Subgroups — #28363170

Order 13952.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295050	C₂	1	2	yes	maximal
2	#20294791	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	C₂²	1	4	yes	—
5	C₄⋊#11724784	—	2	4	no	—
6	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₀₉	C₄	1	2	yes	—
9	C₄×C₈×C₁₀₉	C₄	1	2	yes	—
10	#13182013	—	2	8	no	—
11	C₂×#11724784	—	2	4	no	—
12	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
13	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₁₀₉	—	2	4	no	—
15	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
16	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₀₉	D₄	1	2	yes	—
19	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	—	2	4	no	—
21	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
22	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	#41	1	4	yes	—
26	C₈×C₁₀₉	—	2	4	no	—
27	#11724784	—	4	4	no	—
28	C₂²×C₁₀₉	—	2	2	no	—
29	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₀₉	#125	1	2	yes	—
31	C₄×C₁₀₉	—	2	2	no	—
32	C₄×C₁₀₉	#108	1	4	yes	—
33	C₄×C₁₀₉	#117	1	2	yes	—
34	C₄×C₁₀₉	#124	1	2	yes	—
35	C₂×C₁₀₉	—	2	2	no	—
36	C₂×C₁₀₉	#382	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂×#108	1	2	yes	—
38	#1837	—	109	1	no	maximal, Sylow
39	C₁₀₉	#1837	1	1	yes	characteristic, complemented, Sylow
40	#423	—	109	2	no	—
41	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
42	#341	—	109	4	no	—
43	C₂×M₁₆	—	109	4	no	—
44	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
45	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
46	C₄⋊C₈	—	218	4	no	—
47	C₄×C₈	Dic₁₀₉	1	2	yes	—
48	C₄×C₈	Dic₁₀₉	1	2	yes	—
49	C₂²×C₄	Dic₂₁₈	1	2	yes	—
50	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
51	M₁₆	—	218	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	218	4	no	—
54	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
55	C₄²	D₄₃₆	1	2	yes	—
56	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
57	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
58	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₁₀₉	1	2	yes	—
62	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
63	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
64	C₂×C₄	#13182029	1	4	yes	—
65	C₈	—	2	4	no	—
66	C₈	—	436	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
69	C₂²	#16812691	1	2	yes	—
70	C₄	—	2	2	no	—
71	C₄	#16812635	1	4	yes	—
72	C₄	#16812658	1	2	yes	—
73	C₄	#16812687	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	#20294892	1	1	yes	characteristic
76	C₂	C₂×#16812635	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#1837
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
39. C₁₀₉	#1837	semidirect

Maximal subgroups

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

subgroup	index
#20295050	2
#20294791	2
C₂×C₄×C₈×C₁₀₉	2
#1837	109