Group #30662824 — subgroups

Subgroups — #2641×C₁₃₁ #30662824

Order 16768.

584 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 200 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×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#394×C₁₃₁	C₂	1	8	yes	maximal, complemented
4	#390×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#376×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×D₄×C₁₃₁	C₂²	1	4	yes	—
8	#127×C₁₃₁	—	2	4	no	—
9	#125×C₁₃₁	—	2	4	no	—
10	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	16	yes	—
13	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	#95×C₁₃₁	C₂²	1	4	yes	—
16	C₂⁴×C₁₃₁	C₂³	1	4	yes	—
17	C₂×D₄×C₁₃₁	—	2	8	no	—
18	C₂×D₄×C₁₃₁	—	2	4	no	—
19	C₂×D₄×C₁₃₁	—	2	8	no	—
20	C₂×D₄×C₁₃₁	—	4	8	no	—
21	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
22	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
24	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₃₁	—	2	8	no	—
26	C₄⋊C₄×C₁₃₁	—	2	4	no	—
27	C₂²⋊C₄×C₁₃₁	—	4	16	no	—
28	C₄²×C₁₃₁	—	2	4	no	—
29	C₄²×C₁₃₁	D₄	1	4	yes	—
30	C₂³×C₁₃₁	—	2	4	no	—
31	C₂³×C₁₃₁	—	2	8	no	—
32	C₂³×C₁₃₁	—	2	16	no	—
33	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
34	D₄×C₁₃₁	—	4	16	no	—
35	D₄×C₁₃₁	—	4	8	no	—
36	C₂×C₄×C₁₃₁	—	2	4	no	—
37	C₂×C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄×C₁₃₁	—	2	8	no	—
40	C₂×C₄×C₁₃₁	—	2	8	no	—
41	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₁	#41	1	4	yes	—
44	C₂²×C₁₃₁	—	4	4	no	—
45	C₂²×C₁₃₁	—	4	8	no	—
46	C₂²×C₁₃₁	—	4	16	no	—
47	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
48	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
49	C₂²×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
50	C₄×C₁₃₁	—	2	4	no	—
51	C₄×C₁₃₁	—	4	4	no	—
52	C₄×C₁₃₁	—	4	4	no	—
53	C₄×C₁₃₁	#124	1	4	yes	—
54	C₂×C₁₃₁	—	8	4	no	—
55	C₂×C₁₃₁	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₃₁	#535	1	2	yes	—
57	C₂×C₁₃₁	#540	1	4	yes	—
58	C₁₃₁	#2641	1	1	yes	characteristic, direct product factor, Sylow
59	#2641	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×#127	C₂×C₁₃₁	1	1	yes	characteristic
61	C₂×#125	C₂×C₁₃₁	1	1	yes	characteristic, complemented
62	#394	C₂×C₁₃₁	1	8	yes	complemented
63	#390	C₂×C₁₃₁	1	4	yes	complemented
64	#376	C₂×C₁₃₁	1	1	yes	characteristic, complemented
65	C₂²×D₄	C₂²×C₁₃₁	1	2	yes	—
66	C₂²×D₄	C₂²×C₁₃₁	1	4	yes	—
67	#127	—	2	4	no	—
68	#125	—	2	4	no	—
69	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
70	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
71	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	16	yes	—
72	C₂×C₄²	C₂²×C₁₃₁	1	2	yes	—
73	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
74	#95	C₂²×C₁₃₁	1	4	yes	—
75	C₂⁴	C₂³×C₁₃₁	1	4	yes	—
76	C₂×D₄	—	2	8	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	2	8	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
81	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
82	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
83	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	4	16	no	—
87	C₄²	—	2	4	no	—
88	C₄²	D₄×C₁₃₁	1	4	yes	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	8	no	—
91	C₂³	—	2	16	no	—
92	C₂³	C₂⁴×C₁₃₁	1	1	yes	characteristic
93	D₄	—	4	16	no	—
94	D₄	—	4	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
101	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
102	C₂×C₄	#41×C₁₃₁	1	4	yes	—
103	C₂²	—	4	4	no	—
104	C₂²	—	4	8	no	—
105	C₂²	—	4	16	no	—
106	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
107	C₂²	C₂×#41×C₁₃₁	1	2	yes	—
108	C₂²	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	4	no	—
112	C₄	#124×C₁₃₁	1	4	yes	—
113	C₂	—	8	4	no	—
114	C₂	C₂×#124×C₁₃₁	1	1	yes	characteristic
115	C₂	#535×C₁₃₁	1	2	yes	—
116	C₂	#540×C₁₃₁	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#125×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#394×C₁₃₁

