Subgroups — #28363152

Order 13952.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11724784	C₂	1	2	yes	maximal
2	#20294781	C₂	1	4	yes	maximal
3	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11724784	C₂²	1	4	yes	—
5	C₄⋊#11724784	—	2	4	no	—
6	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
8	#95×C₁₀₉	C₄	1	4	yes	—
9	C₂×#11724784	—	2	4	no	—
10	C₂×#11724784	—	2	8	no	—
11	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₄²×C₁₀₉	—	2	4	no	—
15	C₄²×C₁₀₉	D₄	1	2	yes	—
16	C₄²×C₁₀₉	Q₈	1	2	yes	—
17	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	—	2	4	no	—
19	C₂×C₄×C₁₀₉	—	2	4	no	—
20	C₂×C₄×C₁₀₉	—	2	8	no	—
21	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	#41	1	4	yes	—
25	#11724784	—	4	4	no	—
26	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₀₉	#117	1	2	yes	—
28	C₂²×C₁₀₉	#131	1	4	yes	—
29	C₄×C₁₀₉	—	2	4	no	—
30	C₄×C₁₀₉	—	4	4	no	—
31	C₄×C₁₀₉	#124	1	2	yes	—
32	C₄×C₁₀₉	#125	1	2	yes	—
33	C₂×C₁₀₉	#382	1	1	yes	characteristic
34	C₂×C₁₀₉	#424	1	2	yes	—
35	C₂×C₁₀₉	#433	1	4	yes	—
36	#1832	—	109	1	no	maximal, Sylow
37	C₁₀₉	#1832	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄⋊C₈	—	109	2	no	—
39	#376	D₁₀₉	1	1	yes	characteristic
40	#336	—	109	4	no	—
41	C₂²×C₈	—	109	4	no	—
42	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
43	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
44	C₄⋊C₈	—	218	4	no	—
45	#95	Dic₁₀₉	1	4	yes	—
46	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
47	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
48	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
49	C₂×C₈	—	218	4	no	—
50	C₂×C₈	—	218	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	Dic₂₁₈	1	2	yes	—
53	C₄²	D₄₃₆	1	2	yes	—
54	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
59	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
61	C₂×C₄	#13182029	1	4	yes	—
62	C₈	—	436	4	no	—
63	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
64	C₂²	#16812658	1	2	yes	—
65	C₂²	#16812706	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	4	4	no	—
68	C₄	#16812687	1	2	yes	—
69	C₄	#16812691	1	2	yes	—
70	C₂	#20294892	1	1	yes	characteristic
71	C₂	#20295053	1	2	yes	—
72	C₂	#20295087	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator, socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
Fitting	#376×C₁₀₉	C₂
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#376	D₁₀₉
hypocenter	C₁₀₉	#1832
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
37. C₁₀₉	#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11724784	2
#20294781	2
#376×C₁₀₉	2
#1832	109