Group #31814762 — subgroups

Subgroups — #31814762

Order 17792.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#17984839	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984831	C₂	1	4	yes	maximal, complemented
3	#22580854	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₅₆	C₂²	1	4	yes	—
7	C₂²×D₅₅₆	C₂²	1	2	yes	—
8	#17984839	—	2	4	no	—
9	#17984831	—	2	16	no	—
10	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
16	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂³×D₁₃₉	C₂³	1	4	yes	—
18	C₂×D₅₅₆	—	2	8	no	—
19	C₂×D₅₅₆	—	2	8	no	—
20	C₂×D₅₅₆	—	2	4	no	—
21	C₂×D₅₅₆	—	2	16	no	—
22	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
23	(C₂×D₁₃₉)⋊C₄	—	2	32	no	—
24	C₄×Dic₁₃₉	—	2	8	no	—
25	C₄×Dic₁₃₉	D₄	1	8	yes	complemented
26	C₂×Q₈×C₁₃₉	—	2	8	no	—
27	C₂×Q₈×C₁₃₉	—	2	4	no	—
28	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
31	C₄⋊C₄×C₁₃₉	—	2	8	no	—
32	C₄²×C₁₃₉	D₄	1	4	yes	—
33	C₂²×D₁₃₉	—	2	8	no	—
34	C₂²×D₁₃₉	—	2	4	no	—
35	C₂²×D₁₃₉	—	2	16	no	—
36	D₅₅₆	—	4	16	no	—
37	D₅₅₆	—	4	8	no	—
38	C₂×Dic₁₃₉	—	2	8	no	—
39	C₂×Dic₁₃₉	C₂×D₄	1	8	yes	—
40	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
41	Q₈×C₁₃₉	—	4	8	no	—
42	C₂×C₄×C₁₃₉	—	2	4	no	—
43	C₂×C₄×C₁₃₉	—	2	8	no	—
44	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₃₉	#41	1	8	yes	—
48	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
49	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₂×#41	1	4	yes	—
51	C₂×D₁₃₉	—	4	8	no	—
52	C₂×D₁₃₉	—	4	4	no	—
53	C₂×D₁₃₉	—	4	16	no	—
54	C₄×C₁₃₉	—	4	4	no	—
55	C₄×C₁₃₉	#124	1	8	yes	complemented
56	C₄×C₁₃₉	#127	1	4	yes	—
57	Dic₁₃₉	—	2	8	no	—
58	C₂×C₁₃₉	C₂×#124	1	2	yes	—
59	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
60	C₂×C₁₃₉	#532	1	4	yes	—
61	D₁₃₉	—	8	4	no	—
62	C₁₃₉	#2858	1	1	yes	characteristic, complemented, Sylow
63	#2858	—	139	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic, complemented
65	C₂×#127	—	139	1	no	—
66	C₂×#124	—	139	4	no	—
67	#390	—	139	8	no	—
68	C₄³	—	139	1	no	—
69	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	—
70	C₂²×D₄	—	139	2	no	—
71	C₂²×D₄	—	139	4	no	—
72	C₄⋊Q₈	—	2	4	no	—
73	#127	—	278	4	no	—
74	#124	—	278	16	no	—
75	C₂×C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
76	C₂×C₂²⋊C₄	—	139	16	no	—
77	C₂×C₄²	—	139	4	no	—
78	C₂×C₄²	—	139	2	no	—
79	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
80	C₂⁴	—	139	4	no	—
81	C₂×Q₈	—	2	8	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×D₄	—	278	8	no	—
84	C₂×D₄	—	278	8	no	—
85	C₂×D₄	—	278	4	no	—
86	C₂×D₄	—	278	16	no	—
87	C₂²×C₄	—	139	4	no	—
88	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
91	C₄⋊C₄	—	2	8	no	—
92	C₂²⋊C₄	—	278	32	no	—
93	C₄²	—	278	8	no	—
94	C₄²	—	139	8	no	—
95	C₄²	C₂²⋊D₁₃₉	1	4	yes	—
96	C₂³	—	278	8	no	—
97	C₂³	—	278	4	no	—
98	C₂³	—	278	16	no	—
99	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	556	16	no	—
102	D₄	—	556	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	278	8	no	—
106	C₂×C₄	—	139	8	no	—
107	C₂×C₄	D₄×D₁₃₉	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
110	C₂×C₄	#14555786	1	8	yes	—
111	C₂²	—	556	16	no	—
112	C₂²	—	556	8	no	—
113	C₂²	—	556	4	no	—
114	C₂²	C₂×D₄×D₁₃₉	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
116	C₂²	C₂×#14555786	1	4	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	278	8	no	—
119	C₄	#17984831	1	8	yes	complemented
120	C₄	#17984838	1	4	yes	—
121	C₂	—	1112	4	no	—
122	C₂	C₂×#17984831	1	2	yes	—
123	C₂	C₂×#17984838	1	1	yes	characteristic
124	C₂	#22581504	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984839

