Group #31814656 — subgroups

Subgroups — #31814656

Order 17792.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#22580866	C₂	1	6	yes	maximal, complemented
2	#22580865	C₂	1	3	yes	maximal, complemented
3	#22580864	C₂	1	2	yes	maximal, complemented
4	#22580808	C₂	1	3	yes	maximal
5	#393×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₁₃₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₇₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₁₃₉	C₂²	1	3	yes	—
9	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	6	yes	—
11	C₁₃₉⋊#95	C₂²	1	6	yes	complemented
12	C₁₃₉⋊#95	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	3	yes	—
15	#95×C₁₃₉	C₂²	1	3	yes	complemented
16	C₂³×D₁₃₉	—	4	3	no	—
17	C₂³×D₁₃₉	—	4	3	no	—
18	C₂³×D₁₃₉	Q₈	1	4	yes	complemented
19	C₂³×D₁₃₉	C₂³	1	2	yes	—
20	C₂×Dic₂₇₈	—	4	3	no	—
21	C₂×Dic₂₇₈	—	4	6	no	—
22	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
23	C₂²⋊Dic₁₃₉	—	4	3	no	—
24	(C₂×D₁₃₉)⋊C₄	—	4	6	no	—
25	(C₂×D₁₃₉)⋊C₄	—	4	6	no	—
26	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₁₃₉	—	4	3	no	—
28	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
29	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
30	C₂²⋊C₄×C₁₃₉	—	4	3	no	—
31	C₂²×D₁₃₉	—	2	6	no	—
32	C₂²×D₁₃₉	—	2	6	no	—
33	C₂²×D₁₃₉	—	4	12	no	—
34	C₂²×D₁₃₉	—	4	12	no	—
35	C₂²×D₁₃₉	—	4	4	no	—
36	C₂²×D₁₃₉	C₂×Q₈	1	4	yes	—
37	Dic₂₇₈	—	4	12	no	—
38	C₂×Dic₁₃₉	—	2	6	no	—
39	C₂×Dic₁₃₉	—	2	6	no	—
40	C₂×Dic₁₃₉	—	2	6	no	—
41	C₂³×C₁₃₉	—	2	3	no	—
42	C₂³×C₁₃₉	—	2	3	no	—
43	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
44	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₃₉	—	4	4	no	—
46	C₂×C₄×C₁₃₉	—	2	3	no	—
47	C₂×C₄×C₁₃₉	—	2	3	no	—
48	C₂×C₄×C₁₃₉	—	2	6	no	—
49	C₂×C₄×C₁₃₉	—	2	6	no	—
50	C₂²×C₁₃₉	—	2	6	no	—
51	C₂²×C₁₃₉	—	4	3	no	—
52	C₂²×C₁₃₉	—	4	1	no	—
53	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₁₃₉	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₁₃₉	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₁₃₉	—	2	12	no	—
57	C₂×D₁₃₉	—	4	16	no	—
58	C₂×D₁₃₉	—	4	6	no	—
59	C₂×D₁₃₉	—	4	2	no	—
60	C₄×C₁₃₉	—	4	3	no	—
61	C₄×C₁₃₉	—	4	3	no	—
62	Dic₁₃₉	—	4	6	no	—
63	C₂×C₁₃₉	—	4	2	no	—
64	C₂×C₁₃₉	#543	1	3	yes	—
65	C₂×C₁₃₉	H(3,𝔽₄)	1	3	yes	—
66	C₂×C₁₃₉	H(3,𝔽₄)	1	1	yes	characteristic
67	D₁₃₉	—	4	4	no	—
68	C₁₃₉	#2840	1	1	yes	characteristic, complemented, Sylow
69	#2840	—	139	1	no	maximal, Sylow
70	#393	—	139	2	no	—
71	#393	—	139	3	no	—
72	#393	—	139	6	no	—
73	#393	D₁₃₉	1	1	yes	characteristic, complemented
74	#379	—	139	3	no	—
75	C₂⁵	—	139	1	no	—
76	C₂²×Q₈	—	139	3	no	—
77	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	139	3	no	—
79	C₂×C₂²⋊C₄	—	139	6	no	—
80	C₂×C₂²⋊C₄	—	139	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	3	yes	—
82	#95	—	139	6	no	—
83	#95	—	139	3	no	—
84	#95	C₂×D₁₃₉	1	3	yes	complemented
85	C₂⁴	—	556	3	no	—
86	C₂⁴	—	556	3	no	—
87	C₂⁴	—	139	2	no	—
88	C₂⁴	—	139	4	no	—
89	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	556	3	no	—
92	C₂×Q₈	—	556	6	no	—
93	C₂²×C₄	—	139	6	no	—
94	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
95	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	556	6	no	—
98	C₂²⋊C₄	—	556	6	no	—
99	C₂²⋊C₄	—	556	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	278	6	no	—
103	C₂³	—	278	6	no	—
104	C₂³	—	556	12	no	—
105	C₂³	—	556	12	no	—
106	C₂³	—	556	4	no	—
107	C₂³	—	139	4	no	—
108	C₂³	Q₈×D₁₃₉	1	2	yes	—
109	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	556	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	278	6	no	—
117	C₂×C₄	—	278	6	no	—
118	C₂×C₄	—	278	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	278	12	no	—
123	C₂²	—	556	16	no	—
124	C₂²	—	556	6	no	—
125	C₂²	—	556	2	no	—
126	C₂²	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
127	C₂²	#17984887	1	3	yes	—
128	C₂²	#17984888	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	556	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	556	4	no	—
134	C₂	#22581578	1	3	yes	—
135	C₂	#22581695	1	3	yes	—
136	C₂	#22581696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580866

