Subgroups — C₂×#27212284 #41002819

Order 26944.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269755	C₂	1	1	yes	maximal, characteristic
3	#27212284	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₄₂	—	2	1	no	—
6	C₂×Dic₈₄₂	—	2	1	no	—
7	C₂×Dic₈₄₂	—	2	1	no	—
8	C₂×Dic₈₄₂	—	2	1	no	—
9	C₂×Dic₈₄₂	C₄	1	2	yes	—
10	C₂×Dic₈₄₂	C₂²	1	2	yes	—
11	C₂²×Dic₄₂₁	C₄	1	2	yes	—
12	#20269755	—	2	1	no	—
13	#20269755	C₂²	1	2	yes	—
14	C₂×#16809890	—	2	1	no	—
15	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
16	M₁₆×C₄₂₁	—	2	1	no	—
17	M₁₆×C₄₂₁	C₂²	1	2	yes	—
18	C₂×C₈×C₄₂₁	—	2	1	no	—
19	Dic₈₄₂	—	2	2	no	—
20	Dic₈₄₂	—	2	2	no	—
21	Dic₈₄₂	—	2	2	no	—
22	Dic₈₄₂	—	2	2	no	—
23	C₂×Dic₄₂₁	—	2	2	no	—
24	C₂×Dic₄₂₁	—	2	2	no	—
25	C₂×Dic₄₂₁	C₂×C₄	1	4	yes	—
26	#16809890	—	2	2	no	—
27	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
30	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
32	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
33	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
34	C₈×C₄₂₁	—	2	2	no	—
35	C₂²×C₄₂₁	—	2	2	no	—
36	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
42	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
43	Dic₄₂₁	—	2	4	no	—
44	C₂×C₄₂₁	—	2	1	no	—
45	C₂×C₄₂₁	—	2	1	no	—
46	C₂×C₄₂₁	#101	1	2	yes	complemented
47	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄₂₁	C₂×#101	1	1	yes	characteristic, complemented, Sylow
49	C₂×#101	—	421	1	no	maximal, Sylow
50	C₂²×Q₈	—	421	1	no	—
51	C₂×M₁₆	—	421	1	no	—
52	C₂×M₁₆	D₄₂₁	1	1	yes	characteristic
53	#101	—	421	4	no	—
54	C₂×Q₈	—	842	1	no	—
55	C₂×Q₈	—	842	1	no	—
56	C₂×Q₈	—	842	1	no	—
57	C₂×Q₈	—	842	1	no	—
58	C₂×Q₈	—	421	2	no	—
59	C₂×Q₈	—	421	2	no	—
60	C₂²×C₄	—	421	2	no	—
61	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
62	M₁₆	—	2	1	no	—
63	M₁₆	—	842	1	no	—
64	M₁₆	—	421	2	no	—
65	M₁₆	C₂×D₄₂₁	1	2	yes	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	842	1	no	—
68	C₂³	C₄×D₄₂₁	1	1	yes	characteristic
69	Q₈	—	842	2	no	—
70	Q₈	—	842	2	no	—
71	Q₈	—	842	2	no	—
72	Q₈	—	842	2	no	—
73	C₂×C₄	—	842	2	no	—
74	C₂×C₄	—	842	2	no	—
75	C₂×C₄	—	421	4	no	—
76	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
79	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
80	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	842	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	#20269748	1	1	yes	characteristic
86	C₂²	#20269748	1	1	yes	characteristic
87	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
88	C₄	—	842	4	no	—
89	C₄	#20269748	1	1	yes	characteristic
90	C₄	#20269748	1	1	yes	characteristic
91	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
92	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	#27212284	1	2	yes	direct product factor
96	C₂	C₂×#20269748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269748
commutator	C₂²×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×M₁₆×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	C₂³×C₄₂₁	C₂×C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₂×M₁₆	D₄₂₁
hypocenter	C₄₂₁	C₂×#101
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27212284	C₂	direct (N × H)
46. C₂×C₄₂₁	#101	semidirect
48. C₄₂₁	C₂×#101	semidirect
95. C₂	#27212284	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×Dic₈₄₂	2
C₂×#20269755	2
#27212284	2
C₂×M₁₆×C₄₂₁	2
C₂×#101	421

Subgroups — C2×#27212284 #41002819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212284 #41002819