Subgroups — #28360997

Order 13952.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic
2	#20294779	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic
5	C₄×Dic₂₁₈	—	2	1	no	—
6	#16812633	C₄	1	1	yes	characteristic
7	C₄⋊#11724784	—	2	1	no	—
8	#16812624	—	2	1	no	—
9	C₂×C₁₆×C₁₀₉	—	2	1	no	—
10	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₈	—	2	1	no	—
12	Dic₄₃₆	—	2	1	no	—
13	Dic₄₃₆	C₈	1	2	yes	complemented
14	C₂×#11724784	—	4	1	no	—
15	C₄⋊Dic₁₀₉	—	2	1	no	—
16	#13182008	—	4	1	no	—
17	C₄×Dic₁₀₉	—	4	1	no	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
21	C₁₆×C₁₀₉	—	2	2	no	—
22	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₁₀₉	—	4	1	no	—
26	C₂×Dic₁₀₉	—	4	1	no	—
27	Dic₂₁₈	—	2	2	no	—
28	Dic₂₁₈	—	4	1	no	—
29	C₈×C₁₀₉	—	2	1	no	—
30	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
32	#11724784	—	8	1	no	—
33	C₂²×C₁₀₉	#102	1	1	yes	characteristic
34	C₄×C₁₀₉	—	2	1	no	—
35	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₀₉	#104	1	1	yes	characteristic
37	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
38	C₄×C₁₀₉	Q₃₂	1	1	yes	characteristic
39	Dic₁₀₉	—	4	2	no	—
40	Dic₁₀₉	—	8	1	no	—
41	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₀₉	#358	1	1	yes	characteristic
43	C₂×C₁₀₉	#359	1	1	yes	characteristic
44	#1332	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1332	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	109	1	no	—
47	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
48	#335	—	109	1	no	—
49	C₂×Q₁₆	—	109	1	no	—
50	C₄×Q₈	—	218	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	#107	—	109	1	no	—
53	C₄⋊C₈	—	218	1	no	—
54	#103	—	218	1	no	—
55	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
56	C₂×Q₈	—	218	1	no	—
57	Q₁₆	—	109	2	no	—
58	Q₁₆	—	218	1	no	—
59	C₂×C₈	—	436	1	no	—
60	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
62	C₄⋊C₄	—	436	1	no	—
63	C₄⋊C₄	—	218	1	no	—
64	C₄²	—	436	1	no	—
65	C₄²	D₄₃₆	1	1	yes	characteristic
66	C₁₆	—	2	2	no	—
67	Q₈	—	436	1	no	—
68	Q₈	—	218	2	no	—
69	C₂×C₄	—	436	1	no	—
70	C₂×C₄	—	436	1	no	—
71	C₂×C₄	#13182006	1	1	yes	characteristic
72	C₂×C₄	D₈₇₂	1	1	yes	characteristic
73	C₂×C₄	#13182018	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	872	1	no	—
76	C₈	C₈×D₁₀₉	1	1	yes	characteristic
77	C₈	#13182012	1	1	yes	characteristic
78	C₂²	#16812621	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	872	1	no	—
81	C₄	—	436	2	no	—
82	C₄	#16812611	1	1	yes	characteristic
83	C₄	#16812627	1	1	yes	characteristic
84	C₄	#16812644	1	1	yes	characteristic
85	C₄	Dic₈₇₂	1	1	yes	characteristic
86	C₂	#20294755	1	1	yes	characteristic
87	C₂	#20294831	1	1	yes	characteristic
88	C₂	#20294834	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈₇₂
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#102
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	#1332
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. Dic₄₃₆	C₈	semidirect
45. C₁₀₉	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₃₆	2
#20294779	2
C₄×C₁₆×C₁₀₉	2
#1332	109