Subgroups — C₂×#27212388 #41003622

Order 26944.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269771	C₂	1	2	yes	maximal, complemented
2	#27212388	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₁₆₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#20269760	C₂	1	2	yes	maximal, complemented
5	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#20269771	—	2	4	no	—
8	#20269771	C₂²	1	8	yes	complemented
9	C₂×Dic₈₄₂	C₂²	1	2	yes	—
10	C₂×D₁₆₈₄	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₄₂₁	—	2	2	no	—
12	C₂×C₄×D₄₂₁	—	2	2	no	—
13	Dic₁₆₈₄	C₂²	1	4	yes	complemented
14	#20269760	C₂²	1	8	yes	complemented
15	D₃₃₆₈	C₂²	1	4	yes	complemented
16	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₄₂₁	C₂²	1	4	yes	complemented
20	C₂²×D₄₂₁	—	2	2	no	—
21	Dic₈₄₂	—	2	2	no	—
22	Dic₈₄₂	C₂³	1	4	yes	—
23	D₁₆₈₄	—	2	2	no	—
24	D₁₆₈₄	C₂³	1	4	yes	—
25	C₂²⋊D₄₂₁	—	2	8	no	—
26	C₂×Dic₄₂₁	—	2	2	no	—
27	C₄×D₄₂₁	—	2	8	no	—
28	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
30	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
33	C₈×C₄₂₁	C₂³	1	2	yes	—
34	C₈×C₄₂₁	C₂³	1	2	yes	—
35	C₂²×C₄₂₁	—	2	2	no	—
36	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
37	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
38	C₂×D₄₂₁	—	2	4	no	—
39	C₂×D₄₂₁	—	4	4	no	—
40	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
41	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
42	C₄×C₄₂₁	C₂⁴	1	1	yes	characteristic
43	Dic₄₂₁	—	2	4	no	—
44	C₂×C₄₂₁	—	2	2	no	—
45	C₂×C₄₂₁	#135	1	2	yes	complemented
46	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
47	D₄₂₁	—	4	4	no	—
48	C₄₂₁	C₂×#135	1	1	yes	characteristic, complemented, Sylow
49	C₂×#135	—	421	1	no	maximal, Sylow
50	C₂×#41	—	421	2	no	—
51	#135	—	421	8	no	—
52	C₂×Q₁₆	—	421	1	no	—
53	C₂×SD₁₆	—	421	2	no	—
54	C₂×D₈	—	421	1	no	—
55	C₂²×C₈	D₄₂₁	1	1	yes	characteristic, complemented
56	#41	—	842	4	no	—
57	#41	—	421	8	no	—
58	C₂×Q₈	—	421	2	no	—
59	C₂×D₄	—	842	2	no	—
60	C₂×D₄	—	421	2	no	—
61	C₂²×C₄	—	842	2	no	—
62	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
63	Q₁₆	—	421	4	no	—
64	SD₁₆	—	421	8	no	—
65	D₈	—	421	4	no	—
66	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
68	C₂×C₈	C₂×D₄₂₁	1	4	yes	complemented
69	C₂³	—	842	2	no	—
70	C₂³	D₁₆₈₄	1	1	yes	characteristic
71	Q₈	—	842	2	no	—
72	Q₈	—	421	4	no	—
73	D₄	—	842	8	no	—
74	D₄	—	842	2	no	—
75	D₄	—	421	4	no	—
76	C₂×C₄	—	842	2	no	—
77	C₂×C₄	—	842	8	no	—
78	C₂×C₄	D₁₆₈₄	1	2	yes	—
79	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
82	C₈	C₂²×D₄₂₁	1	2	yes	—
83	C₈	C₂²×D₄₂₁	1	2	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	842	4	no	—
86	C₂²	—	1684	4	no	—
87	C₂²	C₂×D₁₆₈₄	1	2	yes	—
88	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
89	C₄	—	842	4	no	—
90	C₄	C₂×D₁₆₈₄	1	2	yes	—
91	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
92	C₄	C₂³×D₄₂₁	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	—	1684	4	no	—
95	C₂	#27212388	1	2	yes	direct product factor
96	C₂	C₂²×D₁₆₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₆₈₄
commutator, cosocle	C₄×C₄₂₁	C₂⁴
Frattini	C₄	C₂³×D₄₂₁
Fitting	C₂²×C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂²×C₈	D₄₂₁
hypocenter	C₄₂₁	C₂×#135
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#20269771	C₂	semidirect
2. #27212388	C₂	direct (N × H)
3. C₂×Dic₁₆₈₄	C₂	semidirect
4. C₂×#20269760	C₂	semidirect
5. C₂×D₃₃₆₈	C₂	semidirect
6. C₂²×C₈×C₄₂₁	C₂	semidirect
8. #20269771	C₂²	semidirect
13. Dic₁₆₈₄	C₂²	semidirect
14. #20269760	C₂²	semidirect
15. D₃₃₆₈	C₂²	semidirect
19. C₂×C₈×C₄₂₁	C₂²	semidirect
45. C₂×C₄₂₁	#135	semidirect
48. C₄₂₁	C₂×#135	semidirect
55. C₂²×C₈	D₄₂₁	semidirect
68. C₂×C₈	C₂×D₄₂₁	semidirect
95. C₂	#27212388	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20269771	2
#27212388	2
C₂×Dic₁₆₈₄	2
C₂×#20269760	2
C₂×D₃₃₆₈	2
C₂²×C₈×C₄₂₁	2
C₂×#135	421

Subgroups — C2×#27212388 #41003622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212388 #41003622