Group #30760327 — subgroups

Subgroups — #30760327

Order 17152.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22560378	C₂	1	3	yes	maximal
2	#22558090	C₂	1	3	yes	maximal
3	C₆₇×#2844	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523976	—	2	3	no	—
5	C₂×#14523974	—	2	3	no	—
6	C₂×#14523954	C₂²	1	3	yes	—
7	#16863266	C₂²	1	1	yes	characteristic
8	#400×C₆₇	—	2	3	no	—
9	#395×C₆₇	—	2	3	no	—
10	#382×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₆₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₆₇	—	2	3	no	—
13	#14523976	—	4	6	no	—
14	#14523974	—	4	6	no	—
15	#14523954	—	2	3	no	—
16	#14523954	Q₈	1	6	yes	complemented
17	C₂²×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
19	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
20	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
21	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
22	#95×C₆₇	—	2	3	no	—
23	#95×C₆₇	—	2	1	no	—
24	#95×C₆₇	—	2	6	no	—
25	C₂²×Dic₆₇	—	4	1	no	—
26	C₂×C₆₇⋊C₈	—	2	6	no	—
27	C₂×C₆₇⋊C₈	—	2	6	no	—
28	C₂×C₆₇⋊C₈	—	2	6	no	—
29	C₄⋊Dic₆₇	—	4	3	no	—
30	C₄⋊Dic₆₇	—	4	3	no	—
31	C₂×Q₈×C₆₇	—	8	3	no	—
32	C₂²×C₄×C₆₇	—	2	3	no	—
33	C₂²×C₄×C₆₇	—	2	3	no	—
34	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
35	C₄⋊C₄×C₆₇	—	4	3	no	—
36	C₄⋊C₄×C₆₇	—	4	3	no	—
37	C₄⋊C₄×C₆₇	—	8	3	no	—
38	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
39	C₄²×C₆₇	#41	1	1	yes	characteristic
40	C₂³×C₆₇	#120	1	1	yes	characteristic
41	Q₈×C₆₇	—	8	4	no	—
42	C₂×C₄×C₆₇	—	2	3	no	—
43	C₂×C₄×C₆₇	—	2	3	no	—
44	C₂×C₄×C₆₇	—	4	3	no	—
45	C₂×C₄×C₆₇	—	4	3	no	—
46	C₂×C₄×C₆₇	—	4	6	no	—
47	C₂×C₄×C₆₇	—	4	6	no	—
48	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
49	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
50	C₂×Dic₆₇	—	8	3	no	—
51	C₆₇⋊C₈	—	2	12	no	—
52	C₂²×C₆₇	#395	1	1	yes	characteristic
53	C₂²×C₆₇	#453	1	3	yes	—
54	C₂²×C₆₇	#456	1	3	yes	—
55	C₄×C₆₇	—	2	3	no	—
56	C₄×C₆₇	—	2	3	no	—
57	C₄×C₆₇	—	8	3	no	—
58	C₄×C₆₇	—	8	3	no	—
59	Dic₆₇	—	16	1	no	—
60	#10583	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1655	1	1	yes	characteristic
62	C₂×C₆₇	#1660	1	3	yes	—
63	C₂×C₆₇	#2060	1	3	yes	—
64	#2844	D₆₇	1	1	yes	characteristic
65	#1847	—	67	3	no	—
66	#1295	—	67	3	no	—
67	C₆₇	#10583	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	134	3	no	—
69	C₂×#106	—	134	3	no	—
70	C₂×#97	—	67	3	no	—
71	#400	—	2	3	no	—
72	#395	—	2	3	no	—
73	#384	—	67	1	no	—
74	#382	C₂×D₆₇	1	3	yes	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	134	3	no	—
77	C₂×C₄⋊C₄	—	2	3	no	—
78	C₂×C₄⋊C₄	—	134	3	no	—
79	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
80	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
81	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
82	#107	—	268	6	no	—
83	#106	—	268	6	no	—
84	#97	—	67	6	no	—
85	#97	—	134	3	no	—
86	#95	—	2	3	no	—
87	#95	—	2	1	no	—
88	#95	—	2	6	no	—
89	C₂×Q₈	—	8	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	268	1	no	—
93	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
94	C₂×C₈	—	134	6	no	—
95	C₂×C₈	—	134	6	no	—
96	C₂×C₈	—	134	6	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	4	3	no	—
99	C₄⋊C₄	—	8	3	no	—
100	C₄⋊C₄	—	268	3	no	—
101	C₄⋊C₄	—	268	3	no	—
102	C₄²	Q₈×D₆₇	1	3	yes	—
103	C₄²	#11761629	1	1	yes	characteristic
104	C₂³	#14524012	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	536	3	no	—
113	C₂×C₄	#14524019	1	3	yes	—
114	C₂×C₄	#14524041	1	3	yes	—
115	C₈	—	134	12	no	—
116	C₂²	#16863315	1	1	yes	characteristic
117	C₂²	#16863564	1	3	yes	—
118	C₂²	#16863579	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	1072	1	no	—
124	C₂	#22559498	1	1	yes	characteristic
125	C₂	#22559520	1	3	yes	—
126	C₂	#22561551	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560378

C₂

yes

maximal

#22558090

C₂

yes

maximal

C₆₇×#2844

C₂

yes

maximal, characteristic

C₂×#14523976

—

C₂×#14523974

—

C₂×#14523954

C₂²

yes

—

#16863266

C₂²

yes

characteristic

#400×C₆₇

—

#395×C₆₇

—

#382×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#395

yes

characteristic

C₂²×C₆₇

#453

yes

—

C₂²×C₆₇

#456

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2060

yes

—

#2844

D₆₇

yes

characteristic

#1847

—

#1295

—

C₆₇

#10583

yes

characteristic, complemented, Sylow

C₂×#107

—

134

—

C₂×#106

—

134

—

C₂×#97

—

—

—

—

C₂×D₆₇

yes

—

C₂²×Q₈

—

C₂²×C₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#107

—

268

—

#106

—

268

—

#97

—

#97

—

134

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

268

—

101

C₄⋊C₄

—

268

—

102

C₄²

Q₈×D₆₇

yes

—

103

C₄²

#11761629

yes

characteristic

104

C₂³

#14524012

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

536

—

113

C₂×C₄

#14524019

yes

—

114

C₂×C₄

#14524041

yes

—

115

C₈

—

134

—

116

C₂²

#16863315

yes

characteristic

117

C₂²

#16863564

yes

—

118

C₂²

#16863579

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

1072

—

124

C₂

#22559498

yes

characteristic

125

C₂

#22559520

yes

—

126

C₂

#22561551

yes

—

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2844	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#2844	D₆₇
hypocenter	C₆₇	#10583
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2844

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
16. #14523954	Q₈	semidirect
67. C₆₇	#10583	semidirect

H ≅ G/N

type

16. #14523954

Q₈

semidirect

67. C₆₇

#10583

semidirect

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

subgroup	index
#22560378	2
#22558090	2
C₆₇×#2844	2
#10583	67

subgroup

index

#22560378

#22558090

C₆₇×#2844

#10583

Subgroups — #30760327

Conjugacy classes of subgroups

Complements

Maximal subgroups