Group #30758908 — subgroups

Subgroups — #30758908

Order 17152.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22561032	C₂	1	3	yes	maximal
2	#22558090	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524040	C₂²	1	1	yes	characteristic
5	C₂×#14523963	—	2	6	no	—
6	C₂×#14523954	C₂²	1	3	yes	—
7	#398×C₆₇	—	2	6	no	—
8	#382×C₆₇	C₂²	1	3	yes	—
9	C₂²×D₂₆₈	—	2	3	no	—
10	C₂²×C₆₇⋊C₈	—	2	3	no	—
11	#14524040	—	2	3	no	—
12	#14524040	Q₈	1	2	yes	—
13	#14523963	—	4	12	no	—
14	#14523954	—	2	6	no	—
15	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
16	C₂×C₄⋊C₄×C₆₇	D₄	1	6	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	#95×C₆₇	—	2	2	no	—
19	#95×C₆₇	—	2	6	no	—
20	C₂³×D₆₇	—	4	1	no	—
21	C₂×D₂₆₈	—	2	6	no	—
22	C₂×D₂₆₈	—	4	6	no	—
23	C₂×D₂₆₈	—	4	6	no	—
24	C₂×D₂₆₈	—	4	3	no	—
25	C₂×C₆₇⋊C₈	—	2	6	no	—
26	C₂×C₆₇⋊C₈	—	4	3	no	—
27	C₂×C₆₇⋊C₈	—	4	3	no	—
28	C₂²×C₄×C₆₇	—	2	6	no	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
30	C₄⋊C₄×C₆₇	—	4	6	no	—
31	C₄⋊C₄×C₆₇	—	8	6	no	—
32	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₆₇	#41	1	3	yes	—
34	C₂³×C₆₇	#120	1	1	yes	characteristic
35	C₂²×D₆₇	—	4	2	no	—
36	C₂²×D₆₇	—	8	3	no	—
37	C₂²×D₆₇	—	8	3	no	—
38	C₂×C₄×C₆₇	—	2	3	no	—
39	C₂×C₄×C₆₇	—	2	3	no	—
40	C₂×C₄×C₆₇	—	4	6	no	—
41	C₂×C₄×C₆₇	—	4	6	no	—
42	C₂×C₄×C₆₇	—	4	6	no	—
43	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₆₇	#124	1	3	yes	—
45	D₂₆₈	—	4	12	no	—
46	D₂₆₈	—	8	6	no	—
47	C₆₇⋊C₈	—	4	6	no	—
48	C₂²×C₆₇	#393	1	1	yes	characteristic
49	C₂²×C₆₇	#453	1	3	yes	—
50	C₂²×C₆₇	#454	1	3	yes	—
51	C₂×D₆₇	—	8	6	no	—
52	C₂×D₆₇	—	16	3	no	—
53	C₂×D₆₇	—	16	1	no	—
54	C₄×C₆₇	—	2	3	no	—
55	C₄×C₆₇	—	2	3	no	—
56	C₄×C₆₇	—	8	6	no	—
57	#10360	—	67	1	no	maximal, Sylow
58	C₂×C₆₇	#1656	1	3	yes	—
59	C₂×C₆₇	#1658	1	1	yes	characteristic
60	C₂×C₆₇	#2027	1	3	yes	—
61	D₆₇	—	16	2	no	—
62	#2845	D₆₇	1	1	yes	characteristic, complemented
63	#1961	—	67	3	no	—
64	#1295	—	67	3	no	—
65	C₆₇	#10360	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	67	1	no	—
67	C₂×#102	—	134	6	no	—
68	C₂×#97	—	67	3	no	—
69	#398	—	2	6	no	—
70	#382	C₂×D₆₇	1	3	yes	—
71	C₂²×D₄	—	134	3	no	—
72	C₂²×C₈	—	134	3	no	—
73	#127	—	67	2	no	—
74	#127	—	134	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	6	yes	—
77	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
78	#102	—	268	12	no	—
79	#97	—	134	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	268	1	no	—
83	C₂×D₄	—	268	6	no	—
84	C₂×D₄	—	268	6	no	—
85	C₂×D₄	—	268	3	no	—
86	C₂×D₄	—	134	6	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
89	C₂×C₈	—	268	3	no	—
90	C₂×C₈	—	268	3	no	—
91	C₂×C₈	—	134	6	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₆₇	1	1	yes	characteristic
95	C₄²	#11761629	1	3	yes	—
96	C₂³	—	268	2	no	—
97	C₂³	—	536	3	no	—
98	C₂³	—	536	3	no	—
99	C₂³	#14524012	1	1	yes	characteristic
100	D₄	—	268	12	no	—
101	D₄	—	536	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#14524019	1	3	yes	—
108	C₂×C₄	#14524032	1	3	yes	—
109	C₈	—	268	6	no	—
110	C₂²	—	536	6	no	—
111	C₂²	—	1072	3	no	—
112	C₂²	—	1072	1	no	—
113	C₂²	#16863307	1	1	yes	characteristic
114	C₂²	#16863564	1	3	yes	—
115	C₂²	#16863569	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	1072	2	no	—
120	C₂	#22559502	1	3	yes	—
121	C₂	#22559511	1	1	yes	characteristic
122	C₂	#22561362	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561032

