Subgroups — C₂²×#16812691 #28376285

Order 13952.

1192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812691	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal
3	C₂²×#13182008	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₉	C₂²	1	4	yes	—
6	#16812691	C₂²	1	64	yes	direct product factor
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	24	yes	—
8	C₂×#13182008	C₂²	1	48	yes	—
9	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₉	C₂²	1	12	yes	—
12	C₂²×Dic₁₀₉	—	2	16	no	—
13	C₂²×Dic₁₀₉	C₂³	1	24	yes	—
14	C₄⋊Dic₁₀₉	C₂³	1	32	yes	—
15	#13182008	C₂³	1	64	yes	—
16	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₉	—	2	8	no	—
18	C₂²×C₄×C₁₀₉	Q₈	1	8	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
20	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
21	C₄²×C₁₀₉	C₂³	1	16	yes	—
22	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	—
23	C₂³×C₁₀₉	#41	1	8	yes	—
24	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
25	C₂×C₄×C₁₀₉	—	2	24	no	—
26	C₂×C₄×C₁₀₉	C₂×Q₈	1	24	yes	—
27	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
28	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
29	C₂×Dic₁₀₉	—	2	48	no	—
30	C₂×Dic₁₀₉	C₂⁴	1	16	yes	—
31	C₂²×C₁₀₉	#125	1	16	yes	complemented
32	C₂²×C₁₀₉	C₂²×Q₈	1	6	yes	—
33	C₂²×C₁₀₉	C₂×#41	1	12	yes	—
34	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
35	C₄×C₁₀₉	—	2	8	no	—
36	C₄×C₁₀₉	C₂²×Q₈	1	8	yes	—
37	Dic₁₀₉	—	2	16	no	—
38	C₂×C₁₀₉	C₂×#125	1	12	yes	complemented
39	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
40	C₂×C₁₀₉	C₂²×#41	1	2	yes	—
41	C₂²×#125	—	109	1	no	maximal, Sylow
42	C₁₀₉	C₂²×#125	1	1	yes	characteristic, complemented, Sylow
43	C₂×#125	—	109	24	no	—
44	C₂²×C₄⋊C₄	—	109	4	no	—
45	C₂²×C₄⋊C₄	—	109	2	no	—
46	C₂²×C₄²	D₁₀₉	1	1	yes	characteristic
47	C₂³×C₄	—	109	4	no	—
48	C₂³×C₄	C₂×D₁₀₉	1	2	yes	—
49	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
50	#125	—	109	64	no	—
51	C₂×C₄⋊C₄	—	109	48	no	—
52	C₂×C₄⋊C₄	—	109	24	no	—
53	C₂×C₄²	C₂×D₁₀₉	1	12	yes	—
54	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
55	C₂²×C₄	—	2	8	no	—
56	C₂²×C₄	—	109	24	no	—
57	C₂²×C₄	—	218	16	no	—
58	C₂²×C₄	Dic₂₁₈	1	8	yes	—
59	C₂²×C₄	C₂²×D₁₀₉	1	12	yes	—
60	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
61	C₄⋊C₄	—	109	64	no	—
62	C₄⋊C₄	—	109	32	no	—
63	C₄²	C₂²×D₁₀₉	1	16	yes	—
64	C₂³	C₂×Dic₂₁₈	1	4	yes	—
65	C₂³	#13182029	1	8	yes	—
66	C₂³	C₂³×D₁₀₉	1	3	yes	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	—	109	16	no	—
69	C₂×C₄	—	218	48	no	—
70	C₂×C₄	C₂×Dic₂₁₈	1	24	yes	—
71	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
72	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
73	C₂²	#16812691	1	16	yes	direct product factor
74	C₂²	C₂²×Dic₂₁₈	1	6	yes	—
75	C₂²	C₂×#13182029	1	12	yes	—
76	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
77	C₄	—	2	8	no	—
78	C₄	—	218	16	no	—
79	C₄	C₂²×Dic₂₁₈	1	8	yes	—
80	C₂	C₂×#16812691	1	12	yes	direct product factor
81	C₂	C₂³×Dic₂₁₈	1	1	yes	characteristic
82	C₂	C₂²×#13182029	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂²×C₄²×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₂³
hypercenter	C₂²×C₄²	D₁₀₉
hypocenter	C₁₀₉	C₂²×#125
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#16812691	C₂	direct (N × H)
6. #16812691	C₂²	direct (N × H)
31. C₂²×C₁₀₉	#125	semidirect
38. C₂×C₁₀₉	C₂×#125	semidirect
42. C₁₀₉	C₂²×#125	semidirect
73. C₂²	#16812691	direct (N × H)
80. C₂	C₂×#16812691	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×#16812691	2
C₂²×C₄⋊Dic₁₀₉	2
C₂²×#13182008	2
C₂²×C₄²×C₁₀₉	2
C₂²×#125	109

Subgroups — C22×#16812691 #28376285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16812691 #28376285