Group #30856219 — subgroups

Subgroups — #30856219

Order 17152.

1118 conjugacy classes of subgroups; only the 571 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 571 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×#16863420	C₂	1	6	yes	maximal
2	#22560363	C₂	1	8	yes	maximal, complemented
3	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	C₂²	1	4	yes	—
5	#16863420	—	2	24	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	24	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
8	#376×C₆₇	C₄	1	4	yes	—
9	#376×C₆₇	C₂²	1	4	yes	—
10	C₂×#95×C₆₇	C₄	1	4	yes	—
11	C₂×#11761612	—	2	24	no	—
12	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
13	C₄⋊(C₆₇⋊C₈)	—	2	48	no	—
14	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₆₇	D₄	1	12	yes	—
18	C₂×C₄²×C₆₇	Q₈	1	12	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
20	#95×C₆₇	C₂×C₄	1	16	yes	—
21	#11761612	—	4	16	no	—
22	C₂×C₆₇⋊C₈	—	2	24	no	—
23	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	—	4	12	no	—
25	C₂²×C₄×C₆₇	C₄⋊C₄	1	24	yes	—
26	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
29	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
30	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₄²×C₆₇	C₄⋊C₄	1	24	yes	—
32	C₄²×C₆₇	C₂×D₄	1	12	yes	—
33	C₄²×C₆₇	C₂×Q₈	1	12	yes	—
34	C₂³×C₆₇	—	2	4	no	—
35	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
36	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	24	no	—
38	C₂×C₄×C₆₇	—	4	24	no	—
39	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
40	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
41	C₂×C₄×C₆₇	#127	1	4	yes	—
42	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
43	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₆₇	C₂²×Q₈	1	3	yes	—
45	C₆₇⋊C₈	—	4	8	no	—
46	C₂²×C₆₇	—	2	12	no	—
47	C₂²×C₆₇	#384	1	4	yes	—
48	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
49	C₂²×C₆₇	#568	1	4	yes	—
50	C₄×C₆₇	—	2	24	no	—
51	C₄×C₆₇	#384	1	4	yes	—
52	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
53	C₄×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
54	#20792	—	67	1	no	maximal, Sylow
55	C₂×C₆₇	—	2	4	no	—
56	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
57	C₂×C₆₇	#2905	1	6	yes	—
58	C₂×#423	—	67	6	no	—
59	C₂×#376	D₆₇	1	1	yes	characteristic
60	#1843	—	67	8	no	—
61	C₆₇	#20792	1	1	yes	characteristic, complemented, Sylow
62	C₂²×M₁₆	—	67	4	no	—
63	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
64	#423	—	134	24	no	—
65	C₂×C₄⋊C₈	—	67	24	no	—
66	#376	Dic₆₇	1	4	yes	—
67	#376	C₂×D₆₇	1	4	yes	—
68	C₂×#95	Dic₆₇	1	4	yes	—
69	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
70	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
71	C₂×M₁₆	—	134	24	no	—
72	C₂²×C₈	—	67	8	no	—
73	C₂×C₄²	Dic₁₃₄	1	12	yes	—
74	C₂×C₄²	D₂₆₈	1	12	yes	—
75	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
76	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
77	C₄⋊C₈	—	134	48	no	—
78	#95	C₂×Dic₆₇	1	16	yes	—
79	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
80	C₂²×C₄	—	4	12	no	—
81	C₂²×C₄	C₄⋊Dic₆₇	1	24	yes	—
82	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
83	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
85	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
86	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
87	M₁₆	—	268	16	no	—
88	C₂×C₈	—	134	24	no	—
89	C₄²	C₄⋊Dic₆₇	1	24	yes	—
90	C₄²	C₂×D₂₆₈	1	12	yes	—
91	C₄²	C₂×Dic₁₃₄	1	12	yes	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₄⋊Dic₆₇	1	6	yes	—
94	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	4	24	no	—
97	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
99	C₂×C₄	#14524040	1	4	yes	—
100	C₂×C₄	#14524043	1	12	yes	—
101	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
102	C₂×C₄	C₂²×Dic₁₃₄	1	3	yes	—
103	C₈	—	268	8	no	—
104	C₂²	—	2	12	no	—
105	C₂²	#16863266	1	4	yes	—
106	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
107	C₂²	#16864163	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	#16863266	1	4	yes	—
110	C₄	C₂×#14524040	1	1	yes	characteristic
111	C₄	C₂×#14524043	1	3	yes	—
112	C₂	—	2	4	no	—
113	C₂	C₂×#16863266	1	1	yes	characteristic
114	C₂	#22568773	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863420