C₂

yes

maximal

#22558090

C₂

yes

maximal, complemented

C₆₇×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

characteristic

C₂×#14523963

—

C₂×#14523954

C₂²

yes

—

#398×C₆₇

—

#382×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

C₂²×C₆₇⋊C₈

—

#14524040

—

#14524040

Q₈

yes

—

#14523963

—

#14523954

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#120

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#124

yes

—

D₂₆₈

—

D₂₆₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#393

yes

characteristic

C₂²×C₆₇

#453

yes

—

C₂²×C₆₇

#454

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

#10360

—

maximal, Sylow

C₂×C₆₇

#1656

yes

—

C₂×C₆₇

#1658

yes

characteristic

C₂×C₆₇

#2027

yes

—

D₆₇

—

#2845

D₆₇

yes

characteristic, complemented

#1961

—

#1295

—

C₆₇

#10360

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

134

—

C₂×#97

—

#398

—

#382

C₂×D₆₇

yes

—

C₂²×D₄

—

134

—

C₂²×C₈

—

134

—

#127

—

#127

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#102

—

268

—

#97

—

134

—

#95

—

#95

—

C₂⁴

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₆₇

yes

characteristic

C₄²

#11761629

yes

—

C₂³

—

268

—

C₂³

—

536

—

C₂³

—

536

—

C₂³

#14524012

yes

characteristic

100

D₄

—

268

—

101

D₄

—

536

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#14524019

yes

—

108

C₂×C₄

#14524032

yes

—

109

C₈

—

268

—

110

C₂²

—

536

—

111

C₂²

—

1072

—

112

C₂²

—

1072

—

113

C₂²

#16863307

yes

characteristic

114

C₂²

#16863564

yes

—

115

C₂²

#16863569

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

1072

—

120

C₂

#22559502

yes

—

121

C₂

#22559511

yes

characteristic

122

C₂

#22561362

yes

—

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2845	C₂
socle	C₂³×C₆₇	#120
prime radical	C₂×#14524040	C₂²
hypercenter	#2845	D₆₇
hypocenter	C₆₇	#10360
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2845

C₂

C₂³×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #22558090	C₂	semidirect
3. C₆₇×#2845	C₂	semidirect
62. #2845	D₆₇	semidirect
65. C₆₇	#10360	semidirect

H ≅ G/N

type

2. #22558090

C₂

semidirect

3. C₆₇×#2845

C₂

semidirect

62. #2845

D₆₇

semidirect

65. C₆₇

#10360

semidirect

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

subgroup	index
#22561032	2
#22558090	2
C₆₇×#2845	2
#10360	67

subgroup

index

#22561032

#22558090

C₆₇×#2845

#10360

Subgroups — #30758908

Conjugacy classes of subgroups

Complements

Maximal subgroups