Group #31771181 — subgroups

Subgroups — #31771181

Order 17216.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16864402	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575453	C₂	1	2	yes	maximal, complemented
4	#22575441	C₂	1	2	yes	maximal, complemented
5	#22575437	C₂	1	4	yes	maximal, complemented
6	#22575433	C₂	1	4	yes	maximal, complemented
7	#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864402	—	2	4	no	—
9	C₂×Dic₅₃₈	—	2	4	no	—
10	C₂×Dic₅₃₈	C₂²	1	2	yes	—
11	C₂×Dic₅₃₈	C₂²	1	2	yes	—
12	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
15	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
16	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
17	C₄⋊Dic₂₆₉	C₂²	1	8	yes	—
18	#16864379	C₂²	1	8	yes	—
19	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
22	C₄²×C₂₆₉	C₂²	1	2	yes	—
23	C₂²×D₂₆₉	C₂³	1	2	yes	—
24	Dic₅₃₈	—	2	4	no	—
25	Dic₅₃₈	—	2	4	no	—
26	Dic₅₃₈	—	2	2	no	—
27	D₁₀₇₆	—	2	2	no	—
28	C₂²⋊D₂₆₉	—	2	4	no	—
29	C₂×Dic₂₆₉	—	2	4	no	—
30	C₂×Dic₂₆₉	C₂³	1	4	yes	—
31	C₂×Dic₂₆₉	C₂³	1	2	yes	—
32	C₄×D₂₆₉	—	2	4	no	—
33	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₂₆₉	D₄	1	4	yes	—
35	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
36	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
38	C₂²×C₂₆₉	—	2	2	no	—
39	C₂²×C₂₆₉	C₂×D₄	1	2	yes	—
40	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
41	C₂×D₂₆₉	—	2	2	no	—
42	C₂×D₂₆₉	—	2	4	no	—
43	C₄×C₂₆₉	—	2	4	no	—
44	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
45	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
46	Dic₂₆₉	—	2	2	no	—
47	Dic₂₆₉	—	2	4	no	—
48	C₂×C₂₆₉	—	2	2	no	—
49	C₂×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
51	D₂₆₉	—	4	2	no	—
52	C₂₆₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	269	1	no	maximal, Sylow
54	C₂×#41	—	269	1	no	—
55	C₂²×Q₈	—	269	1	no	—
56	C₄⋊Q₈	—	269	2	no	—
57	#124	—	269	2	no	—
58	#123	—	269	4	no	—
59	C₂²⋊Q₈	—	269	4	no	—
60	#117	D₂₆₉	1	1	yes	characteristic, complemented
61	#41	—	538	4	no	—
62	C₂×Q₈	—	269	2	no	—
63	C₂×Q₈	—	269	2	no	—
64	C₂×Q₈	—	269	1	no	—
65	C₂×Q₈	—	538	4	no	—
66	C₂×D₄	—	269	2	no	—
67	C₂×D₄	—	269	1	no	—
68	C₂²×C₄	—	269	2	no	—
69	C₂²×C₄	—	269	2	no	—
70	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
71	C₄⋊C₄	—	269	8	no	—
72	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
73	C₂²⋊C₄	—	269	8	no	—
74	C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
75	C₄²	C₂×D₂₆₉	1	2	yes	—
76	C₂³	—	269	2	no	—
77	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
78	Q₈	—	538	4	no	—
79	Q₈	—	538	4	no	—
80	Q₈	—	538	2	no	—
81	D₄	—	538	4	no	—
82	D₄	—	538	2	no	—
83	C₂×C₄	—	269	4	no	—
84	C₂×C₄	—	269	2	no	—
85	C₂×C₄	—	538	4	no	—
86	C₂×C₄	—	538	4	no	—
87	C₂×C₄	D₁₀₇₆	1	4	yes	—
88	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
89	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	538	2	no	—
93	C₂²	—	538	4	no	—
94	C₂²	C₂×D₁₀₇₆	1	2	yes	—
95	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	538	2	no	—
98	C₄	—	538	4	no	—
99	C₄	C₂×D₁₀₇₆	1	2	yes	—
100	C₄	C₂×D₁₀₇₆	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	1076	2	no	—
103	C₂	C₂²×D₁₀₇₆	1	1	yes	characteristic
104	C₂	#22575500	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16864402

