Group #22578478 — subgroups

Subgroups — C₂×#17984175 #22578478

Order 8768.

266 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 140 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	#17984175	C₂	1	8	yes	maximal, direct product factor
2	C₂×#14555520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14555519	C₂	1	2	yes	maximal, complemented
4	C₂×#14555518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#11764047	C₂	1	2	yes	maximal, complemented
6	C₂×#14555509	C₂	1	1	yes	maximal, characteristic
7	#14555520	C₂²	1	4	yes	complemented
8	#14555519	C₂²	1	8	yes	complemented
9	#14555518	C₂²	1	4	yes	complemented
10	C₂×#11764047	C₂²	1	1	yes	characteristic
11	C₂×#11764047	C₂²	1	2	yes	—
12	C₂×#11764047	C₂²	1	1	yes	characteristic
13	C₂×#11764047	C₂²	1	8	yes	complemented
14	#14555509	C₄	1	4	yes	—
15	#14555509	C₂²	1	4	yes	—
16	C₂×Dic₂₇₄	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₁₃₇	C₄	1	2	yes	—
18	C₂²×Dic₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₁₃₇	C₄	1	1	yes	characteristic
21	#11764047	C₂³	1	2	yes	—
22	#11764047	C₂³	1	4	yes	—
23	#11764047	C₂³	1	2	yes	—
24	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
25	Dic₂₇₄	C₂×C₄	1	4	yes	—
26	C₂²⋊D₁₃₇	C₂×C₄	1	8	yes	—
27	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₂³	1	4	yes	—
31	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₃₇	C₂³	1	2	yes	—
33	C₂³×C₁₃₇	C₂×C₄	1	2	yes	—
34	D₄×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	—	2	4	no	—
37	C₂²×C₁₃₇	C₂²×C₄	1	4	yes	—
38	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₃₇	—	2	2	no	—
40	C₂×D₁₃₇	C₂²×C₄	1	2	yes	—
41	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
42	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	Dic₁₃₇	C₂²×C₄	1	4	yes	—
44	Dic₁₃₇	C₂⁴	1	1	yes	characteristic
45	C₂×C₁₃₇	—	2	4	no	—
46	C₂×C₁₃₇	#131	1	2	yes	complemented
47	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
48	D₁₃₇	—	2	2	no	—
49	C₁₃₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	137	1	no	maximal, Sylow
51	C₂×#41	—	137	1	no	—
52	#131	—	137	8	no	—
53	C₂×M₁₆	—	137	2	no	—
54	C₂×M₁₆	—	137	1	no	—
55	C₂²×C₈	—	137	2	no	—
56	C₂²×C₈	—	137	1	no	—
57	#41	—	137	4	no	—
58	#41	—	137	4	no	—
59	C₂×Q₈	—	137	1	no	—
60	C₂×D₄	—	137	2	no	—
61	C₂×D₄	#10587739	1	1	yes	characteristic
62	C₂²×C₄	—	137	2	no	—
63	C₂²×C₄	—	137	1	no	—
64	M₁₆	—	137	8	no	—
65	M₁₆	—	137	4	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	2	no	—
68	C₂×C₈	—	137	1	no	—
69	C₂×C₈	—	137	8	no	—
70	C₂×C₈	—	137	4	no	—
71	C₂³	—	137	1	no	—
72	C₂³	C₂×#10587739	1	2	yes	—
73	Q₈	—	137	4	no	—
74	D₄	—	137	8	no	—
75	D₄	C₂×#10587739	1	4	yes	—
76	C₂×C₄	—	137	4	no	—
77	C₂×C₄	—	137	4	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	—	137	2	no	—
80	C₂×C₄	—	137	2	no	—
81	C₂×C₄	—	137	2	no	—
82	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
83	C₈	—	137	2	no	—
84	C₈	—	137	4	no	—
85	C₈	—	137	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	274	2	no	—
88	C₂²	—	137	2	no	—
89	C₂²	C₂²×#10587739	1	4	yes	—
90	C₂²	C₂²×#10587739	1	1	yes	characteristic
91	C₄	—	137	1	no	—
92	C₄	—	137	1	no	—
93	C₄	—	137	4	no	—
94	C₄	C₂²×#10587739	1	2	yes	—
95	C₂	—	2	4	no	—
96	C₂	—	274	2	no	—
97	C₂	#17984175	1	2	yes	direct product factor
98	C₂	C₂³×#10587739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984175

