Group #31814777 — subgroups

Subgroups — #31814777

Order 17792.

524 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 157 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×#17984830	C₂	1	1	yes	maximal, characteristic
2	#22580858	C₂	1	2	yes	maximal, complemented
3	#22580854	C₂	1	2	yes	maximal
4	#22580843	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
8	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
9	#17984830	—	2	4	no	—
10	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	8	yes	—
11	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
14	C₁₃₉⋊#95	C₂²	1	8	yes	—
15	#125×C₁₃₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂³×D₁₃₉	C₂³	1	2	yes	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	—	2	4	no	—
22	C₂×D₅₅₆	—	2	2	no	—
23	C₂×D₅₅₆	—	2	4	no	—
24	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
25	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
26	(C₂×D₁₃₉)⋊C₄	—	2	16	no	—
27	(C₂×D₁₃₉)⋊C₄	—	2	8	no	—
28	C₄×Dic₁₃₉	—	2	8	no	—
29	C₄×Dic₁₃₉	—	2	4	no	—
30	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₉	—	2	8	no	—
34	C₄⋊C₄×C₁₃₉	—	2	4	no	—
35	C₄²×C₁₃₉	D₄	1	4	yes	—
36	C₂²×D₁₃₉	—	2	2	no	—
37	C₂²×D₁₃₉	—	2	8	no	—
38	C₂²×D₁₃₉	—	2	4	no	—
39	Dic₂₇₈	—	4	4	no	—
40	D₅₅₆	—	4	4	no	—
41	C₂×Dic₁₃₉	—	2	4	no	—
42	C₂×Dic₁₃₉	—	2	4	no	—
43	C₂×Dic₁₃₉	—	2	2	no	—
44	C₂×Dic₁₃₉	—	2	4	no	—
45	C₂×Dic₁₃₉	#41	1	8	yes	—
46	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	—	2	4	no	—
48	C₂×C₄×C₁₃₉	—	2	4	no	—
49	C₂×C₄×C₁₃₉	—	2	8	no	—
50	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃₉	#41	1	4	yes	—
53	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₃₉	C₂×#41	1	4	yes	—
55	C₂²×C₁₃₉	C₂×#41	1	2	yes	—
56	C₂×D₁₃₉	—	4	2	no	—
57	C₂×D₁₃₉	—	4	8	no	—
58	C₂×D₁₃₉	—	4	4	no	—
59	C₄×C₁₃₉	—	2	4	no	—
60	C₄×C₁₃₉	—	4	4	no	—
61	C₄×C₁₃₉	#124	1	4	yes	—
62	Dic₁₃₉	—	2	8	no	—
63	Dic₁₃₉	—	4	2	no	—
64	C₂×C₁₃₉	C₂×#124	1	1	yes	characteristic
65	C₂×C₁₃₉	#529	1	4	yes	—
66	C₂×C₁₃₉	#532	1	2	yes	—
67	D₁₃₉	—	8	2	no	—
68	C₁₃₉	#2861	1	1	yes	characteristic, complemented, Sylow
69	#2861	—	139	1	no	maximal, Sylow
70	C₂×#125	D₁₃₉	1	1	yes	characteristic, complemented
71	C₂×#124	—	139	1	no	—
72	#391	—	139	2	no	—
73	#390	—	139	2	no	—
74	#388	—	139	8	no	—
75	C₄³	—	139	1	no	—
76	C₂²×Q₈	—	139	1	no	—
77	C₂²×D₄	—	139	1	no	—
78	#125	—	2	4	no	—
79	#124	—	278	4	no	—
80	C₂×C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
81	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
82	C₂×C₂²⋊C₄	—	139	4	no	—
83	C₂×C₂²⋊C₄	—	139	8	no	—
84	C₂×C₄²	—	139	4	no	—
85	C₂×C₄²	—	139	2	no	—
86	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
87	#95	—	139	8	no	—
88	C₂⁴	—	139	2	no	—
89	C₂×Q₈	—	278	2	no	—
90	C₂×Q₈	—	278	4	no	—
91	C₂×D₄	—	278	2	no	—
92	C₂×D₄	—	278	4	no	—
93	C₂²×C₄	—	139	2	no	—
94	C₂²×C₄	—	139	4	no	—
95	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
96	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₄⋊C₄	—	2	8	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₂²⋊C₄	—	278	16	no	—
101	C₂²⋊C₄	—	278	8	no	—
102	C₄²	—	278	8	no	—
103	C₄²	—	278	4	no	—
104	C₄²	C₂²⋊D₁₃₉	1	4	yes	—
105	C₂³	—	278	2	no	—
106	C₂³	—	278	8	no	—
107	C₂³	—	278	4	no	—
108	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
109	Q₈	—	556	4	no	—
110	D₄	—	556	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	278	4	no	—
115	C₂×C₄	—	278	4	no	—
116	C₂×C₄	—	278	2	no	—
117	C₂×C₄	—	278	4	no	—
118	C₂×C₄	—	139	8	no	—
119	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
121	C₂×C₄	#14555786	1	4	yes	—
122	C₂²	—	556	2	no	—
123	C₂²	—	556	8	no	—
124	C₂²	—	556	4	no	—
125	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
126	C₂²	#41×D₁₃₉	1	4	yes	—
127	C₂²	C₂×#14555786	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	4	4	no	—
130	C₄	—	278	8	no	—
131	C₄	—	556	2	no	—
132	C₄	#17984831	1	4	yes	—
133	C₂	—	1112	2	no	—
134	C₂	C₂×#17984831	1	1	yes	characteristic
135	C₂	#22581488	1	4	yes	—
136	C₂	#22581499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984830

