Group #22581668 — subgroups

Subgroups — #22581668

Order 8896.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#17984842	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984828	C₂	1	4	yes	maximal, complemented
3	#17984816	C₂	1	2	yes	maximal, complemented
4	#17984813	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984801	C₂	1	2	yes	maximal, complemented
7	#17984800	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₇₈	C₂²	1	2	yes	—
10	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄×D₁₃₉	C₂²	1	2	yes	—
13	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
14	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₃₉	C₂²	1	4	yes	—
17	(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	—
19	C₄×Dic₁₃₉	C₂²	1	4	yes	—
20	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₉	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂²×D₁₃₉	C₂³	1	2	yes	—
26	Dic₂₇₈	—	4	2	no	—
27	D₅₅₆	—	2	2	no	—
28	C₂²⋊D₁₃₉	—	4	4	no	—
29	C₂×Dic₁₃₉	C₂³	1	2	yes	—
30	C₂×Dic₁₃₉	C₂³	1	4	yes	—
31	C₄×D₁₃₉	—	2	4	no	—
32	C₂³×C₁₃₉	C₂³	1	2	yes	—
33	D₄×C₁₃₉	—	2	2	no	—
34	C₂×C₄×C₁₃₉	—	2	4	no	—
35	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
37	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
39	C₂²×C₁₃₉	—	2	2	no	—
40	C₂²×C₁₃₉	—	2	2	no	—
41	C₂²×C₁₃₉	—	2	2	no	—
42	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₃₉	—	2	2	no	—
44	C₂×D₁₃₉	—	2	2	no	—
45	C₂×D₁₃₉	—	2	2	no	—
46	C₄×C₁₃₉	—	2	2	no	—
47	C₄×C₁₃₉	—	2	1	no	—
48	C₄×C₁₃₉	#41	1	2	yes	—
49	C₄×C₁₃₉	#41	1	2	yes	—
50	Dic₁₃₉	—	2	2	no	—
51	Dic₁₃₉	—	2	4	no	—
52	C₂×C₁₃₉	—	4	2	no	—
53	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₃₉	—	4	2	no	—
57	C₁₃₉	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	139	1	no	maximal, Sylow
59	C₄⋊Q₈	—	139	1	no	—
60	#124	—	139	4	no	—
61	#121	—	139	2	no	—
62	#121	—	139	2	no	—
63	C₄×D₄	—	139	1	no	—
64	C₄×D₄	D₁₃₉	1	1	yes	characteristic, complemented
65	#117	—	139	2	no	—
66	#117	—	139	2	no	—
67	C₂×Q₈	—	139	2	no	—
68	C₂×D₄	—	139	4	no	—
69	C₂×D₄	—	139	1	no	—
70	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
71	C₂²×C₄	—	139	2	no	—
72	C₂²×C₄	C₂×D₁₃₉	1	2	yes	—
73	C₄⋊C₄	—	139	1	no	—
74	C₄⋊C₄	—	139	2	no	—
75	C₄⋊C₄	—	139	2	no	—
76	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂²⋊C₄	—	139	4	no	—
78	C₂²⋊C₄	—	139	4	no	—
79	C₂²⋊C₄	—	139	2	no	—
80	C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
81	C₄²	—	139	4	no	—
82	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
83	C₂³	—	139	2	no	—
84	C₂³	C₂²×D₁₃₉	1	2	yes	—
85	Q₈	—	556	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	278	2	no	—
88	D₄	—	556	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	278	4	no	—
91	C₂×C₄	—	139	2	no	—
92	C₂×C₄	—	139	4	no	—
93	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
94	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	278	2	no	—
101	C₂²	—	278	2	no	—
102	C₂²	—	278	2	no	—
103	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	278	2	no	—
107	C₄	—	278	4	no	—
108	C₄	#14555784	1	2	yes	—
109	C₄	#14555785	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	556	2	no	—
112	C₂	C₂×#14555784	1	1	yes	characteristic
113	C₂	C₂×#14555785	1	1	yes	characteristic
114	C₂	#17984888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984842

C₂

yes

maximal, characteristic, complemented

#17984828

C₂

yes

maximal, complemented

#17984816

C₂

yes

maximal, complemented

#17984813

C₂

yes

maximal, complemented

C₄×D₅₅₆

C₂

yes

maximal, characteristic, complemented

#17984801

C₂

yes

maximal, complemented

#17984800

C₂

yes

maximal, complemented

C₄×D₄×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

—

C₂²⋊Dic₁₃₉

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

—

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

—

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

—

Dic₂₇₈

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

C₂³

yes

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#41

yes

—

C₄×C₁₃₉

#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

H(5,𝔽₂)

yes

characteristic

D₁₃₉

—

C₁₃₉

#555

yes

characteristic, complemented, Sylow

#555

—

139

maximal, Sylow

C₄⋊Q₈

—

139

—

#124

—

139

—

#121

—

139

—

#121

—

139

—

C₄×D₄

—

139

—

C₄×D₄

D₁₃₉

yes

characteristic, complemented

#117

—

139

—

#117

—

139

—

C₂×Q₈

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

C₂×D₁₃₉

yes

—

C₄²

—

139

—

C₄²

C₂×D₁₃₉

yes

characteristic

C₂³

—

139

—

C₂³

C₂²×D₁₃₉

yes

—

Q₈

—

556

—

D₄

—

D₄

—

278

—

D₄

—

556

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

278

—

101

C₂²

—

278

—

102

C₂²

—

278

—

103

C₂²

C₂³×D₁₃₉

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

278

—

107

C₄

—

278

—

108

C₄

#14555784

yes

—

109

C₄

#14555785

yes

—

110

C₂

—

111

C₂

—

556

—

112

C₂

C₂×#14555784

yes

characteristic

113

C₂

C₂×#14555785

yes

characteristic

114

C₂

#17984888

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	C₄×D₄×C₁₃₉	C₂
hypercenter	C₄×D₄	D₁₃₉
hypocenter	C₁₃₉	#555
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

C₄×D₄×C₁₃₉

C₂

hypercenter

C₄×D₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #17984842	C₂	semidirect
2. #17984828	C₂	semidirect
3. #17984816	C₂	semidirect
4. #17984813	C₂	semidirect
5. C₄×D₅₅₆	C₂	semidirect
6. #17984801	C₂	semidirect
7. #17984800	C₂	semidirect
8. C₄×D₄×C₁₃₉	C₂	semidirect
57. C₁₃₉	#555	semidirect
64. C₄×D₄	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984842

C₂

semidirect

2. #17984828

C₂

semidirect

3. #17984816

C₂

semidirect

4. #17984813

C₂

semidirect

5. C₄×D₅₅₆

C₂

semidirect

6. #17984801

C₂

semidirect

7. #17984800

C₂

semidirect

8. C₄×D₄×C₁₃₉

C₂

semidirect

57. C₁₃₉

#555

semidirect

64. C₄×D₄

D₁₃₉

semidirect

Subgroups — #22581668

Conjugacy classes of subgroups

Complements