C₂

yes

maximal, direct product factor

C₂×#14555520

C₂

yes

maximal, characteristic, complemented

C₂×#14555519

C₂

yes

maximal, complemented

C₂×#14555518

C₂

yes

maximal, characteristic, complemented

C₂²×#11764047

C₂

yes

maximal, complemented

C₂×#14555509

C₂

yes

maximal, characteristic

#14555520

C₂²

yes

complemented

#14555519

C₂²

yes

complemented

#14555518

C₂²

yes

complemented

C₂×#11764047

C₂²

yes

characteristic

C₂×#11764047

C₂²

yes

—

C₂×#11764047

C₂²

yes

characteristic

C₂×#11764047

C₂²

yes

complemented

#14555509

C₄

yes

—

#14555509

C₂²

yes

—

C₂×Dic₂₇₄

C₄

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₄

yes

—

C₂²×Dic₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂×D₄×C₁₃₇

C₄

yes

characteristic

#11764047

C₂³

yes

—

#11764047

C₂³

yes

—

#11764047

C₂³

yes

—

C₂²×D₁₃₇

C₂×C₄

yes

characteristic

Dic₂₇₄

C₂×C₄

yes

—

C₂²⋊D₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

C₂×C₄

yes

—

C₄×D₁₃₇

C₂³

yes

—

C₂³×C₁₃₇

C₂×C₄

yes

—

D₄×C₁₃₇

C₂×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂²×C₄

yes

—

C₄×C₁₃₇

C₂²×C₄

yes

—

Dic₁₃₇

C₂²×C₄

yes

characteristic

Dic₁₃₇

C₂²×C₄

yes

—

Dic₁₃₇

C₂⁴

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

#131

yes

complemented

C₂×C₁₃₇

C₂³×C₄

yes

characteristic

D₁₃₇

—

C₁₃₇

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

137

maximal, Sylow

C₂×#41

—

137

—

#131

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

#41

—

137

—

#41

—

137

—

C₂×Q₈

—

137

—

C₂×D₄

—

137

—

C₂×D₄

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

137

—

M₁₆

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂³

—

137

—

C₂³

C₂×#10587739

yes

—

Q₈

—

137

—

D₄

—

137

—

D₄

C₂×#10587739

yes

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₂²

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂²×#10587739

yes

—

C₂²

C₂²×#10587739

yes

characteristic

C₄

—

137

—

C₄

—

137

—

C₄

—

137

—

C₄

C₂²×#10587739

yes

—

C₂

—

C₂

—

274

—

C₂

#17984175

yes

direct product factor

C₂

C₂³×#10587739

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂³×C₄
Frattini	C₂	C₂³×#10587739
Fitting	C₂×D₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₄
hypercenter	C₂×D₄	#10587739
hypocenter	C₁₃₇	C₂×#131
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×#10587739

commutator

C₂×C₁₃₇

C₂³×C₄

Frattini

C₂

C₂³×#10587739

Fitting

C₂×D₄×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²×C₄

C₂×D₄

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. #17984175	C₂	direct (N × H)
2. C₂×#14555520	C₂	semidirect
3. C₂×#14555519	C₂	semidirect
4. C₂×#14555518	C₂	semidirect
5. C₂²×#11764047	C₂	semidirect
7. #14555520	C₂²	semidirect
8. #14555519	C₂²	semidirect
9. #14555518	C₂²	semidirect
13. C₂×#11764047	C₂²	semidirect
46. C₂×C₁₃₇	#131	semidirect
49. C₁₃₇	C₂×#131	semidirect
97. C₂	#17984175	direct (N × H)

H ≅ G/N

type

1. #17984175

C₂

direct (N × H)

2. C₂×#14555520

C₂

semidirect

3. C₂×#14555519

C₂

semidirect

4. C₂×#14555518

C₂

semidirect

5. C₂²×#11764047

C₂

semidirect

7. #14555520

C₂²

semidirect

8. #14555519

C₂²

semidirect

9. #14555518

C₂²

semidirect

13. C₂×#11764047

C₂²

semidirect

46. C₂×C₁₃₇

#131

semidirect

49. C₁₃₇

C₂×#131

semidirect

97. C₂

#17984175

direct (N × H)

Subgroups — C₂×#17984175 #22578478

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17984175 #22578478

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17984175 #22578478