Subgroups — C₂×#26014988 #36378326

Order 22592.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₄₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊D₃₅₃	C₂	1	4	yes	maximal, complemented
3	#26014988	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₅₃	C₂²	1	2	yes	—
7	C₂×D₁₄₁₂	—	2	4	no	—
8	C₂×D₁₄₁₂	C₂²	1	4	yes	complemented
9	C₂×C₂²⋊D₃₅₃	—	2	16	no	—
10	C₂×C₂²⋊D₃₅₃	C₂²	1	16	yes	complemented
11	C₂²×Dic₃₅₃	C₂²	1	2	yes	—
12	C₄×Dic₃₅₃	C₂²	1	4	yes	complemented
13	C₂⁴×C₃₅₃	C₂²	1	2	yes	—
14	C₂×D₄×C₃₅₃	—	2	4	no	—
15	C₂×D₄×C₃₅₃	C₂²	1	4	yes	complemented
16	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₃₅₃	—	2	8	no	—
18	C₂²×D₃₅₃	—	2	4	no	—
19	C₂²×D₃₅₃	C₂³	1	4	yes	—
20	D₁₄₁₂	—	2	8	no	—
21	C₂²⋊D₃₅₃	—	2	32	no	—
22	C₂×Dic₃₅₃	D₄	1	8	yes	complemented
23	C₂×Dic₃₅₃	C₂³	1	4	yes	—
24	C₂³×C₃₅₃	—	2	4	no	—
25	C₂³×C₃₅₃	—	2	8	no	—
26	C₂³×C₃₅₃	C₂³	1	4	yes	—
27	C₂³×C₃₅₃	C₂³	1	1	yes	characteristic
28	D₄×C₃₅₃	—	2	8	no	—
29	C₂×C₄×C₃₅₃	D₄	1	4	yes	complemented
30	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
31	C₂²×C₃₅₃	—	2	4	no	—
32	C₂²×C₃₅₃	—	2	8	no	—
33	C₂²×C₃₅₃	—	4	8	no	—
34	C₂²×C₃₅₃	C₂×D₄	1	4	yes	—
35	C₂²×C₃₅₃	C₂×D₄	1	2	yes	—
36	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
37	C₂×D₃₅₃	—	2	8	no	—
38	C₂×D₃₅₃	—	2	4	no	—
39	C₂×D₃₅₃	—	4	8	no	—
40	C₄×C₃₅₃	C₂×D₄	1	4	yes	complemented
41	Dic₃₅₃	C₂×D₄	1	8	yes	complemented
42	C₂×C₃₅₃	—	4	4	no	—
43	C₂×C₃₅₃	#127	1	4	yes	complemented
44	C₂×C₃₅₃	C₂²×D₄	1	2	yes	—
45	C₂×C₃₅₃	C₂²×D₄	1	1	yes	characteristic
46	D₃₅₃	—	4	4	no	—
47	C₃₅₃	C₂×#127	1	1	yes	characteristic, complemented, Sylow
48	C₂×#127	—	353	1	no	maximal, Sylow
49	C₂²×D₄	—	353	4	no	—
50	C₂²×D₄	—	353	1	no	—
51	C₂²×D₄	D₃₅₃	1	1	yes	characteristic, complemented
52	#127	—	353	8	no	—
53	C₂×C₄²	—	353	1	no	—
54	C₂⁴	—	353	2	no	—
55	C₂⁴	C₂×D₃₅₃	1	2	yes	—
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	353	16	no	—
58	C₂×D₄	—	353	4	no	—
59	C₂×D₄	—	706	16	no	—
60	C₂×D₄	—	706	4	no	—
61	C₂×D₄	C₂×D₃₅₃	1	4	yes	complemented
62	C₂²×C₄	—	353	2	no	—
63	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
64	C₄²	—	353	4	no	—
65	C₂³	—	2	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	—	353	4	no	—
68	C₂³	—	706	8	no	—
69	C₂³	—	706	4	no	—
70	C₂³	C₂²×D₃₅₃	1	4	yes	—
71	C₂³	C₂²×D₃₅₃	1	1	yes	characteristic
72	D₄	—	2	8	no	—
73	D₄	—	706	32	no	—
74	D₄	—	706	8	no	—
75	C₂×C₄	—	353	8	no	—
76	C₂×C₄	—	353	4	no	—
77	C₂×C₄	C₂²⋊D₃₅₃	1	4	yes	complemented
78	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
79	C₂²	—	2	8	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	8	no	—
82	C₂²	—	1412	8	no	—
83	C₂²	—	706	8	no	—
84	C₂²	—	706	4	no	—
85	C₂²	D₄×D₃₅₃	1	4	yes	—
86	C₂²	C₂×C₂²⋊D₃₅₃	1	2	yes	—
87	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
88	C₄	—	353	8	no	—
89	C₄	C₂×C₂²⋊D₃₅₃	1	4	yes	complemented
90	C₂	—	4	4	no	—
91	C₂	—	1412	4	no	—
92	C₂	#26014988	1	4	yes	direct product factor
93	C₂	C₂×D₄×D₃₅₃	1	2	yes	—
94	C₂	C₂²×C₂²⋊D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₃
commutator, cosocle	C₂²×C₃₅₃	C₂⁴
Frattini	C₂²	C₂³×D₃₅₃
Fitting	C₂²×D₄×C₃₅₃	C₂
socle	C₂³×C₃₅₃	C₂³
hypercenter	C₂²×D₄	D₃₅₃
hypocenter	C₃₅₃	C₂×#127
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×D₁₄₁₂	C₂	semidirect
2. C₂²×C₂²⋊D₃₅₃	C₂	semidirect
3. #26014988	C₂	direct (N × H)
4. C₂×C₄×Dic₃₅₃	C₂	semidirect
5. C₂²×D₄×C₃₅₃	C₂	semidirect
8. C₂×D₁₄₁₂	C₂²	semidirect
10. C₂×C₂²⋊D₃₅₃	C₂²	semidirect
12. C₄×Dic₃₅₃	C₂²	semidirect
15. C₂×D₄×C₃₅₃	C₂²	semidirect
22. C₂×Dic₃₅₃	D₄	semidirect
29. C₂×C₄×C₃₅₃	D₄	semidirect
40. C₄×C₃₅₃	C₂×D₄	semidirect
41. Dic₃₅₃	C₂×D₄	semidirect
43. C₂×C₃₅₃	#127	semidirect
47. C₃₅₃	C₂×#127	semidirect
51. C₂²×D₄	D₃₅₃	semidirect
61. C₂×D₄	C₂×D₃₅₃	semidirect
77. C₂×C₄	C₂²⋊D₃₅₃	semidirect
89. C₄	C₂×C₂²⋊D₃₅₃	semidirect
92. C₂	#26014988	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₄₁₂	2
C₂²×C₂²⋊D₃₅₃	2
#26014988	2
C₂×C₄×Dic₃₅₃	2
C₂²×D₄×C₃₅₃	2
C₂×#127	353

Subgroups — C2×#26014988 #36378326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26014988 #36378326