Group #20295602 — subgroups

Subgroups — #20295602

Order 6976.

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×#13182029	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812699	C₂	1	2	yes	maximal, complemented
4	#16812687	C₂	1	2	yes	maximal, complemented
5	#16812683	C₂	1	4	yes	maximal, complemented
6	#16812679	C₂	1	4	yes	maximal, complemented
7	#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182029	—	2	4	no	—
9	C₂×Dic₂₁₈	—	2	4	no	—
10	C₂×Dic₂₁₈	C₂²	1	2	yes	—
11	C₂×Dic₂₁₈	C₂²	1	2	yes	—
12	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
13	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₉	C₂²	1	8	yes	—
18	#13182006	C₂²	1	8	yes	—
19	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₄²×C₁₀₉	C₂²	1	2	yes	—
23	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₉	C₂³	1	2	yes	—
25	C₂×C₄×C₁₀₉	D₄	1	4	yes	—
26	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₉	—	2	4	no	—
30	C₂×Dic₁₀₉	—	2	4	no	—
31	C₂×Dic₁₀₉	C₂³	1	4	yes	—
32	C₂×Dic₁₀₉	C₂³	1	2	yes	—
33	D₄₃₆	—	2	2	no	—
34	C₄×D₁₀₉	—	2	4	no	—
35	Dic₂₁₈	—	2	4	no	—
36	Dic₂₁₈	—	2	4	no	—
37	Dic₂₁₈	—	2	2	no	—
38	C₂²×C₁₀₉	—	2	2	no	—
39	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	2	no	—
42	C₂×D₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	—	2	4	no	—
44	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	Dic₁₀₉	—	2	4	no	—
47	Dic₁₀₉	—	2	2	no	—
48	C₂×C₁₀₉	—	2	2	no	—
49	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
51	D₁₀₉	—	4	2	no	—
52	C₁₀₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	109	1	no	maximal, Sylow
54	C₂×#41	—	109	1	no	—
55	C₂²×Q₈	—	109	1	no	—
56	C₄⋊Q₈	—	109	2	no	—
57	#124	—	109	2	no	—
58	#123	—	109	4	no	—
59	C₂²⋊Q₈	—	109	4	no	—
60	#117	D₁₀₉	1	1	yes	characteristic, complemented
61	#41	—	218	4	no	—
62	C₂×Q₈	—	109	2	no	—
63	C₂×Q₈	—	109	2	no	—
64	C₂×Q₈	—	109	1	no	—
65	C₂×Q₈	—	218	4	no	—
66	C₂×D₄	—	109	2	no	—
67	C₂×D₄	—	109	1	no	—
68	C₂²×C₄	—	109	2	no	—
69	C₂²×C₄	—	109	2	no	—
70	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	C₄⋊C₄	—	109	8	no	—
72	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
73	C₂²⋊C₄	—	109	8	no	—
74	C₂²⋊C₄	C₂×D₁₀₉	1	2	yes	—
75	C₄²	C₂×D₁₀₉	1	2	yes	—
76	C₂³	—	109	2	no	—
77	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
78	Q₈	—	218	4	no	—
79	Q₈	—	218	4	no	—
80	Q₈	—	218	2	no	—
81	D₄	—	218	4	no	—
82	D₄	—	218	2	no	—
83	C₂×C₄	—	109	4	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	218	4	no	—
86	C₂×C₄	—	218	4	no	—
87	C₂×C₄	D₄₃₆	1	4	yes	—
88	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
89	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	218	2	no	—
93	C₂²	—	218	4	no	—
94	C₂²	C₂×D₄₃₆	1	2	yes	—
95	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	218	2	no	—
98	C₄	—	218	4	no	—
99	C₄	C₂×D₄₃₆	1	2	yes	—
100	C₄	C₂×D₄₃₆	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	436	2	no	—
103	C₂	C₂²×D₄₃₆	1	1	yes	characteristic
104	C₂	#16812746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂

yes

maximal, characteristic, complemented

#16812699

C₂

yes

maximal, complemented

#16812687

C₂

yes

maximal, complemented

#16812683

C₂

yes

maximal, complemented

#16812679

C₂

yes

maximal, complemented

#117×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

—

C₂×Dic₂₁₈

C₂²

yes

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182006

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

—

D₁₀₉

—

C₁₀₉

#536

yes

characteristic, complemented, Sylow

#536

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

109

—

#124

—

109

—

#123

—

109

—

C₂²⋊Q₈

—

109

—

#117

D₁₀₉

yes

characteristic, complemented

#41

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₄²

C₂×D₁₀₉

yes

—

C₂³

—

109

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

218

—

Q₈

—

218

—

Q₈

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄₃₆

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×D₄₃₆

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂×D₄₃₆

yes

—

100

C₄

C₂×D₄₃₆

yes

—

101

C₂

—

102

C₂

—

436

—

103

C₂

C₂²×D₄₃₆

yes

characteristic

104

C₂

#16812746

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#117×C₁₀₉	C₂
hypercenter	#117	D₁₀₉
hypocenter	C₁₀₉	#536
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

#117×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₂²×Dic₂₁₈	C₂	semidirect
3. #16812699	C₂	semidirect
4. #16812687	C₂	semidirect
5. #16812683	C₂	semidirect
6. #16812679	C₂	semidirect
7. #117×C₁₀₉	C₂	semidirect
52. C₁₀₉	#536	semidirect
60. #117	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₂₁₈

C₂

semidirect

3. #16812699

C₂

semidirect

4. #16812687

C₂

semidirect

5. #16812683

C₂

semidirect

6. #16812679

C₂

semidirect

7. #117×C₁₀₉

C₂

semidirect

52. C₁₀₉

#536

semidirect

60. #117

D₁₀₉

semidirect

Subgroups — #20295602

Conjugacy classes of subgroups

Complements