C₂

yes

maximal, complemented

#22580865

C₂

yes

maximal, complemented

#22580864

C₂

yes

maximal, complemented

#22580808

C₂

yes

maximal

#393×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₃₉

C₂²

yes

characteristic

C₂²×Dic₂₇₈

C₂²

yes

complemented

C₂×C₂²⋊Dic₁₃₉

C₂²

yes

—

C₂×(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

complemented

C₁₃₉⋊#95

C₂²

yes

complemented

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₁₃₉

C₂²

yes

—

#95×C₁₃₉

C₂²

yes

complemented

C₂³×D₁₃₉

—

C₂³×D₁₃₉

—

C₂³×D₁₃₉

Q₈

yes

complemented

C₂³×D₁₃₉

C₂³

yes

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₂⁴×C₁₃₉

C₂³

yes

characteristic

C₂×Q₈×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²⋊C₄×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂×Q₈

yes

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×Q₈

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₉

H(5,𝔽₂)

yes

—

C₂²×C₁₃₉

H(5,𝔽₂)

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#543

yes

—

C₂×C₁₃₉

H(3,𝔽₄)

yes

—

C₂×C₁₃₉

H(3,𝔽₄)

yes

characteristic

D₁₃₉

—

C₁₃₉

#2840

yes

characteristic, complemented, Sylow

#2840

—

139

maximal, Sylow

#393

—

139

—

#393

—

139

—

#393

—

139

—

#393

D₁₃₉

yes

characteristic, complemented

#379

—

139

—

C₂⁵

—

139

—

C₂²×Q₈

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

C₂×D₁₃₉

yes

—

#95

—

139

—

#95

—

139

—

#95

C₂×D₁₃₉

yes

complemented

C₂⁴

—

556

—

C₂⁴

—

556

—

C₂⁴

—

139

—

C₂⁴

—

139

—

C₂⁴

C₂²×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

556

—

C₂×Q₈

—

556

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

556

—

C₂²⋊C₄

—

556

—

C₂²⋊C₄

—

556

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

278

—

103

C₂³

—

278

—

104

C₂³

—

556

—

105

C₂³

—

556

—

106

C₂³

—

556

—

107

C₂³

—

139

—

108

C₂³

Q₈×D₁₃₉

yes

—

109

C₂³

C₂³×D₁₃₉

yes

characteristic

110

Q₈

—

111

Q₈

—

556

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

278

—

117

C₂×C₄

—

278

—

118

C₂×C₄

—

278

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

278

—

123

C₂²

—

556

—

124

C₂²

—

556

—

125

C₂²

—

556

—

126

C₂²

C₂×Q₈×D₁₃₉

yes

characteristic

127

C₂²

#17984887

yes

—

128

C₂²

#17984888

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

556

—

132

C₂

—

133

C₂

—

556

—

134

C₂

#22581578

yes

—

135

C₂

#22581695

yes

—

136

C₂

#22581696

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	#393×C₁₃₉	C₂
prime radical	C₂⁴×D₁₃₉	C₂²
hypercenter	#393	D₁₃₉
hypocenter	C₁₃₉	#2840
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator, socle, cosocle

C₂³×C₁₃₉

C₂⁴

Fitting

#393×C₁₃₉

C₂

prime radical

C₂⁴×D₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #22580866	C₂	semidirect
2. #22580865	C₂	semidirect
3. #22580864	C₂	semidirect
5. #393×C₁₃₉	C₂	semidirect
7. C₂²×Dic₂₇₈	C₂²	semidirect
11. C₁₃₉⋊#95	C₂²	semidirect
12. C₁₃₉⋊#95	C₂²	semidirect
13. C₂²×Q₈×C₁₃₉	C₂²	semidirect
15. #95×C₁₃₉	C₂²	semidirect
18. C₂³×D₁₃₉	Q₈	semidirect
68. C₁₃₉	#2840	semidirect
73. #393	D₁₃₉	semidirect
77. C₂²×Q₈	C₂×D₁₃₉	semidirect
84. #95	C₂×D₁₃₉	semidirect

H ≅ G/N

type

1. #22580866

C₂

semidirect

2. #22580865

C₂

semidirect

3. #22580864

C₂

semidirect

5. #393×C₁₃₉

C₂

semidirect

7. C₂²×Dic₂₇₈

C₂²

semidirect

11. C₁₃₉⋊#95

C₂²

semidirect

12. C₁₃₉⋊#95

C₂²

semidirect

13. C₂²×Q₈×C₁₃₉

C₂²

semidirect

15. #95×C₁₃₉

C₂²

semidirect

18. C₂³×D₁₃₉

Q₈

semidirect

68. C₁₃₉

#2840

semidirect

73. #393

D₁₃₉

semidirect

77. C₂²×Q₈

C₂×D₁₃₉

semidirect

84. #95

C₂×D₁₃₉

semidirect

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

subgroup	index
#22580866	2
#22580865	2
#22580864	2
#22580808	2
#393×C₁₃₉	2
#2840	139

subgroup

index

#393×C₁₃₉

139

Subgroups — #31814656

Conjugacy classes of subgroups

Complements

Maximal subgroups