C₂

yes

maximal, complemented

#390×C₁₃₁

C₂

yes

maximal, complemented

#376×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃₁

C₂²

yes

—

C₂²×D₄×C₁₃₁

C₂²

yes

—

#127×C₁₃₁

—

#125×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

#95×C₁₃₁

C₂²

yes

—

C₂⁴×C₁₃₁

C₂³

yes

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂⁴

yes

characteristic

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₂²×C₁₃₁

C₂×#41

yes

—

C₂²×C₁₃₁

H(5,𝔽₂)

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#124

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂×#124

yes

characteristic

C₂×C₁₃₁

#535

yes

—

C₂×C₁₃₁

#540

yes

—

C₁₃₁

#2641

yes

characteristic, direct product factor, Sylow

#2641

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#127

C₂×C₁₃₁

yes

characteristic

C₂×#125

C₂×C₁₃₁

yes

characteristic, complemented

#394

C₂×C₁₃₁

yes

complemented

#390

C₂×C₁₃₁

yes

complemented

#376

C₂×C₁₃₁

yes

characteristic, complemented

C₂²×D₄

C₂²×C₁₃₁

yes

—

C₂²×D₄

C₂²×C₁₃₁

yes

—

#127

—

#125

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

characteristic

#95

C₂²×C₁₃₁

yes

—

C₂⁴

C₂³×C₁₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₄×C₁₃₁

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₁₃₁

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×D₄×C₁₃₁

yes

—

101

C₂×C₄

C₂×D₄×C₁₃₁

yes

—

102

C₂×C₄

#41×C₁₃₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₄×C₁₃₁

yes

characteristic

107

C₂²

C₂×#41×C₁₃₁

yes

—

108

C₂²

H(5,𝔽₂)×C₁₃₁

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#124×C₁₃₁

yes

—

113

C₂

—

114

C₂

C₂×#124×C₁₃₁

yes

characteristic

115

C₂

#535×C₁₃₁

yes

—

116

C₂

#540×C₁₃₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₃₁

prime radical

C₂×#127×C₁₃₁

C₂

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ G

N	H ≅ G/N	type
2. C₂×#125×C₁₃₁	C₂	semidirect
3. #394×C₁₃₁	C₂	semidirect
4. #390×C₁₃₁	C₂	semidirect
5. #376×C₁₃₁	C₂	semidirect
58. C₁₃₁	#2641	direct (N × H)
59. #2641	C₁₃₁	direct (N × H)
61. C₂×#125	C₂×C₁₃₁	semidirect
62. #394	C₂×C₁₃₁	semidirect
63. #390	C₂×C₁₃₁	semidirect
64. #376	C₂×C₁₃₁	semidirect

H ≅ G/N

type

2. C₂×#125×C₁₃₁

C₂

semidirect

3. #394×C₁₃₁

C₂

semidirect

4. #390×C₁₃₁

C₂

semidirect

5. #376×C₁₃₁

C₂

semidirect

58. C₁₃₁

#2641

direct (N × H)

59. #2641

C₁₃₁

direct (N × H)

61. C₂×#125

C₂×C₁₃₁

semidirect

62. #394

C₂×C₁₃₁

semidirect

63. #390

C₂×C₁₃₁

semidirect

64. #376

C₂×C₁₃₁

semidirect

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

subgroup	index
C₂×#127×C₁₃₁	2
C₂×#125×C₁₃₁	2
#394×C₁₃₁	2
#390×C₁₃₁	2
#376×C₁₃₁	2
#2641	131

subgroup

index

C₂×#127×C₁₃₁

C₂×#125×C₁₃₁

#394×C₁₃₁

#390×C₁₃₁

#376×C₁₃₁

#2641

131

Subgroups — #2641×C₁₃₁ #30662824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2641×C131 #30662824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2641×C₁₃₁ #30662824