Group #30856082 — subgroups

Subgroups — #30856082

Order 17152.

830 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 243 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	#376×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#22560354	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560353	C₂	1	6	yes	maximal, complemented
4	#22559901	C₂	1	6	yes	maximal
5	C₆₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11761611	C₂²	1	4	yes	—
7	C₂×C₄²×D₆₇	C₂²	1	3	yes	—
8	#16863242	C₂²	1	3	yes	—
9	#16863241	C₄	1	4	yes	—
10	#95×D₆₇	C₄	1	4	yes	—
11	#16863151	C₂²	1	6	yes	—
12	#16863150	C₂²	1	12	yes	—
13	#376×C₆₇	C₂²	1	1	yes	characteristic
14	#336×C₆₇	C₂²	1	6	yes	—
15	C₂²×C₄×D₆₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
17	C₂×#11761611	—	2	24	no	—
18	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
21	C₄²×D₆₇	—	2	12	no	—
22	C₆₇⋊#95	C₂×C₄	1	12	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
25	#95×C₆₇	C₂×C₄	1	4	yes	—
26	C₂³×D₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
28	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×D₆₇	—	2	12	no	—
30	C₂×C₄×D₆₇	—	4	12	no	—
31	C₂×C₄×D₆₇	#41	1	12	yes	—
32	#11761611	—	4	16	no	—
33	C₂×C₆₇⋊C₈	—	2	12	no	—
34	C₄×Dic₆₇	—	2	6	no	—
35	C₄×Dic₆₇	—	2	12	no	—
36	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	12	no	—
39	C₄²×C₆₇	—	2	6	no	—
40	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
41	C₂²×D₆₇	—	2	4	no	—
42	C₂²×D₆₇	#117	1	6	yes	—
43	C₂×C₄×C₆₇	—	2	6	no	—
44	C₂×C₄×C₆₇	—	2	12	no	—
45	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
46	C₂×Dic₆₇	—	2	6	no	—
47	C₂×Dic₆₇	—	2	12	no	—
48	C₂×Dic₆₇	#117	1	6	yes	—
49	C₄×D₆₇	—	4	24	no	—
50	C₄×D₆₇	#126	1	16	yes	—
51	C₈×C₆₇	—	4	4	no	—
52	C₆₇⋊C₈	—	4	4	no	—
53	C₂²×C₆₇	C₂×#117	1	3	yes	—
54	C₂²×C₆₇	#568	1	4	yes	—
55	C₂×D₆₇	—	2	12	no	—
56	C₂×D₆₇	#383	1	4	yes	—
57	C₄×C₆₇	—	4	6	no	—
58	C₄×C₆₇	C₂×#126	1	4	yes	—
59	Dic₆₇	—	4	6	no	—
60	Dic₆₇	#383	1	4	yes	—
61	#20773	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
63	C₂×C₆₇	#2921	1	6	yes	—
64	D₆₇	—	2	4	no	—
65	C₂×#376	—	67	1	no	—
66	#1840	—	67	1	no	—
67	#1840	—	67	6	no	—
68	#1840	D₆₇	1	1	yes	characteristic, complemented
69	#1728	—	67	6	no	—
70	C₆₇	#20773	1	1	yes	characteristic, complemented, Sylow
71	C₂²×M₁₆	—	67	4	no	—
72	C₂²×C₄²	—	67	3	no	—
73	#376	—	67	3	no	—
74	#376	—	67	4	no	—
75	#376	C₂×D₆₇	1	1	yes	characteristic
76	C₂×#95	—	67	4	no	—
77	#336	—	67	12	no	—
78	#336	—	67	6	no	—
79	#336	C₂×D₆₇	1	6	yes	—
80	C₂³×C₄	—	67	6	no	—
81	C₂³×C₄	—	67	1	no	—
82	C₂×M₁₆	—	134	24	no	—
83	C₂²×C₈	—	67	4	no	—
84	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
85	C₂×C₄²	—	67	6	no	—
86	C₂×C₄²	—	67	3	no	—
87	C₂×C₄²	—	134	12	no	—
88	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
89	#95	—	67	12	no	—
90	#95	C₄×D₆₇	1	4	yes	—
91	C₂⁴	—	67	1	no	—
92	C₂²×C₄	—	268	12	no	—
93	C₂²×C₄	—	67	1	no	—
94	C₂²×C₄	—	67	12	no	—
95	C₂²×C₄	—	67	6	no	—
96	C₂²×C₄	—	134	12	no	—
97	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
98	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
99	M₁₆	—	268	16	no	—
100	C₂×C₈	—	2	12	no	—
101	C₂×C₈	—	134	12	no	—
102	C₄²	—	2	6	no	—
103	C₄²	—	134	12	no	—
104	C₄²	—	134	6	no	—
105	C₂³	—	67	6	no	—
106	C₂³	—	134	4	no	—
107	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	268	24	no	—
111	C₂×C₄	—	67	6	no	—
112	C₂×C₄	—	67	16	no	—
113	C₂×C₄	—	134	6	no	—
114	C₂×C₄	—	134	12	no	—
115	C₂×C₄	#41×D₆₇	1	6	yes	—
116	C₈	—	4	4	no	—
117	C₈	—	268	4	no	—
118	C₂²	—	67	4	no	—
119	C₂²	—	134	12	no	—
120	C₂²	#117×D₆₇	1	3	yes	—
121	C₂²	#16864161	1	4	yes	—
122	C₄	—	4	6	no	—
123	C₄	—	268	6	no	—
124	C₄	—	67	4	no	—
125	C₄	#126×D₆₇	1	4	yes	—
126	C₂	—	134	4	no	—
127	C₂	#383×D₆₇	1	1	yes	characteristic
128	C₂	#22568843	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#376×D₆₇