C₂

yes

maximal, characteristic

C₂×#17984831

C₂

yes

maximal, complemented

#22580854

C₂

yes

maximal, complemented

C₄²×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₅₆

C₂²

yes

—

C₂²×D₅₅₆

C₂²

yes

—

#17984839

—

#17984831

—

C₂×(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

C₂³

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂²×Dic₁₃₉

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

complemented

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×#41

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#124

yes

complemented

C₄×C₁₃₉

#127

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂×#124

yes

—

C₂×C₁₃₉

C₂×#127

yes

characteristic

C₂×C₁₃₉

#532

yes

—

D₁₃₉

—

C₁₃₉

#2858

yes

characteristic, complemented, Sylow

#2858

—

139

maximal, Sylow

C₂×C₄⋊Q₈

D₁₃₉

yes

characteristic, complemented

C₂×#127

—

139

—

C₂×#124

—

139

—

#390

—

139

—

C₄³

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

—

C₂²×D₄

—

139

—

C₂²×D₄

—

139

—

C₄⋊Q₈

—

#127

—

278

—

#124

—

278

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₂⁴

—

139

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₄²

C₂²⋊D₁₃₉

yes

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

C₂³×D₁₃₉

yes

characteristic

100

Q₈

—

101

D₄

—

556

—

102

D₄

—

556

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

139

—

107

C₂×C₄

D₄×D₁₃₉

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

110

C₂×C₄

#14555786

yes

—

111

C₂²

—

556

—

112

C₂²

—

556

—

113

C₂²

—

556

—

114

C₂²

C₂×D₄×D₁₃₉

yes

—

115

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

116

C₂²

C₂×#14555786

yes

—

117

C₄

—

118

C₄

—

278

—

119

C₄

#17984831

yes

complemented

120

C₄

#17984838

yes

—

121

C₂

—

1112

—

122

C₂

C₂×#17984831

yes

—

123

C₂

C₂×#17984838

yes

characteristic

124

C₂

#22581504

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₁₃₉	C₂
prime radical	C₂×#17984839	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	#2858
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator, socle, cosocle

C₂³×C₁₃₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₁₃₉

C₂

prime radical

C₂×#17984839

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#17984831	C₂	semidirect
3. #22580854	C₂	semidirect
4. C₄²×Dic₁₃₉	C₂	semidirect
5. C₂×C₄⋊Q₈×C₁₃₉	C₂	semidirect
25. C₄×Dic₁₃₉	D₄	semidirect
55. C₄×C₁₃₉	#124	semidirect
62. C₁₃₉	#2858	semidirect
64. C₂×C₄⋊Q₈	D₁₃₉	semidirect
119. C₄	#17984831	semidirect

H ≅ G/N

type

2. C₂×#17984831

C₂

semidirect

3. #22580854

C₂

semidirect

4. C₄²×Dic₁₃₉

C₂

semidirect

5. C₂×C₄⋊Q₈×C₁₃₉

C₂

semidirect

25. C₄×Dic₁₃₉

D₄

semidirect

55. C₄×C₁₃₉

#124

semidirect

62. C₁₃₉

#2858

semidirect

64. C₂×C₄⋊Q₈

D₁₃₉

semidirect

119. C₄

#17984831

semidirect

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

subgroup	index
C₂×#17984839	2
C₂×#17984831	2
#22580854	2
C₄²×Dic₁₃₉	2
C₂×C₄⋊Q₈×C₁₃₉	2
#2858	139

subgroup

index

C₂×#17984839

C₂×#17984831

#22580854

C₄²×Dic₁₃₉

C₂×C₄⋊Q₈×C₁₃₉

#2858

139

Subgroups — #31814762

Conjugacy classes of subgroups

Complements

Maximal subgroups