Subgroups — #27171215

Order 12928.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265141	C₂	1	2	yes	maximal
2	#20264946	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108058	C₂²	1	4	yes	—
5	C₂²⋊#11717579	—	2	4	no	—
6	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
7	#117×C₁₀₁	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	2	yes	—
10	C₂×C₄²×C₁₀₁	C₄	1	2	yes	—
11	#13108058	—	2	8	no	—
12	C₂×#11717579	—	2	4	no	—
13	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₁	—	2	4	no	—
15	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
18	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₀₁	—	2	4	no	—
20	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
21	C₄²×C₁₀₁	—	2	4	no	—
22	C₂³×C₁₀₁	—	2	2	no	—
23	C₂³×C₁₀₁	—	4	2	no	—
24	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
25	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	—	2	2	no	—
27	C₂×C₄×C₁₀₁	—	2	8	no	—
28	C₂×C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄×C₁₀₁	—	4	4	no	—
30	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₀₁	#41	1	4	yes	—
33	#11717579	—	4	4	no	—
34	C₂²×C₁₀₁	—	2	2	no	—
35	C₂²×C₁₀₁	—	2	2	no	—
36	C₂²×C₁₀₁	—	2	4	no	—
37	C₂²×C₁₀₁	—	4	4	no	—
38	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₀₁	#117	1	2	yes	—
40	C₄×C₁₀₁	—	2	2	no	—
41	C₄×C₁₀₁	—	4	4	no	—
42	C₄×C₁₀₁	#123	1	4	yes	—
43	C₂×C₁₀₁	—	2	2	no	—
44	C₂×C₁₀₁	—	4	2	no	—
45	C₂×C₁₀₁	#381	1	1	yes	characteristic
46	C₂×C₁₀₁	#413	1	2	yes	—
47	#1822	—	101	1	no	maximal, Sylow
48	C₁₀₁	#1822	1	1	yes	characteristic, complemented, Sylow
49	C₂×#117	D₁₀₁	1	1	yes	characteristic
50	#407	—	101	2	no	—
51	#343	—	101	4	no	—
52	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
53	C₂×M₁₆	—	101	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₁₀₁	1	2	yes	—
57	C₂×C₄²	Dic₁₀₁	1	2	yes	—
58	C₂²⋊C₈	—	202	4	no	—
59	C₂⁴	C₂×Dic₁₀₁	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
62	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
63	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
64	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
65	M₁₆	—	202	8	no	—
66	C₂×C₈	—	202	4	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	4	2	no	—
72	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
73	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
80	C₂×C₄	#13108074	1	4	yes	—
81	C₈	—	404	4	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	4	no	—
86	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
87	C₂²	#15692214	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#15692238	1	4	yes	—
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#20265040	1	1	yes	characteristic
94	C₂	#20265161	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#117×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	#1822
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #20264946	C₂	semidirect
48. C₁₀₁	#1822	semidirect

Maximal subgroups

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

subgroup	index
#20265141	2
#20264946	2
C₂×#117×C₁₀₁	2
#1822	101