Group #31807184 — subgroups

Subgroups — #31807184

Order 17792.

196 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 93 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×#17984778	C₂	1	1	yes	maximal, characteristic
2	#22580996	C₂	1	2	yes	maximal
3	C₂×#17984774	C₂	1	1	yes	maximal, characteristic
4	#22580721	C₂	1	2	yes	maximal
5	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
8	#17984778	C₄	1	4	yes	complemented
9	#17984775	—	2	2	no	—
10	#17984774	—	2	1	no	—
11	#17984774	—	2	1	no	—
12	#17984773	—	2	2	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#117×C₁₃₉	C₂²	1	1	yes	characteristic
16	#97×C₁₃₉	—	2	1	no	—
17	C₄×C₈×C₁₃₉	—	2	1	no	—
18	C₂²×Dic₁₃₉	—	2	2	no	—
19	#14555769	—	2	2	no	—
20	#14555769	C₂×C₄	1	4	yes	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₃₉	—	2	1	no	—
26	M₁₆×C₁₃₉	Q₈	1	2	yes	—
27	C₂×C₈×C₁₃₉	—	2	1	no	—
28	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
32	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
34	C₄²×C₁₃₉	—	2	1	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₂×Dic₁₃₉	—	8	2	no	—
37	C₁₃₉⋊C₈	—	2	4	no	—
38	C₂³×C₁₃₉	#41	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	—	2	2	no	—
40	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
46	C₈×C₁₃₉	—	2	2	no	—
47	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
48	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₁₃₉	—	2	2	no	—
50	C₂²×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₃₉	#121	1	1	yes	characteristic
52	C₂²×C₁₃₉	#125	1	1	yes	characteristic
53	C₄×C₁₃₉	—	4	2	no	—
54	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
56	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₃₉	—	8	2	no	—
59	C₂×C₁₃₉	—	2	1	no	—
60	C₂×C₁₃₉	—	2	1	no	—
61	C₂×C₁₃₉	#389	1	1	yes	characteristic
62	C₂×C₁₃₉	#471	1	2	yes	—
63	C₁₃₉	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	139	1	no	maximal, Sylow
65	C₂×#108	—	139	1	no	—
66	#428	—	139	2	no	—
67	C₂×#106	—	139	1	no	—
68	#405	D₁₃₉	1	1	yes	characteristic
69	#343	—	139	2	no	—
70	C₂×M₁₆	—	139	2	no	—
71	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
73	#117	C₂×D₁₃₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	139	2	no	—
75	#108	—	139	4	no	—
76	#107	—	278	2	no	—
77	#106	—	278	1	no	—
78	#106	—	278	1	no	—
79	#106	—	278	2	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂²×C₄	—	278	2	no	—
83	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	278	2	no	—
86	M₁₆	—	139	4	no	—
87	M₁₆	Dic₂₇₈	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
91	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
93	C₂×C₈	Dic₂₇₈	1	2	yes	—
94	C₄⋊C₄	—	278	2	no	—
95	C₄⋊C₄	—	278	2	no	—
96	C₄⋊C₄	D₅₅₆	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#14555785	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	556	2	no	—
102	C₂×C₄	—	1112	2	no	—
103	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
105	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
108	C₂×C₄	#14555785	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	278	4	no	—
111	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
112	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₂₇₈	1	1	yes	characteristic
115	C₂²	#17984816	1	1	yes	characteristic
116	C₂²	#17984834	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	1112	2	no	—
119	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
120	C₄	C₄×D₅₅₆	1	1	yes	characteristic
121	C₄	#17984821	1	1	yes	characteristic
122	C₄	#17984842	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#22580851	1	1	yes	characteristic
126	C₂	#22581216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984778

C₂

yes

maximal, characteristic

#22580996

C₂

yes

maximal

C₂×#17984774

C₂

yes

maximal, characteristic

#22580721

C₂

yes

maximal

#405×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

#17984778

C₄

yes

—

—

—

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

—

C₄×C₈×C₁₃₉

—

C₂²×Dic₁₃₉

—

#14555769

—

#14555769

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

Q₈

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×Q₈

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×D₄

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

#471

yes

—

C₁₃₉

#1951

yes

characteristic, complemented, Sylow

#1951

—

139

maximal, Sylow

C₂×#108

—

139

—

#428

—

139

—

C₂×#106

—

139

—

#405

D₁₃₉

yes

characteristic

#343

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

139

—

#108

—

139

—

#107

—

278

—

#106

—

278

—

#106

—

278

—

#106

—

278

—

#97

—

C₄×C₈

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

139

—

M₁₆

Dic₂₇₈

yes

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

D₅₅₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#14555785

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

556

—

102

C₂×C₄

—

1112

—

103

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

105

C₂×C₄

C₂×D₅₅₆

yes

characteristic

106

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

107

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

108

C₂×C₄

#14555785

yes

characteristic

109

C₈

—

110

C₈

—

278

—

111

C₈

Dic₁₃₉⋊C₄

yes

—

112

C₈

Dic₁₃₉⋊C₄

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₂₇₈

yes

characteristic

115

C₂²

#17984816

yes

characteristic

116

C₂²

#17984834

yes

characteristic

117

C₄

—

118

C₄

—

1112

—

119

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

120

C₄

C₄×D₅₅₆

yes

characteristic

121

C₄

#17984821

yes

characteristic

122

C₄

#17984842

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#22580851

yes

characteristic

126

C₂

#22581216

yes

—

	subgroup	G/H
center	C₂²	#17984816
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#405×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#121
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#1951
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#17984816

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#405×C₁₃₉

C₂

C₂²×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #17984778	C₄	semidirect
63. C₁₃₉	#1951	semidirect

H ≅ G/N

type

8. #17984778

C₄

semidirect

63. C₁₃₉

#1951

semidirect

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

subgroup	index
C₂×#17984778	2
#22580996	2
C₂×#17984774	2
#22580721	2
#405×C₁₃₉	2
#1951	139

subgroup

index

#405×C₁₃₉

139

Subgroups — #31807184

Conjugacy classes of subgroups

Complements

Maximal subgroups