Subgroups — C₂×#27212888 #41011982

Order 27008.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269888	C₂	1	2	yes	maximal
2	#27212888	C₂	1	4	yes	maximal, direct product factor
3	C₂×#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₁₁⋊C₈	—	2	2	no	—
5	#20269888	—	2	2	no	—
6	#20269888	C₂²	1	4	yes	—
7	#125×C₂₁₁	—	2	2	no	—
8	#125×C₂₁₁	C₄	1	2	yes	—
9	#125×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₂₁₁⋊C₈	—	2	8	no	—
14	C₂×C₂₁₁⋊C₈	—	2	4	no	—
15	C₂²×C₄×C₂₁₁	—	2	2	no	—
16	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
18	C₄⋊C₄×C₂₁₁	—	2	4	no	—
19	C₄⋊C₄×C₂₁₁	—	2	2	no	—
20	C₄⋊C₄×C₂₁₁	—	2	4	no	—
21	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
22	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₁₁	D₄	1	2	yes	—
24	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂₁₁⋊C₈	—	2	8	no	—
26	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	—	2	2	no	—
28	C₂×C₄×C₂₁₁	—	2	4	no	—
29	C₂×C₄×C₂₁₁	—	2	4	no	—
30	C₂×C₄×C₂₁₁	—	4	4	no	—
31	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
35	C₂²×C₂₁₁	#101	1	2	yes	—
36	C₂²×C₂₁₁	#104	1	4	yes	—
37	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	—	2	2	no	—
41	C₄×C₂₁₁	—	2	2	no	—
42	C₄×C₂₁₁	—	4	4	no	—
43	C₂×C₂₁₁	#330	1	4	yes	complemented
44	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
45	C₂×C₂₁₁	C₂×#104	1	2	yes	—
46	C₂₁₁	C₂×#330	1	1	yes	characteristic, complemented, Sylow
47	C₂×#330	—	211	1	no	maximal, Sylow
48	C₂×#125	D₂₁₁	1	1	yes	characteristic
49	C₂×#97	—	211	2	no	—
50	#330	—	211	4	no	—
51	C₂²×C₈	—	422	2	no	—
52	#125	—	2	2	no	—
53	#125	Dic₂₁₁	1	2	yes	—
54	#125	C₂×D₂₁₁	1	2	yes	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	Dic₂₁₁	1	2	yes	—
57	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
58	#97	—	422	2	no	—
59	#97	—	211	4	no	—
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
62	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
63	C₂×C₈	—	422	8	no	—
64	C₂×C₈	—	422	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
69	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
70	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
71	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
78	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
79	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
80	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
81	C₈	—	422	8	no	—
82	C₂²	#20269896	1	2	yes	—
83	C₂²	#20269904	1	4	yes	—
84	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₂	#27212888	1	4	yes	direct product factor
91	C₂	C₂×#20269896	1	1	yes	characteristic
92	C₂	C₂×#20269904	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	C₂×#125×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	C₂×#125	D₂₁₁
hypocenter	C₂₁₁	C₂×#330
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
2. #27212888	C₂	direct (N × H)
43. C₂×C₂₁₁	#330	semidirect
46. C₂₁₁	C₂×#330	semidirect
90. C₂	#27212888	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×#20269888	2
#27212888	2
C₂×#125×C₂₁₁	2
C₂×#330	211

Subgroups — C2×#27212888 #41011982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212888 #41011982