Group #22598838 — subgroups

Subgroups — C₂×#17986396 #22598838

Order 9088.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14556127	C₂	1	2	yes	maximal
2	#17986396	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14556060	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556059	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₄₂	—	2	2	no	—
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#14556127	—	2	4	no	—
10	#14556127	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
12	C₂×Dic₇₁⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
14	#14556060	C₂²	1	4	yes	—
15	#14556059	C₂²	1	8	yes	—
16	C₈×Dic₇₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
19	#106×C₇₁	C₂²	1	4	yes	—
20	C₂×Dic₁₄₂	—	2	8	no	—
21	C₂×Dic₁₄₂	—	4	8	no	—
22	C₂²×Dic₇₁	—	2	2	no	—
23	C₂²×Dic₇₁	D₄	1	2	yes	—
24	C₂×C₇₁⋊C₈	Q₈	1	4	yes	complemented
25	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
26	C₄⋊Dic₇₁	—	2	2	no	—
27	C₄⋊Dic₇₁	C₂³	1	4	yes	—
28	Dic₇₁⋊C₄	—	2	8	no	—
29	C₄×Dic₇₁	D₄	1	2	yes	—
30	C₄×Dic₇₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₇₁	—	2	2	no	—
32	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₇₁	Q₈	1	4	yes	complemented
34	C₂×C₈×C₇₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₇₁	—	2	2	no	—
36	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
37	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	4	no	—
39	C₂×C₄×C₇₁	—	4	4	no	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₇₁	—	2	4	no	—
45	C₂×Dic₇₁	—	4	4	no	—
46	C₂×Dic₇₁	SD₁₆	1	8	yes	—
47	C₂×Dic₇₁	C₂×D₄	1	4	yes	—
48	Dic₁₄₂	—	4	16	no	—
49	C₈×C₇₁	C₂×Q₈	1	4	yes	complemented
50	C₇₁⋊C₈	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₇₁	C₄⋊Q₈	1	2	yes	—
52	C₂²×C₇₁	C₂×SD₁₆	1	4	yes	—
53	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₇₁	—	4	4	no	—
55	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
56	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
58	Dic₇₁	—	4	4	no	—
59	Dic₇₁	C₂×SD₁₆	1	8	yes	—
60	C₂×C₇₁	#498	1	4	yes	complemented
61	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₇₁	C₂²×SD₁₆	1	2	yes	—
63	C₂×#498	—	71	1	no	maximal, Sylow
64	C₇₁	C₂×#498	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	71	2	no	—
66	#498	—	71	8	no	—
67	C₂×#106	—	71	1	no	—
68	C₂×#106	—	71	2	no	—
69	C₂×#106	D₇₁	1	1	yes	characteristic
70	C₂×C₄×C₈	—	71	1	no	—
71	C₂²×Q₈	—	142	2	no	—
72	C₂²×C₈	—	71	1	no	—
73	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
74	C₄⋊Q₈	—	71	8	no	—
75	C₄⋊Q₈	—	142	4	no	—
76	C₂×C₄⋊C₄	—	71	2	no	—
77	C₂×C₄⋊C₄	—	142	2	no	—
78	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
79	C₂×C₄²	—	71	1	no	—
80	#106	—	71	4	no	—
81	#106	—	71	8	no	—
82	#106	C₂×D₇₁	1	4	yes	—
83	C₄×C₈	—	71	4	no	—
84	C₂×Q₈	—	284	8	no	—
85	C₂×Q₈	—	142	8	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	71	2	no	—
88	C₂²×C₄	—	142	2	no	—
89	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
90	C₂×C₈	—	71	4	no	—
91	C₂×C₈	—	71	2	no	—
92	C₂×C₈	Dic₁₄₂	1	4	yes	complemented
93	C₂×C₈	C₂²×D₇₁	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	71	4	no	—
96	C₄⋊C₄	—	142	8	no	—
97	C₄⋊C₄	—	142	2	no	—
98	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
99	C₄²	—	71	2	no	—
100	C₄²	—	71	2	no	—
101	C₂³	D₄×D₇₁	1	1	yes	characteristic
102	Q₈	—	284	16	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	284	4	no	—
106	C₂×C₄	—	71	4	no	—
107	C₂×C₄	—	71	8	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₇₁	1	2	yes	—
111	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
112	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
113	C₈	—	71	4	no	—
114	C₈	C₂×Dic₁₄₂	1	4	yes	complemented
115	C₂²	#14556127	1	2	yes	—
116	C₂²	SD₁₆×D₇₁	1	4	yes	—
117	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	284	4	no	—
120	C₄	—	71	8	no	—
121	C₄	#14556127	1	2	yes	—
122	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
123	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
124	C₂	#17986396	1	4	yes	direct product factor
125	C₂	C₂×#14556127	1	1	yes	characteristic
126	C₂	C₂×SD₁₆×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556127