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

C₂

yes

maximal, characteristic, complemented

#22575453

C₂

yes

maximal, complemented

#22575441

C₂

yes

maximal, complemented

#22575437

C₂

yes

maximal, complemented

#22575433

C₂

yes

maximal, complemented

#117×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#16864402

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂²

yes

—

C₂×Dic₅₃₈

C₂²

yes

—

C₂×Dic₅₃₈

C₂²

yes

characteristic

C₂×D₁₀₇₆

C₂²

yes

characteristic

C₂×C₂²⋊D₂₆₉

C₂²

yes

—

C₂²×Dic₂₆₉

C₂²

yes

—

C₂×C₄×D₂₆₉

C₂²

yes

—

C₄⋊Dic₂₆₉

C₂²

yes

—

#16864379

C₂²

yes

—

C₂²×C₄×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₂²⋊C₄×C₂₆₉

C₂²

yes

—

C₄²×C₂₆₉

C₂²

yes

—

C₂²×D₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₄×D₂₆₉

—

C₂³×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

D₄

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

C₂⁴

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×D₄

yes

—

C₄×C₂₆₉

C₂×D₄

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×C₂₆₉

2₋¹⁺⁴

yes

—

D₂₆₉

—

C₂₆₉

#536

yes

characteristic, complemented, Sylow

#536

—

269

maximal, Sylow

C₂×#41

—

269

—

C₂²×Q₈

—

269

—

C₄⋊Q₈

—

269

—

#124

—

269

—

#123

—

269

—

C₂²⋊Q₈

—

269

—

#117

D₂₆₉

yes

characteristic, complemented

#41

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

C₂×D₄

—

269

—

C₂×D₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂×D₂₆₉

yes

characteristic

C₄⋊C₄

—

269

—

C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

C₂×D₂₆₉

yes

—

C₄²

C₂×D₂₆₉

yes

—

C₂³

—

269

—

C₂³

C₂²×D₂₆₉

yes

characteristic

Q₈

—

538

—

Q₈

—

538

—

Q₈

—

538

—

D₄

—

538

—

D₄

—

538

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

538

—

C₂×C₄

—

538

—

C₂×C₄

D₁₀₇₆

yes

—

C₂×C₄

C₂²×D₂₆₉

yes

—

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

C₂²

—

C₂²

—

538

—

C₂²

—

538

—

C₂²

C₂×D₁₀₇₆

yes

—

C₂²

C₂³×D₂₆₉

yes

characteristic

C₄

—

C₄

—

538

—

C₄

—

538

—

C₄

C₂×D₁₀₇₆

yes

—

100

C₄

C₂×D₁₀₇₆

yes

—

101

C₂

—

102

C₂

—

1076

—

103

C₂

C₂²×D₁₀₇₆

yes

characteristic

104

C₂

#22575500

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	#117×C₂₆₉	C₂
prime radical	C₂×#16864402	C₂
hypercenter	#117	D₂₆₉
hypocenter	C₂₆₉	#536
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₆₉

commutator, socle, cosocle

C₂²×C₂₆₉

C₂⁴

Fitting

#117×C₂₆₉

C₂

C₂

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₂²×Dic₅₃₈	C₂	semidirect
3. #22575453	C₂	semidirect
4. #22575441	C₂	semidirect
5. #22575437	C₂	semidirect
6. #22575433	C₂	semidirect
7. #117×C₂₆₉	C₂	semidirect
52. C₂₆₉	#536	semidirect
60. #117	D₂₆₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₅₃₈

C₂

semidirect

3. #22575453

C₂

semidirect

4. #22575441

C₂

semidirect

5. #22575437

C₂

semidirect

6. #22575433

C₂

semidirect

7. #117×C₂₆₉

C₂

semidirect

52. C₂₆₉

#536

semidirect

60. #117

D₂₆₉

semidirect

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

subgroup	index
C₂×#16864402	2
C₂²×Dic₅₃₈	2
#22575453	2
#22575441	2
#22575437	2
#22575433	2
#117×C₂₆₉	2
#536	269

subgroup

index

C₂²×Dic₅₃₈

#117×C₂₆₉

269

Subgroups — #31771181

Conjugacy classes of subgroups

Complements

Maximal subgroups