Subgroups — #27187054

Order 12928.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266121	C₂	1	2	yes	maximal
2	#20266051	C₂	1	4	yes	maximal
3	C₄×C₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#87938	C₂²	1	4	yes	—
5	(C₂×#87938)⋊C₄	—	2	4	no	—
6	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
7	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
8	C₄×#11717579	C₄	1	1	yes	characteristic, complemented
9	C₈×Dic₁₀₁	C₄	1	2	yes	complemented
10	C₄×C₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
11	C₂²×#87938	—	2	4	no	—
12	C₄⋊#87938	—	2	8	no	—
13	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
14	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
15	C₂×#11717579	C₂×C₄	1	2	yes	—
16	C₈×D₁₀₁	—	2	4	no	—
17	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₁₀₁	Q₈	1	2	yes	—
19	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
20	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
22	C₂×#87938	—	4	4	no	—
23	C₂×#87938	—	4	8	no	—
24	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
27	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
28	C₄×D₁₀₁	—	2	4	no	—
29	C₄×D₁₀₁	#41	1	4	yes	—
30	C₈×C₁₀₁	—	2	2	no	—
31	#11717579	—	2	2	no	—
32	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₁₀₁	#124	1	2	yes	—
34	C₂×D₁₀₁	#135	1	4	yes	—
35	#87938	—	8	4	no	—
36	C₄×C₁₀₁	—	2	2	no	—
37	C₄×C₁₀₁	#117	1	2	yes	—
38	Dic₁₀₁	—	2	2	no	—
39	Dic₁₀₁	#125	1	2	yes	—
40	C₂×C₁₀₁	#382	1	1	yes	characteristic
41	C₂×C₁₀₁	#427	1	2	yes	—
42	D₁₀₁	#488	1	4	yes	—
43	#1835	—	101	1	no	maximal, Sylow
44	C₁₀₁	#1835	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₈	—	101	1	no	—
46	#382	—	101	2	no	—
47	#340	—	101	4	no	—
48	C₂²×C₈	—	101	2	no	—
49	C₂×C₄⋊C₄	—	101	4	no	—
50	C₂×C₄²	—	101	1	no	—
51	C₄×C₈	—	101	2	no	—
52	C₄×C₈	—	101	1	no	—
53	C₄×C₈	#87938	1	1	yes	characteristic, complemented
54	#95	—	202	4	no	—
55	C₂²×C₄	—	101	1	no	—
56	C₂²×C₄	—	101	2	no	—
57	C₂²×C₄	—	202	4	no	—
58	C₂×C₈	—	101	2	no	—
59	C₂×C₈	—	202	4	no	—
60	C₂×C₈	C₂×#87938	1	2	yes	—
61	C₄⋊C₄	—	202	8	no	—
62	C₄²	—	101	2	no	—
63	C₄²	—	101	1	no	—
64	C₄²	C₂×#87938	1	1	yes	characteristic
65	C₂³	—	101	1	no	—
66	C₂×C₄	—	101	1	no	—
67	C₂×C₄	—	101	4	no	—
68	C₂×C₄	—	101	2	no	—
69	C₂×C₄	—	404	4	no	—
70	C₂×C₄	—	404	8	no	—
71	C₂×C₄	—	202	4	no	—
72	C₂×C₄	C₄⋊#87938	1	2	yes	—
73	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	202	2	no	—
76	C₂²	—	101	2	no	—
77	C₂²	—	101	4	no	—
78	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	808	4	no	—
81	C₄	—	101	2	no	—
82	C₄	—	202	2	no	—
83	C₄	#15692334	1	2	yes	—
84	C₂	—	101	4	no	—
85	C₂	#20266121	1	1	yes	characteristic
86	C₂	#20266187	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₄×C₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₁	C₂×D₄
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	C₄×C₈	#87938
hypocenter	C₁₀₁	#1835
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×#11717579	C₄	semidirect
9. C₈×Dic₁₀₁	C₄	semidirect
10. C₄×C₈×C₁₀₁	C₄	semidirect
44. C₁₀₁	#1835	semidirect
53. C₄×C₈	#87938	semidirect

Maximal subgroups

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

subgroup	index
#20266121	2
#20266051	2
C₄×C₈×D₁₀₁	2
#1835	101