Subgroups — #28362778

Order 13952.

296 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	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11724784	C₂²	1	4	yes	—
5	C₄×#11724784	C₄	1	8	yes	complemented
6	C₂×C₄²×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄²×C₁₀₉	C₄	1	2	yes	—
8	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#11724784	—	2	8	no	—
11	C₂×#11724784	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₄²×C₁₀₉	—	2	4	no	—
17	C₄²×C₁₀₉	—	2	4	no	—
18	C₄²×C₁₀₉	C₈	1	8	yes	complemented
19	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
20	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	—	2	4	no	—
22	C₂×C₄×C₁₀₉	—	2	4	no	—
23	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
24	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
25	C₂×C₄×C₁₀₉	C₂×C₈	1	8	yes	—
26	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	—
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	#41	1	4	yes	—
30	#11724784	—	2	8	no	—
31	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
32	C₂²×C₁₀₉	#117	1	2	yes	—
33	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
34	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
35	C₄×C₁₀₉	—	2	8	no	—
36	C₄×C₁₀₉	C₄×C₈	1	4	yes	—
37	C₄×C₁₀₉	#97	1	4	yes	—
38	C₄×C₁₀₉	#117	1	4	yes	—
39	C₂×C₁₀₉	#376	1	1	yes	characteristic
40	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
42	C₂×C₁₀₉	#431	1	4	yes	—
43	#1744	—	109	1	no	maximal, Sylow
44	C₁₀₉	#1744	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₈	—	109	2	no	—
46	C₄³	D₁₀₉	1	1	yes	characteristic
47	#336	—	109	4	no	—
48	C₂²×C₈	—	109	4	no	—
49	C₂×C₄²	Dic₁₀₉	1	2	yes	—
50	C₂×C₄²	Dic₁₀₉	1	2	yes	—
51	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
52	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
53	C₄×C₈	—	109	8	no	—
54	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
55	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
58	C₂×C₈	—	109	8	no	—
59	C₂×C₈	—	218	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	#11724784	1	8	yes	complemented
63	C₄²	C₄×D₁₀₉	1	4	yes	—
64	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
68	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
69	C₂×C₄	C₂×#11724784	1	8	yes	—
70	C₂×C₄	C₂×#11724784	1	4	yes	—
71	C₂×C₄	#13182013	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
73	C₂×C₄	#13182029	1	4	yes	—
74	C₈	—	218	8	no	—
75	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
76	C₂²	#16812658	1	2	yes	—
77	C₂²	C₂²×#11724784	1	2	yes	—
78	C₂²	C₂×#13182013	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×#11724784	1	4	yes	—
81	C₄	#16812610	1	4	yes	—
82	C₄	#16812656	1	4	yes	—
83	C₂	#20294872	1	1	yes	characteristic
84	C₂	C₂×C₄×#11724784	1	1	yes	characteristic
85	C₂	C₂×#16812610	1	1	yes	characteristic
86	C₂	#20295079	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄³×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#1744
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. C₄×#11724784	C₄	semidirect
18. C₄²×C₁₀₉	C₈	semidirect
44. C₁₀₉	#1744	semidirect
62. C₄²	#11724784	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
C₄³×C₁₀₉	2
#1744	109