C₂

yes

maximal, characteristic

#22580858

C₂

yes

maximal, complemented

#22580854

C₂

yes

maximal

#22580843

C₂

yes

maximal, complemented

C₄²×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂²×D₅₅₆

C₂²

yes

characteristic

#17984830

—

C₂×(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

#125×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

C₂³

yes

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

#41

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×#41

yes

—

C₂²×C₁₃₉

C₂×#41

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#124

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂×#124

yes

characteristic

C₂×C₁₃₉

#529

yes

—

C₂×C₁₃₉

#532

yes

—

D₁₃₉

—

C₁₃₉

#2861

yes

characteristic, complemented, Sylow

#2861

—

139

maximal, Sylow

C₂×#125

D₁₃₉

yes

characteristic, complemented

C₂×#124

—

139

—

#391

—

139

—

#390

—

139

—

#388

—

139

—

C₄³

—

139

—

C₂²×Q₈

—

139

—

C₂²×D₄

—

139

—

#125

—

#124

—

278

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#95

—

139

—

C₂⁴

—

139

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

278

—

101

C₂²⋊C₄

—

278

—

102

C₄²

—

278

—

103

C₄²

—

278

—

104

C₄²

C₂²⋊D₁₃₉

yes

—

105

C₂³

—

278

—

106

C₂³

—

278

—

107

C₂³

—

278

—

108

C₂³

C₂³×D₁₃₉

yes

characteristic

109

Q₈

—

556

—

110

D₄

—

556

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

278

—

115

C₂×C₄

—

278

—

116

C₂×C₄

—

278

—

117

C₂×C₄

—

278

—

118

C₂×C₄

—

139

—

119

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

121

C₂×C₄

#14555786

yes

—

122

C₂²

—

556

—

123

C₂²

—

556

—

124

C₂²

—

556

—

125

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

126

C₂²

#41×D₁₃₉

yes

—

127

C₂²

C₂×#14555786

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

278

—

131

C₄

—

556

—

132

C₄

#17984831

yes

—

133

C₂

—

1112

—

134

C₂

C₂×#17984831

yes

characteristic

135

C₂

#22581488

yes

—

136

C₂

#22581499

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	C₂×#125×C₁₃₉	C₂
prime radical	C₂²×D₅₅₆	C₂²
hypercenter	C₂×#125	D₁₃₉
hypocenter	C₁₃₉	#2861
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator, socle, cosocle

C₂³×C₁₃₉

C₂⁴

Fitting

C₂×#125×C₁₃₉

C₂

prime radical

C₂²×D₅₅₆

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #22580858	C₂	semidirect
4. #22580843	C₂	semidirect
5. C₄²×Dic₁₃₉	C₂	semidirect
6. C₂×#125×C₁₃₉	C₂	semidirect
68. C₁₃₉	#2861	semidirect
70. C₂×#125	D₁₃₉	semidirect

H ≅ G/N

type

2. #22580858

C₂

semidirect

4. #22580843

C₂

semidirect

5. C₄²×Dic₁₃₉

C₂

semidirect

6. C₂×#125×C₁₃₉

C₂

semidirect

68. C₁₃₉

#2861

semidirect

70. C₂×#125

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984830	2
#22580858	2
#22580854	2
#22580843	2
C₄²×Dic₁₃₉	2
C₂×#125×C₁₃₉	2
#2861	139

subgroup

index

C₄²×Dic₁₃₉

C₂×#125×C₁₃₉

#2861

139

Subgroups — #31814777

Conjugacy classes of subgroups

Complements

Maximal subgroups