Subgroups — #28361007

Order 13952.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295144	C₂	1	1	yes	maximal, characteristic
2	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812721	C₄	1	1	yes	characteristic
5	#16812705	—	2	1	no	—
6	#16812635	C₄	1	1	yes	characteristic
7	#16812627	—	2	1	no	—
8	C₄×#11724784	—	2	1	no	—
9	M₃₂×C₁₀₉	—	2	1	no	—
10	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	#13182029	—	2	1	no	—
12	Dic₄₃₆	C₈	1	1	yes	characteristic, complemented
13	#13182018	—	2	1	no	—
14	D₈₇₂	C₈	1	1	yes	characteristic, complemented
15	#13182013	—	2	1	no	—
16	#13182012	—	4	1	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₈×D₁₀₉	—	4	1	no	—
19	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
22	C₁₆×C₁₀₉	—	4	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₁₀₉	—	4	1	no	—
26	D₄₃₆	—	2	1	no	—
27	C₄×D₁₀₉	—	4	1	no	—
28	Dic₂₁₈	—	2	1	no	—
29	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
31	C₈×C₁₀₉	D₈	1	1	yes	characteristic
32	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
33	#11724784	—	4	2	no	—
34	#11724784	—	4	1	no	—
35	C₂²×C₁₀₉	#104	1	1	yes	characteristic
36	C₂×D₁₀₉	—	4	1	no	—
37	C₄×C₁₀₉	—	2	2	no	—
38	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	#102	1	1	yes	characteristic
40	Dic₁₀₉	—	4	1	no	—
41	C₂×C₁₀₉	—	2	1	no	—
42	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
43	D₁₀₉	—	8	1	no	—
44	#1336	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1336	1	1	yes	characteristic, complemented, Sylow
46	#443	—	109	1	no	—
47	#364	D₁₀₉	1	1	yes	characteristic, complemented
48	#322	—	109	1	no	—
49	#135	—	109	1	no	—
50	#131	—	218	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	109	1	no	—
53	#104	—	218	1	no	—
54	C₄×C₈	—	218	1	no	—
55	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
56	#41	—	218	1	no	—
57	Q₁₆	—	109	1	no	—
58	SD₁₆	—	218	1	no	—
59	D₈	—	109	1	no	—
60	M₁₆	—	436	1	no	—
61	M₁₆	—	218	1	no	—
62	C₂×C₈	—	436	1	no	—
63	C₂×C₈	—	218	2	no	—
64	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
66	C₄²	D₄₃₆	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	218	1	no	—
69	D₄	—	436	1	no	—
70	D₄	—	218	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	436	1	no	—
73	C₂×C₄	#13182006	1	1	yes	characteristic
74	C₈	—	436	2	no	—
75	C₈	—	436	1	no	—
76	C₈	C₈×D₁₀₉	1	1	yes	characteristic
77	C₈	#13182012	1	1	yes	characteristic
78	C₈	#13182014	1	1	yes	characteristic
79	C₈	#13182016	1	1	yes	characteristic
80	C₂²	—	436	1	no	—
81	C₂²	#16812625	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	436	1	no	—
84	C₄	#16812611	1	1	yes	characteristic
85	C₄	#16812619	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	872	1	no	—
88	C₂	#20294753	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812619
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	#364×C₁₀₉	C₂
socle	C₂×C₁₀₉	D₄⋊C₈
prime radical	#16812721	C₄
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	#364	D₁₀₉
hypocenter	C₁₀₉	#1336
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #20294747	C₂	semidirect
3. #364×C₁₀₉	C₂	semidirect
12. Dic₄₃₆	C₈	semidirect
14. D₈₇₂	C₈	semidirect
45. C₁₀₉	#1336	semidirect
47. #364	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295144	2
#20294747	2
#364×C₁₀₉	2
#1336	109