

—

121

C₄

—

122

C₄

—

123

C₄

#16863262

yes

characteristic

124

C₄

#16863420

yes

characteristic

125

C₄

#16863453

yes

—

126

C₂

#22560326

yes

characteristic

127

C₂

#22561968

yes

—

128

C₂

#22562016

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₆₇×#1726	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#12880
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₇

commutator, socle, prime radical

C₂³×C₆₇

C₂²×C₈

Fitting

C₆₇×#1726

C₂

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₈ ◁ #1726

N	H ≅ G/N	type
68. C₆₇	#12880	semidirect

H ≅ G/N

type

68. C₆₇

#12880

semidirect

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558287	2
C₆₇×#1726	2
#12880	67

subgroup

index

C₂×C₄⋊(C₆₇⋊C₁₆)

#22558287

C₆₇×#1726

#12880

Subgroups — #30772181

Conjugacy classes of subgroups

Complements

Maximal subgroups