Subgroups — #28361314

Order 13952.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295077	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294748	C₂	1	2	yes	maximal, complemented
4	C₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	2	yes	—
7	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
8	#16812628	C₄	1	4	yes	—
9	#16812611	C₄	1	4	yes	—
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₄×#11724784	C₂²	1	2	yes	—
12	C₈×Dic₁₀₉	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	4	yes	—
18	C₂×#11724784	C₂×C₄	1	2	yes	—
19	C₈×D₁₀₉	C₈	1	8	yes	complemented
20	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
28	C₂×Dic₁₀₉	C₄²	1	2	yes	—
29	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
30	C₄×D₁₀₉	—	2	4	no	—
31	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
32	C₈×C₁₀₉	—	2	4	no	—
33	C₈×C₁₀₉	C₂×C₈	1	4	yes	—
34	#11724784	—	2	4	no	—
35	#11724784	C₂×C₈	1	4	yes	—
36	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₁₀₉	—	2	2	no	—
38	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
39	C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
40	C₄×C₁₀₉	#131	1	4	yes	—
41	Dic₁₀₉	—	2	2	no	—
42	Dic₁₀₉	C₄×C₈	1	2	yes	—
43	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
44	C₂×C₁₀₉	#405	1	2	yes	—
45	D₁₀₉	—	2	2	no	—
46	#1450	—	109	1	no	maximal, Sylow
47	C₁₀₉	#1450	1	1	yes	characteristic, complemented, Sylow
48	#431	—	109	2	no	—
49	C₂×C₄×C₈	—	109	1	no	—
50	#322	—	109	2	no	—
51	C₈²	—	109	1	no	—
52	C₈²	D₁₀₉	1	1	yes	characteristic, complemented
53	C₂²×C₈	—	109	2	no	—
54	C₂×C₄²	—	109	1	no	—
55	C₄⋊C₈	—	109	4	no	—
56	C₂²⋊C₈	—	109	4	no	—
57	C₄×C₈	—	109	2	no	—
58	C₄×C₈	—	109	2	no	—
59	C₄×C₈	—	109	1	no	—
60	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
61	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂²×C₄	—	109	1	no	—
63	C₂²×C₄	—	109	2	no	—
64	C₂×C₈	—	109	2	no	—
65	C₂×C₈	—	109	8	no	—
66	C₂×C₈	—	109	4	no	—
67	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
68	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
69	C₄²	—	109	1	no	—
70	C₄²	—	109	2	no	—
71	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
72	C₂³	—	109	1	no	—
73	C₂×C₄	—	109	1	no	—
74	C₂×C₄	—	109	4	no	—
75	C₂×C₄	—	109	2	no	—
76	C₂×C₄	—	218	4	no	—
77	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
78	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	—	109	4	no	—
81	C₈	—	218	4	no	—
82	C₈	C₈×D₁₀₉	1	4	yes	—
83	C₂²	—	109	2	no	—
84	C₂²	—	218	2	no	—
85	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
86	C₄	—	109	2	no	—
87	C₄	—	218	2	no	—
88	C₄	C₂×C₈×D₁₀₉	1	2	yes	—
89	C₄	#16812705	1	4	yes	—
90	C₂	—	218	2	no	—
91	C₂	C₄×C₈×D₁₀₉	1	1	yes	characteristic
92	C₂	#20294980	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₈²×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄²
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₈²	D₁₀₉
hypocenter	C₁₀₉	#1450
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
3. #20294748	C₂	semidirect
4. C₈×#11724784	C₂	semidirect
5. C₈²×C₁₀₉	C₂	semidirect
19. C₈×D₁₀₉	C₈	semidirect
47. C₁₀₉	#1450	semidirect
52. C₈²	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295077	2
C₄×C₈×D₁₀₉	2
#20294748	2
C₈×#11724784	2
C₈²×C₁₀₉	2
#1450	109