Subgroups — #41013528

Order 27008.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#27213012	C₂	1	1	yes	maximal, characteristic
3	#27212908	C₂	1	4	yes	maximal
4	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
7	C₄×C₂₁₁⋊C₈	C₄	1	4	yes	complemented
8	C₂₁₁⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	#95×C₂₁₁	—	2	2	no	—
12	C₂²×Dic₂₁₁	—	2	2	no	—
13	C₂×C₂₁₁⋊C₈	—	2	2	no	—
14	C₂×C₂₁₁⋊C₈	—	2	2	no	—
15	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₂₁₁	—	2	4	no	—
17	C₂²×C₄×C₂₁₁	—	2	2	no	—
18	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₁₁	—	2	4	no	—
21	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₄²×C₂₁₁	Q₈	1	2	yes	—
23	C₂×Dic₂₁₁	—	4	2	no	—
24	C₂×Dic₂₁₁	—	4	4	no	—
25	C₂₁₁⋊C₈	—	2	4	no	—
26	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	—	2	4	no	—
28	C₂×C₄×C₂₁₁	—	4	2	no	—
29	C₂×C₄×C₂₁₁	—	4	4	no	—
30	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	#41	1	2	yes	—
34	C₂×C₄×C₂₁₁	#41	1	2	yes	—
35	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₂₁₁	#124	1	1	yes	characteristic
37	C₂²×C₂₁₁	#124	1	1	yes	characteristic
38	C₂²×C₂₁₁	#135	1	4	yes	—
39	C₄×C₂₁₁	—	2	4	no	—
40	C₄×C₂₁₁	—	8	2	no	—
41	C₄×C₂₁₁	#117	1	2	yes	—
42	C₄×C₂₁₁	#125	1	2	yes	—
43	Dic₂₁₁	—	8	2	no	—
44	C₂×C₂₁₁	#382	1	1	yes	characteristic
45	C₂×C₂₁₁	#427	1	2	yes	—
46	C₂×C₂₁₁	#488	1	2	yes	—
47	C₂×C₂₁₁	#488	1	2	yes	—
48	C₂₁₁	#1835	1	1	yes	characteristic, complemented, Sylow
49	#1835	—	211	1	no	maximal, Sylow
50	C₂×C₄×C₈	—	211	1	no	—
51	#382	—	211	1	no	—
52	#382	D₂₁₁	1	1	yes	characteristic
53	#340	—	211	4	no	—
54	C₂²×C₈	—	211	2	no	—
55	C₂×C₄⋊C₄	—	211	2	no	—
56	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
57	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
58	C₄×C₈	—	211	4	no	—
59	#95	—	2	2	no	—
60	#95	—	422	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	422	2	no	—
63	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
64	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
65	C₂×C₈	—	422	2	no	—
66	C₂×C₈	—	422	2	no	—
67	C₂×C₈	—	211	4	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	422	4	no	—
70	C₄²	C₄×D₂₁₁	1	2	yes	—
71	C₄²	Dic₄₂₂	1	2	yes	—
72	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	844	2	no	—
77	C₂×C₄	—	844	4	no	—
78	C₂×C₄	Dic₂₁₁⋊C₄	1	4	yes	—
79	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
81	C₂×C₄	#16809961	1	2	yes	—
82	C₂×C₄	#16809963	1	2	yes	—
83	C₈	—	422	4	no	—
84	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
85	C₂²	#20269964	1	1	yes	characteristic
86	C₂²	#20269966	1	1	yes	characteristic
87	C₂²	#20269997	1	4	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	—	1688	2	no	—
91	C₄	#20269935	1	2	yes	—
92	C₄	#20269968	1	2	yes	—
93	C₂	#27213011	1	1	yes	characteristic
94	C₂	#27213182	1	2	yes	—
95	C₂	#27213500	1	2	yes	—
96	C₂	#27213503	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#382×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#382	D₂₁₁
hypocenter	C₂₁₁	#1835
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
7. C₄×C₂₁₁⋊C₈	C₄	semidirect
48. C₂₁₁	#1835	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₁₁⋊C₈	2
#27213012	2
#27212908	2
#382×C₂₁₁	2
#1835	211