Subgroups — #28364903

Order 13952.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₀₉	C₂	1	2	yes	maximal
2	#20294854	C₂	1	1	yes	maximal, characteristic
3	#20294830	C₂	1	2	yes	maximal
4	#20294801	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₉	—	2	2	no	—
7	C₂×#13182004	C₂²	1	1	yes	characteristic
8	#16812633	C₂²	1	2	yes	—
9	#16812623	—	2	2	no	—
10	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
12	Q₃₂×C₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	2	no	—
16	C₄⋊Dic₁₀₉	—	4	2	no	—
17	C₄×Dic₁₀₉	—	4	2	no	—
18	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
19	#13182004	—	2	1	no	—
20	C₂×Q₈×C₁₀₉	—	2	2	no	—
21	Q₁₆×C₁₀₉	—	2	2	no	—
22	Q₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	Q₈×C₁₀₉	—	2	4	no	—
26	Q₈×C₁₀₉	—	4	2	no	—
27	C₂×C₄×C₁₀₉	—	4	2	no	—
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	4	2	no	—
30	C₂×Dic₁₀₉	D₈	1	2	yes	—
31	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₀₉	#41	1	1	yes	characteristic
33	#11724784	—	2	1	no	—
34	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₀₉	—	4	4	no	—
36	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₀₉	#135	1	1	yes	characteristic
38	Dic₁₀₉	—	2	2	no	—
39	Dic₁₀₉	—	8	2	no	—
40	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₀₉	#510	1	2	yes	—
42	#2176	—	109	1	no	maximal, Sylow
43	C₁₀₉	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	D₁₀₉	1	1	yes	characteristic
45	C₄×Q₁₆	—	109	2	no	—
46	#367	—	109	1	no	—
47	#358	—	109	2	no	—
48	#346	—	109	1	no	—
49	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
50	C₄×Q₈	—	218	2	no	—
51	Q₃₂	Dic₁₀₉	1	4	yes	complemented
52	C₂×C₁₆	—	109	1	no	—
53	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
54	#107	—	109	2	no	—
55	#103	—	218	2	no	—
56	C₄×C₈	—	109	1	no	—
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×Dic₁₀₉	1	4	yes	—
60	C₂×C₈	—	109	1	no	—
61	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
62	C₄⋊C₄	—	436	2	no	—
63	C₄⋊C₄	—	218	2	no	—
64	C₄²	—	109	1	no	—
65	C₄²	—	436	2	no	—
66	C₁₆	—	218	1	no	—
67	C₁₆	C₂×Dic₁₀₉	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	109	2	no	—
72	C₂×C₄	—	436	2	no	—
73	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
74	C₈	—	218	1	no	—
75	C₈	C₂²×Dic₁₀₉	1	1	yes	characteristic
76	C₈	#13182028	1	1	yes	characteristic
77	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
78	C₄	—	4	4	no	—
79	C₄	—	872	2	no	—
80	C₄	—	218	2	no	—
81	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
82	C₄	#16812718	1	1	yes	characteristic
83	C₂	D₈×Dic₁₀₉	1	1	yes	characteristic
84	C₂	#20295468	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₂×Q₃₂×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂×Q₃₂	D₁₀₉
hypocenter	C₁₀₉	#2176
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₁₀₉	C₄	semidirect
43. C₁₀₉	#2176	semidirect
51. Q₃₂	Dic₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₀₉	2
#20294854	2
#20294830	2
#20294801	2
C₂×Q₃₂×C₁₀₉	2
#2176	109