Subgroups — #28360913

Order 13952.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812610	C₂	1	2	yes	maximal
2	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724784	—	2	2	no	—
4	#16812610	—	2	2	no	—
5	#16812610	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
8	#95×C₁₀₉	—	2	2	no	—
9	C₂×#11724784	—	2	8	no	—
10	C₂×#11724784	—	2	4	no	—
11	C₂²×C₄×C₁₀₉	—	2	2	no	—
12	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
14	C₄⋊C₄×C₁₀₉	—	4	2	no	—
15	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
16	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
17	C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
18	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄×C₁₀₉	—	2	4	no	—
22	C₂×C₄×C₁₀₉	—	4	2	no	—
23	C₂×C₄×C₁₀₉	—	4	4	no	—
24	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	#11724784	—	2	8	no	—
29	C₂²×C₁₀₉	#95	1	1	yes	characteristic
30	C₂²×C₁₀₉	#100	1	1	yes	characteristic
31	C₂²×C₁₀₉	#101	1	1	yes	characteristic
32	C₂²×C₁₀₉	#104	1	4	yes	—
33	C₄×C₁₀₉	—	2	2	no	—
34	C₄×C₁₀₉	—	2	2	no	—
35	C₄×C₁₀₉	—	2	1	no	—
36	C₄×C₁₀₉	—	2	1	no	—
37	C₄×C₁₀₉	—	8	2	no	—
38	C₂×C₁₀₉	#329	1	2	yes	—
39	C₂×C₁₀₉	#330	1	2	yes	—
40	C₂×C₁₀₉	#339	1	2	yes	—
41	C₂×C₁₀₉	#343	1	1	yes	characteristic
42	#1295	—	109	1	no	maximal, Sylow
43	C₁₀₉	#1295	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	109	2	no	—
45	#382	D₁₀₉	1	1	yes	characteristic
46	C₂²×C₈	—	218	2	no	—
47	C₂×C₄⋊C₄	Dic₁₀₉	1	2	yes	—
48	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
49	#97	—	109	4	no	—
50	#97	—	218	2	no	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
55	C₂×C₈	—	218	8	no	—
56	C₂×C₈	—	218	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	Dic₂₁₈	1	1	yes	characteristic
59	C₄²	C₄×D₁₀₉	1	2	yes	—
60	C₄²	D₄₃₆	1	1	yes	characteristic
61	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
68	C₂×C₄	#13182006	1	2	yes	—
69	C₂×C₄	#13182008	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
71	C₈	—	218	8	no	—
72	C₂²	#16812606	1	1	yes	characteristic
73	C₂²	#16812616	1	1	yes	characteristic
74	C₂²	#16812618	1	1	yes	characteristic
75	C₂²	#16812626	1	4	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	8	2	no	—
81	C₂	#20294766	1	2	yes	—
82	C₂	#20294768	1	2	yes	—
83	C₂	#20294785	1	2	yes	—
84	C₂	#20294795	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₀₉
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₉
Fitting	#382×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₀₉	C₂²
hypercenter	#382	D₁₀₉
hypocenter	C₁₀₉	#1295
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
5. #16812610	C₄	semidirect
43. C₁₀₉	#1295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812610	2
#382×C₁₀₉	2
#1295	109