C₂

yes

maximal

#17986396

C₂

yes

maximal, direct product factor

C₂×#14556060

C₂

yes

maximal, characteristic

C₂×#14556059

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

C₂×#106×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556127

—

#14556127

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556060

C₂²

yes

—

#14556059

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#106×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

SD₁₆

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

complemented

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂×SD₁₆

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

C₂×SD₁₆

yes

—

C₂×C₇₁

#498

yes

complemented

C₂×C₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₁

C₂²×SD₁₆

yes

—

C₂×#498

—

maximal, Sylow

C₇₁

C₂×#498

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₇₁

yes

characteristic

C₂×C₄×C₈

—

C₂²×Q₈

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₇₁

yes

—

C₄×C₈

—

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

complemented

C₂×C₈

C₂²×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₂²×D₇₁

yes

—

C₄²

—

100

C₄²

—

101

C₂³

D₄×D₇₁

yes

characteristic

102

Q₈

—

284

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

284

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

142

—

109

C₂×C₄

D₄×D₇₁

yes

characteristic

110

C₂×C₄

Q₈×D₇₁

yes

—

111

C₂×C₄

C₂×Dic₁₄₂

yes

—

112

C₂×C₄

C₂³×D₇₁

yes

characteristic

113

C₈

—

114

C₈

C₂×Dic₁₄₂

yes

complemented

115

C₂²

#14556127

yes

—

116

C₂²

SD₁₆×D₇₁

yes

—

117

C₂²

C₂×D₄×D₇₁

yes

characteristic

118

C₄

—

119

C₄

—

284

—

120

C₄

—

121

C₄

#14556127

yes

—

122

C₄

C₂×Q₈×D₇₁

yes

characteristic

123

C₄

C₂²×Dic₁₄₂

yes

characteristic

124

C₂

#17986396

yes

direct product factor

125

C₂

C₂×#14556127

yes

characteristic

126

C₂

C₂×SD₁₆×D₇₁

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₂×#106×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#106	D₇₁
hypocenter	C₇₁	C₂×#498
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂×#106×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #17986396	C₂	direct (N × H)
24. C₂×C₇₁⋊C₈	Q₈	semidirect
33. C₂×C₈×C₇₁	Q₈	semidirect
49. C₈×C₇₁	C₂×Q₈	semidirect
50. C₇₁⋊C₈	C₂×Q₈	semidirect
60. C₂×C₇₁	#498	semidirect
64. C₇₁	C₂×#498	semidirect
92. C₂×C₈	Dic₁₄₂	semidirect
114. C₈	C₂×Dic₁₄₂	semidirect
124. C₂	#17986396	direct (N × H)

H ≅ G/N

type

2. #17986396

C₂

direct (N × H)

24. C₂×C₇₁⋊C₈

Q₈

semidirect

33. C₂×C₈×C₇₁

Q₈

semidirect

49. C₈×C₇₁

C₂×Q₈

semidirect

50. C₇₁⋊C₈

C₂×Q₈

semidirect

60. C₂×C₇₁

#498

semidirect

64. C₇₁

C₂×#498

semidirect

92. C₂×C₈

Dic₁₄₂

semidirect

114. C₈

C₂×Dic₁₄₂

semidirect

124. C₂

#17986396

direct (N × H)

Subgroups — C₂×#17986396 #22598838

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17986396 #22598838

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17986396 #22598838