Group #31819979 — subgroups

Subgroups — C₂×#22581683 #31819979

Order 17792.

1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 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	#22581683	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984841	C₂	1	4	yes	maximal
3	C₂×#17984840	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₂₇₈	C₂	1	2	yes	maximal
5	C₂×#17984807	C₂	1	4	yes	maximal
6	C₂×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₇₈	C₂²	1	4	yes	—
8	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
9	#17984841	C₂²	1	32	yes	—
10	#17984840	C₂²	1	32	yes	—
11	C₄×Dic₂₇₈	C₂²	1	16	yes	—
12	#17984807	C₂²	1	32	yes	—
13	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
14	C₂×Dic₁₃₉⋊C₄	C₂²	1	8	yes	—
15	C₂×Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
18	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₃₉	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₇₈	—	4	8	no	—
23	C₂×Dic₂₇₈	Q₈	1	16	yes	complemented
24	C₂×Dic₂₇₈	C₂³	1	16	yes	—
25	C₂×Dic₂₇₈	C₂³	1	8	yes	—
26	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
27	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
28	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
29	Dic₁₃₉⋊C₄	C₂³	1	32	yes	—
30	Dic₁₃₉⋊C₄	C₂³	1	16	yes	—
31	C₄×Dic₁₃₉	C₂³	1	16	yes	—
32	C₄×Dic₁₃₉	C₂³	1	16	yes	—
33	C₂×Q₈×C₁₃₉	—	4	4	no	—
34	C₂×Q₈×C₁₃₉	C₂³	1	8	yes	—
35	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₃₉	C₂³	1	16	yes	—
39	C₄²×C₁₃₉	C₂³	1	4	yes	—
40	Dic₂₇₈	—	4	16	no	—
41	Dic₂₇₈	C₂×Q₈	1	32	yes	complemented
42	C₂×Dic₁₃₉	—	2	8	no	—
43	C₂×Dic₁₃₉	C₂×Q₈	1	16	yes	—
44	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
45	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
46	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
47	Q₈×C₁₃₉	—	4	8	no	—
48	C₂×C₄×C₁₃₉	—	2	2	no	—
49	C₂×C₄×C₁₃₉	—	2	8	no	—
50	C₂×C₄×C₁₃₉	C₂×Q₈	1	8	yes	—
51	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
52	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
53	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
54	C₂²×C₁₃₉	C₂²×Q₈	1	4	yes	—
55	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	—	2	8	no	—
59	C₄×C₁₃₉	C₂²×Q₈	1	8	yes	—
60	Dic₁₃₉	—	2	8	no	—
61	Dic₁₃₉	C₂²×Q₈	1	16	yes	—
62	C₂×C₁₃₉	Q₈²	1	4	yes	complemented
63	C₂×C₁₃₉	C₂³×Q₈	1	2	yes	—
64	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
65	C₁₃₉	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈²	—	139	1	no	maximal, Sylow
67	Q₈²	—	139	16	no	—
68	C₂×C₄⋊Q₈	—	139	4	no	—
69	C₂×C₄⋊Q₈	—	139	4	no	—
70	C₂×C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic
71	C₂×C₄×Q₈	—	139	4	no	—
72	C₂×C₄×Q₈	—	139	2	no	—
73	C₂²×Q₈	—	139	4	no	—
74	C₂²×Q₈	—	139	2	no	—
75	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	—
76	C₄⋊Q₈	—	139	32	no	—
77	C₄⋊Q₈	—	139	32	no	—
78	C₄⋊Q₈	C₂×D₁₃₉	1	8	yes	—
79	C₄×Q₈	—	139	32	no	—
80	C₄×Q₈	—	139	16	no	—
81	C₂×C₄⋊C₄	—	139	8	no	—
82	C₂×C₄⋊C₄	—	139	4	no	—
83	C₂×C₄⋊C₄	—	139	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
85	C₂×C₄²	—	139	4	no	—
86	C₂×C₄²	—	139	4	no	—
87	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	556	8	no	—
90	C₂×Q₈	—	139	16	no	—
91	C₂×Q₈	—	139	16	no	—
92	C₂×Q₈	—	139	8	no	—
93	C₂×Q₈	C₂²×D₁₃₉	1	8	yes	—
94	C₂²×C₄	—	139	4	no	—
95	C₂²×C₄	—	139	4	no	—
96	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
97	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
99	C₄⋊C₄	—	139	32	no	—
100	C₄⋊C₄	—	139	16	no	—
101	C₄⋊C₄	—	139	8	no	—
102	C₄⋊C₄	C₂²×D₁₃₉	1	16	yes	—
103	C₄²	—	139	16	no	—
104	C₄²	—	139	16	no	—
105	C₄²	C₂²×D₁₃₉	1	4	yes	—
106	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	556	16	no	—
109	Q₈	—	139	32	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	278	8	no	—
113	C₂×C₄	—	139	8	no	—
114	C₂×C₄	—	139	16	no	—
115	C₂×C₄	—	139	8	no	—
116	C₂×C₄	Q₈×D₁₃₉	1	8	yes	—
117	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
118	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
119	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
120	C₂²	C₂×Q₈×D₁₃₉	1	4	yes	—
121	C₂²	#17984887	1	2	yes	—
122	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	8	no	—
125	C₄	—	278	8	no	—
126	C₄	—	139	16	no	—
127	C₄	C₂×Q₈×D₁₃₉	1	8	yes	—
128	C₂	#22581683	1	4	yes	direct product factor
129	C₂	C₂²×Q₈×D₁₃₉	1	2	yes	—
130	C₂	C₂×#17984887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581683

