Subgroups — #27171262

Order 12928.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
2	#20265043	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	4	yes	maximal
4	#382×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
7	C₄×#11717579	C₄	1	4	yes	complemented
8	#15692162	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
11	#95×C₁₀₁	—	2	2	no	—
12	C₂²×Dic₁₀₁	—	2	2	no	—
13	C₂×#11717579	—	2	2	no	—
14	C₂×#11717579	—	2	2	no	—
15	C₂×#11717579	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₀₁	—	2	4	no	—
17	C₂²×C₄×C₁₀₁	—	2	2	no	—
18	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
19	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₁	—	2	4	no	—
21	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
22	C₄²×C₁₀₁	Q₈	1	2	yes	—
23	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₁	—	2	4	no	—
25	C₂×C₄×C₁₀₁	—	4	2	no	—
26	C₂×C₄×C₁₀₁	—	4	4	no	—
27	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	#41	1	2	yes	—
31	C₂×C₄×C₁₀₁	#41	1	2	yes	—
32	C₂×Dic₁₀₁	—	4	2	no	—
33	C₂×Dic₁₀₁	—	4	4	no	—
34	#11717579	—	2	4	no	—
35	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₀₁	#124	1	1	yes	characteristic
37	C₂²×C₁₀₁	#124	1	1	yes	characteristic
38	C₂²×C₁₀₁	#135	1	4	yes	—
39	C₄×C₁₀₁	—	2	4	no	—
40	C₄×C₁₀₁	—	8	2	no	—
41	C₄×C₁₀₁	#117	1	2	yes	—
42	C₄×C₁₀₁	#125	1	2	yes	—
43	Dic₁₀₁	—	8	2	no	—
44	C₂×C₁₀₁	#382	1	1	yes	characteristic
45	C₂×C₁₀₁	#427	1	2	yes	—
46	C₂×C₁₀₁	#488	1	2	yes	—
47	C₂×C₁₀₁	#488	1	2	yes	—
48	#1835	—	101	1	no	maximal, Sylow
49	C₁₀₁	#1835	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄×C₈	—	101	1	no	—
51	#382	—	101	1	no	—
52	#382	D₁₀₁	1	1	yes	characteristic
53	#340	—	101	4	no	—
54	C₂²×C₈	—	101	2	no	—
55	C₂×C₄⋊C₄	—	101	2	no	—
56	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
57	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
58	C₄×C₈	—	101	4	no	—
59	#95	—	2	2	no	—
60	#95	—	202	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	202	2	no	—
63	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
65	C₂×C₈	—	101	4	no	—
66	C₂×C₈	—	202	2	no	—
67	C₂×C₈	—	202	2	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	202	4	no	—
70	C₄²	Dic₂₀₂	1	2	yes	—
71	C₄²	C₄×D₁₀₁	1	2	yes	—
72	C₂³	C₂×C₂²⋊D₁₀₁	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	—	404	2	no	—
77	C₂×C₄	—	404	4	no	—
78	C₂×C₄	#13108053	1	4	yes	—
79	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
81	C₂×C₄	#13108073	1	2	yes	—
82	C₂×C₄	#13108075	1	2	yes	—
83	C₈	—	202	4	no	—
84	C₂²	C₂×#13108053	1	1	yes	characteristic
85	C₂²	#15692242	1	1	yes	characteristic
86	C₂²	#15692244	1	1	yes	characteristic
87	C₂²	#15692275	1	4	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	—	808	2	no	—
91	C₄	#15692213	1	2	yes	—
92	C₄	#15692246	1	2	yes	—
93	C₂	#20265042	1	1	yes	characteristic
94	C₂	#20265213	1	2	yes	—
95	C₂	#20265531	1	2	yes	—
96	C₂	#20265534	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#382×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#382	D₁₀₁
hypocenter	C₁₀₁	#1835
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
7. C₄×#11717579	C₄	semidirect
49. C₁₀₁	#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×#11717579	2
#20265043	2
#20264939	2
#382×C₁₀₁	2
#1835	101