C₂

yes

maximal, characteristic

#22560354

C₂

yes

maximal, characteristic, complemented

#22560353

C₂

yes

maximal, complemented

#22559901

C₂

yes

maximal

C₆₇×#1840

C₂

yes

maximal, characteristic, complemented

C₂²×#11761611

C₂²

yes

—

C₂×C₄²×D₆₇

C₂²

yes

—

#16863242

C₂²

yes

—

#16863241

C₄

yes

—

#95×D₆₇

C₄

yes

—

#16863151

C₂²

yes

—

#16863150

C₂²

yes

—

#376×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₂²

yes

—

C₂²×C₄×D₆₇

C₂×C₄

yes

—

C₂²×C₄×D₆₇

C₂³

yes

characteristic

C₂×#11761611

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂³

yes

—

C₄²×D₆₇

—

C₆₇⋊#95

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#95×C₆₇

C₂×C₄

yes

—

C₂³×D₆₇

C₂²×C₄

yes

characteristic

C₂²×Dic₆₇

C₂²×C₄

yes

—

C₂²×Dic₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

#41

yes

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂³×C₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

#117

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#117

yes

—

C₄×D₆₇

—

C₄×D₆₇

#126

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×#117

yes

—

C₂²×C₆₇

#568

yes

—

C₂×D₆₇

—

C₂×D₆₇

#383

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#126

yes

—

Dic₆₇

—

Dic₆₇

#383

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

—

—

—

D₆₇

yes

characteristic, complemented

#1728

—

C₆₇

#20773

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

—

#376

—

#376

—

#376

C₂×D₆₇

yes

characteristic

C₂×#95

—

#336

—

#336

—

#336

C₂×D₆₇

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

—

#95

—

#95

C₄×D₆₇

yes

—

C₂⁴

—

C₂²×C₄

—

268

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂×C₄×D₆₇

yes

—

C₂²×C₄

C₂³×D₆₇

yes

characteristic

M₁₆

—

268

—

100

C₂×C₈

—

101

C₂×C₈

—

134

—

102

C₄²

—

103

C₄²

—

134

—

104

C₄²

—

134

—

105

C₂³

—

106

C₂³

—

134

—

107

C₂³

C₂²×C₄×D₆₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

268

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

134

—

114

C₂×C₄

—

134

—

115

C₂×C₄

#41×D₆₇

yes

—

116

C₈

—

117

C₈

—

268

—

118

C₂²

—

119

C₂²

—

134

—

120

C₂²

#117×D₆₇

yes

—

121

C₂²

#16864161

yes

—

122

C₄

—

123

C₄

—

268

—

124

C₄

—

125

C₄

#126×D₆₇

yes

—

126

C₂

—

134

—

127

C₂

#383×D₆₇

yes

characteristic

128

C₂

#22568843

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1840	C₂
prime radical	C₂³×D₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1840	D₆₇
hypocenter	C₆₇	#20773
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1840

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂³×C₄

Fitting

C₆₇×#1840

C₂

prime radical

C₂³×D₆₇

C₂²×C₄

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₄ ◁ #1840

N	H ≅ G/N	type
2. #22560354	C₂	semidirect
3. #22560353	C₂	semidirect
5. C₆₇×#1840	C₂	semidirect
68. #1840	D₆₇	semidirect
70. C₆₇	#20773	semidirect

H ≅ G/N

type

2. #22560354

C₂

semidirect

3. #22560353

C₂

semidirect

5. C₆₇×#1840

C₂

semidirect

68. #1840

D₆₇

semidirect

70. C₆₇

#20773

semidirect

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

subgroup	index
#376×D₆₇	2
#22560354	2
#22560353	2
#22559901	2
C₆₇×#1840	2
#20773	67

subgroup

index

#376×D₆₇

C₆₇×#1840

Subgroups — #30856082

Conjugacy classes of subgroups

Complements

Maximal subgroups