Subgroups — #20294989

Order 6976.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812611	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₀₉	—	2	2	no	—
8	C₂×C₄×D₁₀₉	—	2	2	no	—
9	C₂×C₄×D₁₀₉	C₄	1	2	yes	—
10	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
11	#13182013	—	2	2	no	—
12	C₂×#11724784	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₀₉	—	2	2	no	—
15	C₂×C₈×C₁₀₉	C₂²	1	2	yes	complemented
16	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₁₀₉	—	2	2	no	—
18	C₂²×D₁₀₉	—	2	4	no	—
19	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
22	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₀₉	—	2	2	no	—
25	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
26	C₄×D₁₀₉	—	2	4	no	—
27	C₄×D₁₀₉	—	2	2	no	—
28	C₄×D₁₀₉	—	2	2	no	—
29	C₈×C₁₀₉	—	2	2	no	—
30	#11724784	—	2	2	no	—
31	C₂²×C₁₀₉	—	2	2	no	—
32	C₂²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₀₉	—	2	8	no	—
35	C₂×D₁₀₉	—	2	2	no	—
36	C₂×D₁₀₉	—	2	2	no	—
37	C₂×D₁₀₉	M₁₆	1	4	yes	complemented
38	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
39	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	Dic₁₀₉	—	2	2	no	—
42	C₂×C₁₀₉	—	2	2	no	—
43	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₀₉	C₂×M₁₆	1	2	yes	—
45	D₁₀₉	—	2	4	no	—
46	C₁₀₉	#407	1	1	yes	characteristic, complemented, Sylow
47	#407	—	109	1	no	maximal, Sylow
48	C₂³×C₄	—	109	1	no	—
49	C₂×M₁₆	—	109	1	no	—
50	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	109	4	no	—
52	C₂⁴	—	109	1	no	—
53	C₂²×C₄	—	109	1	no	—
54	C₂²×C₄	—	109	2	no	—
55	C₂²×C₄	—	109	2	no	—
56	C₂²×C₄	—	218	2	no	—
57	C₂²×C₄	—	218	2	no	—
58	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	218	2	no	—
61	C₂×C₈	—	109	2	no	—
62	C₂×C₈	C₂×D₁₀₉	1	2	yes	complemented
63	C₂³	—	109	2	no	—
64	C₂³	—	218	4	no	—
65	C₂³	—	218	2	no	—
66	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
67	C₂×C₄	—	109	2	no	—
68	C₂×C₄	—	218	2	no	—
69	C₂×C₄	—	218	4	no	—
70	C₂×C₄	—	218	2	no	—
71	C₂×C₄	—	218	2	no	—
72	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
73	C₂×C₄	D₄₃₆	1	2	yes	—
74	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
75	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	218	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	109	4	no	—
80	C₂²	—	218	8	no	—
81	C₂²	—	218	2	no	—
82	C₂²	—	218	2	no	—
83	C₂²	#13182006	1	2	yes	—
84	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
85	C₄	—	218	2	no	—
86	C₄	#13182006	1	2	yes	—
87	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
88	C₄	C₂×D₄₃₆	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	218	4	no	—
91	C₂	C₂×#13182006	1	1	yes	characteristic
92	C₂	M₁₆×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×M₁₆×C₁₀₉	C₂
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#407
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#13182013	C₂	semidirect
3. #16812611	C₂	semidirect
4. C₂×M₁₆×C₁₀₉	C₂	semidirect
12. C₂×#11724784	C₂²	semidirect
15. C₂×C₈×C₁₀₉	C₂²	semidirect
37. C₂×D₁₀₉	M₁₆	semidirect
46. C₁₀₉	#407	semidirect
50. C₂×M₁₆	D₁₀₉	semidirect
62. C₂×C₈	C₂×D₁₀₉	semidirect