Subgroups — #28378161

Order 13952.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295935	C₂	1	1	yes	maximal, characteristic
2	#20295933	C₂	1	1	yes	maximal, characteristic
3	#20295418	C₂	1	1	yes	maximal, characteristic
4	#16812760	—	2	1	no	—
5	#16812758	—	2	1	no	—
6	C₂×#13182019	—	2	1	no	—
7	C₂×#13182016	—	2	1	no	—
8	#16812687	C₂²	1	1	yes	characteristic
9	#16812610	C₄	1	1	yes	characteristic, complemented
10	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	#13182038	—	4	1	no	—
12	C₂²⋊#89137	—	4	1	no	—
13	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
14	C₂×D₄₃₆	D₄	1	1	yes	characteristic
15	#13182019	—	4	2	no	—
16	#13182016	—	4	2	no	—
17	C₂×#11724784	—	2	1	no	—
18	#13182006	—	4	1	no	—
19	C₂×Q₈×C₁₀₉	—	2	1	no	—
20	C₄⋊C₄×C₁₀₉	—	4	1	no	—
21	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂×#89137	—	8	1	no	—
24	Q₈×C₁₀₉	—	4	1	no	—
25	Q₈×C₁₀₉	—	4	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	2	1	no	—
30	D₄₃₆	—	4	1	no	—
31	Dic₂₁₈	—	4	1	no	—
32	#11724786	—	8	1	no	—
33	#11724784	—	2	2	no	—
34	C₂²×C₁₀₉	#99	1	1	yes	characteristic
35	C₂×D₁₀₉	—	4	1	no	—
36	C₂×D₁₀₉	—	8	1	no	—
37	#89137	—	16	1	no	—
38	C₄×C₁₀₉	—	2	1	no	—
39	C₄×C₁₀₉	—	4	1	no	—
40	C₄×C₁₀₉	—	8	1	no	—
41	Dic₁₀₉	—	4	1	no	—
42	C₂×C₁₀₉	—	2	1	no	—
43	C₂×C₁₀₉	#353	1	1	yes	characteristic
44	D₁₀₉	—	8	1	no	—
45	#1407	—	109	1	no	maximal, Sylow
46	C₁₀₉	#1407	1	1	yes	characteristic, complemented, Sylow
47	#497	—	109	1	no	—
48	#355	—	109	1	no	—
49	#354	—	109	1	no	—
50	C₂×Q₁₆	—	218	1	no	—
51	C₂×SD₁₆	—	218	1	no	—
52	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
53	#124	—	109	1	no	—
54	#101	—	218	1	no	—
55	#99	—	218	1	no	—
56	#97	—	109	1	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	109	1	no	—
59	C₂×D₄	—	109	1	no	—
60	Q₁₆	—	436	2	no	—
61	SD₁₆	—	436	2	no	—
62	M₁₆	—	436	1	no	—
63	C₂×C₈	—	218	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	436	1	no	—
66	C₂²⋊C₄	—	436	1	no	—
67	C₄²	C₂×#89137	1	1	yes	characteristic
68	C₂³	—	218	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	436	1	no	—
72	D₄	—	436	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	872	1	no	—
76	C₂×C₄	—	218	1	no	—
77	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
78	C₈	—	872	1	no	—
79	C₈	—	218	2	no	—
80	C₂²	—	872	1	no	—
81	C₂²	—	436	1	no	—
82	C₂²	#16812757	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	436	1	no	—
87	C₄	—	1744	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	872	1	no	—
90	C₂	#20295930	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295930
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#353
prime radical	C₂×D₄₃₆	D₄
cosocle	#16812687	C₂²
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1407
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. #16812610	C₄	semidirect
10. C₄⋊Q₈×C₁₀₉	C₄	semidirect
46. C₁₀₉	#1407	semidirect
52. C₄⋊Q₈	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295935	2
#20295933	2
#20295418	2
#1407	109