C₂

yes

maximal, direct product factor

C₂×#17984841

C₂

yes

maximal

C₂×#17984840

C₂

yes

maximal

C₂×C₄×Dic₂₇₈

C₂

yes

maximal

C₂×#17984807

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂²×Dic₂₇₈

C₂²

yes

—

#17984841

C₂²

yes

—

#17984840

C₂²

yes

—

C₄×Dic₂₇₈

C₂²

yes

—

#17984807

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

Q₈

yes

complemented

C₂×Dic₂₇₈

C₂³

yes

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

C₂³

yes

—

Dic₁₃₉⋊C₄

C₂³

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂³

yes

—

Dic₂₇₈

—

Dic₂₇₈

C₂×Q₈

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×Q₈

yes

—

C₂×Dic₁₃₉

C₂⁴

yes

—

C₂×Dic₁₃₉

C₂⁴

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

—

C₂²×C₁₃₉

C₂²×Q₈

yes

—

C₂²×C₁₃₉

H(5,𝔽₂)

yes

—

C₂²×C₁₃₉

C₂⁵

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

C₂²×Q₈

yes

—

C₂×C₁₃₉

Q₈²

yes

complemented

C₂×C₁₃₉

C₂³×Q₈

yes

—

C₂×C₁₃₉

C₂×H(5,𝔽₂)

yes

characteristic

C₁₃₉

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂×Q₈²

—

139

maximal, Sylow

Q₈²

—

139

—

C₂×C₄⋊Q₈

—

139

—

C₂×C₄⋊Q₈

—

139

—

C₂×C₄⋊Q₈

D₁₃₉

yes

characteristic

C₂×C₄×Q₈

—

139

—

C₂×C₄×Q₈

—

139

—

C₂²×Q₈

—

139

—

C₂²×Q₈

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

C₄×Q₈

—

139

—

C₄×Q₈

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

556

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂²×D₁₃₉

yes

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

139

—

100

C₄⋊C₄

—

139

—

101

C₄⋊C₄

—

139

—

102

C₄⋊C₄

C₂²×D₁₃₉

yes

—

103

C₄²

—

139

—

104

C₄²

—

139

—

105

C₄²

C₂²×D₁₃₉

yes

—

106

C₂³

C₂³×D₁₃₉

yes

characteristic

107

Q₈

—

108

Q₈

—

556

—

109

Q₈

—

139

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

278

—

113

C₂×C₄

—

139

—

114

C₂×C₄

—

139

—

115

C₂×C₄

—

139

—

116

C₂×C₄

Q₈×D₁₃₉

yes

—

117

C₂×C₄

C₂³×D₁₃₉

yes

—

118

C₂×C₄

C₂³×D₁₃₉

yes

—

119

C₂×C₄

C₂³×D₁₃₉

yes

—

120

C₂²

C₂×Q₈×D₁₃₉

yes

—

121

C₂²

#17984887

yes

—

122

C₂²

C₂⁴×D₁₃₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

278

—

126

C₄

—

139

—

127

C₄

C₂×Q₈×D₁₃₉

yes

—

128

C₂

#22581683

yes

direct product factor

129

C₂

C₂²×Q₈×D₁₃₉

yes

—

130

C₂

C₂×#17984887

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×C₄⋊Q₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	C₂×Q₈²
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₃₉

commutator, cosocle

C₂²×C₁₃₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₃₉

Fitting

C₂×C₄⋊Q₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₃₉

hypocenter

C₁₃₉

C₂×Q₈²

LCS

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #22581683	C₂	direct (N × H)
23. C₂×Dic₂₇₈	Q₈	semidirect
41. Dic₂₇₈	C₂×Q₈	semidirect
62. C₂×C₁₃₉	Q₈²	semidirect
65. C₁₃₉	C₂×Q₈²	semidirect
128. C₂	#22581683	direct (N × H)

H ≅ G/N

type

1. #22581683

C₂

direct (N × H)

23. C₂×Dic₂₇₈

Q₈

semidirect

41. Dic₂₇₈

C₂×Q₈

semidirect

62. C₂×C₁₃₉

Q₈²

semidirect

65. C₁₃₉

C₂×Q₈²

semidirect

128. C₂

#22581683

direct (N × H)

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

subgroup	index
#22581683	2
C₂×#17984841	2
C₂×#17984840	2
C₂×C₄×Dic₂₇₈	2
C₂×#17984807	2
C₂×C₄⋊Q₈×C₁₃₉	2
C₂×Q₈²	139

subgroup

index

C₂×C₄×Dic₂₇₈

C₂×C₄⋊Q₈×C₁₃₉

C₂×Q₈²

139

Subgroups — C₂×#22581683 #31819979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581683 #31819979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581683 #31819979