C₂

yes

maximal

#22560363

C₂

yes

maximal, complemented

C₂×#376×C₆₇

C₂

yes

maximal, characteristic

C₂²×#11761612

C₂²

yes

—

#16863420

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²×C₄²×C₆₇

C₂²

yes

—

#376×C₆₇

C₄

yes

—

#376×C₆₇

C₂²

yes

—

C₂×#95×C₆₇

C₄

yes

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂³×C₄×C₆₇

C₂×C₄

yes

—

C₂³×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#95×C₆₇

C₂×C₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

—

C₂³×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂²×Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

—

C₂²×C₆₇

C₂²×C₄⋊C₄

yes

—

C₂²×C₆₇

#568

yes

—

C₄×C₆₇

—

C₄×C₆₇

#384

yes

—

C₄×C₆₇

C₂×#127

yes

characteristic

C₄×C₆₇

C₂×C₄⋊Q₈

yes

—

#20792

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₂×#384

yes

characteristic

C₂×C₆₇

#2905

yes

—

C₂×#423

—

C₂×#376

D₆₇

yes

characteristic

#1843

—

C₆₇

#20792

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

C₂×D₆₇

yes

—

#423

—

134

—

C₂×C₄⋊C₈

—

#376

Dic₆₇

yes

—

#376

C₂×D₆₇

yes

—

C₂×#95

Dic₆₇

yes

—

C₂³×C₄

C₂×Dic₆₇

yes

—

C₂³×C₄

C₂²×D₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂×C₄²

Dic₁₃₄

yes

—

C₂×C₄²

D₂₆₈

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

—

C₄⋊C₈

—

134

—

#95

C₂×Dic₆₇

yes

—

C₂⁴

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×D₂₆₈

yes

—

C₂²×C₄

C₂×Dic₁₃₄

yes

—

C₂²×C₄

C₂³×D₆₇

yes

characteristic

M₁₆

—

268

—

C₂×C₈

—

134

—

C₄²

C₄⋊Dic₆₇

yes

—

C₄²

C₂×D₂₆₈

yes

—

C₄²

C₂×Dic₁₃₄

yes

—

C₂³

—

C₂³

C₂×C₄⋊Dic₆₇

yes

—

C₂³

C₂³×Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

C₂×C₄

#14524040

yes

—

100

C₂×C₄

#14524043

yes

—

101

C₂×C₄

C₂²×D₂₆₈

yes

—

102

C₂×C₄

C₂²×Dic₁₃₄

yes

—

103

C₈

—

268

—

104

C₂²

—

105

C₂²

#16863266

yes

—

106

C₂²

C₂²×C₄⋊Dic₆₇

yes

—

107

C₂²

#16864163

yes

—

108

C₄

—

109

C₄

#16863266

yes

—

110

C₄

C₂×#14524040

yes

characteristic

111

C₄

C₂×#14524043

yes

—

112

C₂

—

113

C₂

C₂×#16863266

yes

characteristic

114

C₂

#22568773

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₂×#376×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#376	D₆₇
hypocenter	C₆₇	#20792
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂³×C₄

Fitting

C₂×#376×C₆₇

C₂

prime radical

C₂⁴×C₆₇

C₂²×C₄

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₄ ◁ C₂×#376

N	H ≅ G/N	type
2. #22560363	C₂	semidirect
61. C₆₇	#20792	semidirect

H ≅ G/N

type

2. #22560363

C₂

semidirect

61. C₆₇

#20792

semidirect

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

subgroup	index
C₂×#16863420	2
#22560363	2
C₂×#376×C₆₇	2
#20792	67

subgroup

index

C₂×#16863420

#22560363

C₂×#376×C₆₇

#20792

Subgroups — #30856219

Conjugacy classes of subgroups

Complements

Maximal subgroups