Subgroups — C₄×#16812609 #28362716

Order 13952.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812609	C₂	1	4	yes	maximal
2	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₉	C₄	1	4	yes	—
7	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
8	#16812609	C₄	1	16	yes	direct product factor
9	C₄×#11724784	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₀₉	C₄	1	4	yes	complemented
13	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
14	C₂×#11724784	C₂×C₄	1	4	yes	—
15	C₂×#11724784	C₂×C₄	1	8	yes	—
16	C₄×Dic₁₀₉	—	2	8	no	—
17	C₄×Dic₁₀₉	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
21	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
26	C₂×C₄×C₁₀₉	M₁₆	1	8	yes	—
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	—	2	8	no	—
31	C₂×Dic₁₀₉	C₄²	1	8	yes	—
32	C₈×C₁₀₉	C₄²	1	8	yes	complemented
33	#11724784	C₄²	1	8	yes	complemented
34	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
35	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
36	C₂²×C₁₀₉	C₂×M₁₆	1	4	yes	—
37	C₄×C₁₀₉	#97	1	8	yes	complemented
38	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
39	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
40	Dic₁₀₉	—	2	8	no	—
41	C₂×C₁₀₉	C₄³	1	1	yes	characteristic
42	C₂×C₁₀₉	C₂×#97	1	2	yes	—
43	C₂×C₁₀₉	C₄×M₁₆	1	4	yes	—
44	C₄×#97	—	109	1	no	maximal, Sylow
45	C₁₀₉	C₄×#97	1	1	yes	characteristic, complemented, Sylow
46	C₂×#97	—	109	4	no	—
47	C₂×C₄×C₈	—	109	1	no	—
48	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
49	C₄³	—	109	1	no	—
50	C₂²×C₈	—	109	2	no	—
51	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
52	C₂×C₄²	—	109	2	no	—
53	C₂×C₄²	—	109	4	no	—
54	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
55	#97	—	109	16	no	—
56	C₄×C₈	—	109	4	no	—
57	C₄×C₈	Dic₁₀₉	1	4	yes	complemented
58	C₂²×C₄	—	109	4	no	—
59	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
60	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
61	C₂×C₈	—	109	4	no	—
62	C₂×C₈	—	109	8	no	—
63	C₂×C₈	C₄×D₁₀₉	1	8	yes	—
64	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
65	C₄²	—	109	8	no	—
66	C₄²	—	218	8	no	—
67	C₄²	C₄×D₁₀₉	1	2	yes	—
68	C₄²	C₂×Dic₁₀₉	1	2	yes	—
69	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
70	C₂×C₄	—	109	8	no	—
71	C₂×C₄	—	218	8	no	—
72	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
73	C₂×C₄	#13182012	1	8	yes	—
74	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
76	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
77	C₈	—	109	8	no	—
78	C₈	C₄×Dic₁₀₉	1	8	yes	complemented
79	C₂²	C₄²×D₁₀₉	1	2	yes	—
80	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
81	C₂²	C₂×#13182012	1	4	yes	—
82	C₄	—	218	8	no	—
83	C₄	#16812609	1	8	yes	direct product factor
84	C₄	C₄²×D₁₀₉	1	2	yes	—
85	C₄	C₂×C₄×Dic₁₀₉	1	2	yes	—
86	C₂	C₄²×Dic₁₀₉	1	1	yes	characteristic
87	C₂	C₂×#16812609	1	2	yes	—
88	C₂	C₄×#13182012	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₄³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	C₄×#97
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #16812609	C₄	direct (N × H)
9. C₄×#11724784	C₄	semidirect
12. C₄×C₈×C₁₀₉	C₄	semidirect
32. C₈×C₁₀₉	C₄²	semidirect
33. #11724784	C₄²	semidirect
37. C₄×C₁₀₉	#97	semidirect
45. C₁₀₉	C₄×#97	semidirect
57. C₄×C₈	Dic₁₀₉	semidirect
78. C₈	C₄×Dic₁₀₉	semidirect
83. C₄	#16812609	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×#16812609	2
C₂×C₄×#11724784	2
C₄²×Dic₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
C₄×#97	109

Subgroups — C4×#16812609 #28362716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812609 #28362716