Subgroups — C₂²×#16812774 #28379995

Order 13952.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812774	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#13182032	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812774	C₂²	1	16	yes	direct product factor
5	C₂×#13182032	C₂²	1	12	yes	complemented
6	C₂²×C₄×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	C₄	1	6	yes	—
9	C₂×C₈×D₁₀₉	C₂²	1	6	yes	—
10	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182032	C₂³	1	8	yes	complemented
12	C₂³×D₁₀₉	C₈	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
14	C₂×C₄×D₁₀₉	C₈	1	6	yes	—
15	C₂×C₄×D₁₀₉	C₂×C₄	1	6	yes	—
16	C₂×#11724784	C₂×C₄	1	3	yes	—
17	C₂×#11724784	C₂³	1	3	yes	—
18	C₈×D₁₀₉	C₂×C₄	1	12	yes	—
19	C₈×D₁₀₉	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
22	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	C₂²×D₁₀₉	—	2	4	no	—
24	C₂²×D₁₀₉	C₂×C₈	1	6	yes	—
25	C₂×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
26	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
27	C₂×Dic₁₀₉	C₂×C₈	1	6	yes	—
28	C₄×D₁₀₉	C₂×C₈	1	12	yes	—
29	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
30	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
31	#11724784	C₂²×C₄	1	3	yes	—
32	#11724784	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₀₉	M₃₂	1	4	yes	complemented
34	C₂²×C₁₀₉	C₂²×C₈	1	3	yes	—
35	C₂×D₁₀₉	—	2	12	no	—
36	C₂×D₁₀₉	C₂²×C₈	1	4	yes	—
37	C₄×C₁₀₉	C₂²×C₈	1	3	yes	—
38	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
39	Dic₁₀₉	C₂²×C₈	1	4	yes	—
40	C₂×C₁₀₉	C₂×M₃₂	1	6	yes	complemented
41	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
42	D₁₀₉	—	2	4	no	—
43	C₂²×M₃₂	—	109	1	no	maximal, Sylow
44	C₁₀₉	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₈	—	109	1	no	—
46	C₂×M₃₂	—	109	12	no	—
47	C₂²×C₁₆	—	109	2	no	—
48	C₂³×C₄	—	109	1	no	—
49	C₂²×C₈	—	109	1	no	—
50	C₂²×C₈	—	109	6	no	—
51	C₂²×C₈	—	109	6	no	—
52	C₂²×C₈	#89137	1	1	yes	characteristic
53	M₃₂	—	109	16	no	—
54	C₂×C₁₆	—	109	12	no	—
55	C₂⁴	—	109	1	no	—
56	C₂²×C₄	—	109	1	no	—
57	C₂²×C₄	—	109	6	no	—
58	C₂²×C₄	—	109	6	no	—
59	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
60	C₂×C₈	—	109	3	no	—
61	C₂×C₈	—	109	3	no	—
62	C₂×C₈	—	109	12	no	—
63	C₂×C₈	—	109	4	no	—
64	C₂×C₈	C₂×#89137	1	6	yes	—
65	C₁₆	—	109	8	no	—
66	C₂³	—	109	6	no	—
67	C₂³	—	218	4	no	—
68	C₂³	D₁₀₉⋊C₈	1	1	yes	characteristic
69	C₂×C₄	—	109	6	no	—
70	C₂×C₄	—	109	12	no	—
71	C₂×C₄	—	109	4	no	—
72	C₂×C₄	D₁₀₉⋊C₈	1	3	yes	—
73	C₂×C₄	C₂²×#89137	1	3	yes	—
74	C₈	—	109	1	no	—
75	C₈	—	109	3	no	—
76	C₈	C₂²×#89137	1	4	yes	—
77	C₂²	—	109	4	no	—
78	C₂²	—	218	12	no	—
79	C₂²	#16812774	1	4	yes	direct product factor
80	C₂²	C₂×D₁₀₉⋊C₈	1	3	yes	—
81	C₄	—	109	4	no	—
82	C₄	C₂×D₁₀₉⋊C₈	1	3	yes	—
83	C₄	C₂³×#89137	1	1	yes	characteristic
84	C₂	—	218	4	no	—
85	C₂	C₂×#16812774	1	6	yes	direct product factor
86	C₂	C₂²×D₁₀₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#89137
commutator	C₂×C₁₀₉	C₂³×C₈
Frattini	C₄	C₂³×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂³×C₁₀₉	C₂×C₈
prime radical	C₂³×D₁₀₉	C₈
cosocle	#11724784	C₂⁴
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	C₂²×M₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16812774	C₂	direct (N × H)
2. C₂²×#13182032	C₂	semidirect
4. #16812774	C₂²	direct (N × H)
5. C₂×#13182032	C₂²	semidirect
11. #13182032	C₂³	semidirect
33. C₂²×C₁₀₉	M₃₂	semidirect
40. C₂×C₁₀₉	C₂×M₃₂	semidirect
44. C₁₀₉	C₂²×M₃₂	semidirect
79. C₂²	#16812774	direct (N × H)
85. C₂	C₂×#16812774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16812774	2
C₂²×#13182032	2
C₂²×C₈×D₁₀₉	2
C₂²×M₃₂	109

Subgroups — C22×#16812774 #28379995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16812774 #28379995