Subgroups — #28378004

Order 13952.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295923	C₂	1	1	yes	maximal, characteristic
2	#20295894	C₂	1	1	yes	maximal, characteristic
3	#20295281	C₂	1	1	yes	maximal, characteristic
4	#16812778	—	2	1	no	—
5	#16812773	—	2	1	no	—
6	C₂×#13182012	C₂²	1	1	yes	characteristic
7	#16812675	—	2	1	no	—
8	#16812631	C₄	1	1	yes	characteristic, complemented
9	#16812623	—	2	1	no	—
10	C₂×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#89137	—	4	1	no	—
12	C₂²⋊#89137	—	4	1	no	—
13	C₄×#89137	—	2	2	no	—
14	#13182032	—	4	1	no	—
15	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
16	#13182012	—	4	1	no	—
17	C₂×#11724784	D₄	1	1	yes	characteristic
18	C₄⋊Dic₁₀₉	—	2	1	no	—
19	#13182008	—	4	1	no	—
20	#13182006	—	4	1	no	—
21	C₂×Q₈×C₁₀₉	—	2	1	no	—
22	Q₁₆×C₁₀₉	—	4	1	no	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
25	C₂×#89137	—	4	2	no	—
26	Q₈×C₁₀₉	—	4	1	no	—
27	Q₈×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	—	4	1	no	—
29	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	—	4	1	no	—
31	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
32	C₄×D₁₀₉	—	2	1	no	—
33	C₄×D₁₀₉	—	2	1	no	—
34	C₈×C₁₀₉	—	2	1	no	—
35	#11724784	—	2	1	no	—
36	C₂²×C₁₀₉	#102	1	1	yes	characteristic
37	C₂×D₁₀₉	—	2	1	no	—
38	C₂×D₁₀₉	—	4	1	no	—
39	#89137	—	8	2	no	—
40	C₄×C₁₀₉	—	8	1	no	—
41	C₄×C₁₀₉	#99	1	1	yes	characteristic
42	C₄×C₁₀₉	#104	1	1	yes	characteristic
43	Dic₁₀₉	—	2	1	no	—
44	Dic₁₀₉	—	8	1	no	—
45	C₂×C₁₀₉	—	2	1	no	—
46	C₂×C₁₀₉	#327	1	1	yes	characteristic
47	D₁₀₉	—	4	1	no	—
48	#1340	—	109	1	no	maximal, Sylow
49	C₁₀₉	#1340	1	1	yes	characteristic, complemented, Sylow
50	#470	—	109	1	no	—
51	#349	—	109	1	no	—
52	H(3,ℤ/4)	—	109	1	no	—
53	C₂×Q₁₆	#89137	1	1	yes	characteristic, complemented
54	C₂×M₁₆	—	109	1	no	—
55	C₂²⋊Q₈	—	218	1	no	—
56	#117	—	218	1	no	—
57	M₃₂	—	218	1	no	—
58	#106	—	109	1	no	—
59	#103	—	218	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂²×C₄	—	109	1	no	—
62	Q₁₆	—	4	1	no	—
63	M₁₆	—	436	1	no	—
64	C₂×C₈	—	109	1	no	—
65	C₂×C₈	C₂×#89137	1	1	yes	characteristic
66	C₄⋊C₄	—	436	1	no	—
67	C₄⋊C₄	—	436	1	no	—
68	C₄⋊C₄	—	218	1	no	—
69	C₂²⋊C₄	—	436	1	no	—
70	C₂²⋊C₄	—	436	1	no	—
71	C₄²	—	218	2	no	—
72	C₁₆	—	436	1	no	—
73	C₂³	—	109	1	no	—
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	436	2	no	—
79	C₂×C₄	—	436	1	no	—
80	C₂×C₄	—	218	1	no	—
81	C₂×C₄	—	218	1	no	—
82	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	218	1	no	—
85	C₂²	—	436	1	no	—
86	C₂²	—	218	1	no	—
87	C₂²	#16812761	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	872	1	no	—
90	C₄	—	872	2	no	—
91	C₄	—	218	1	no	—
92	C₄	#16812757	1	1	yes	characteristic
93	C₄	#16812763	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	436	1	no	—
96	C₂	#20295877	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295877
commutator	C₂×C₈×C₁₀₉	C₂×C₄
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#327
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₂×#13182012	C₂²
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	#1340
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
8. #16812631	C₄	semidirect
10. C₂×Q₁₆×C₁₀₉	C₄	semidirect
49. C₁₀₉	#1340	semidirect
53. C₂×Q₁₆	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295923	2
#20295894	2
#20295281	2
#1340	109