Subgroups — C₂³×#13182008 #28376192

Order 13952.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13182008	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₉	—	2	8	no	—
6	C₂³×Dic₁₀₉	C₄	1	16	yes	complemented
7	C₂³×Dic₁₀₉	C₂²	1	14	yes	—
8	C₂³×Dic₁₀₉	C₂²	1	14	yes	—
9	C₂×#13182008	C₂²	1	112	yes	direct product factor
10	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	—	2	8	no	—
12	C₂³×C₄×C₁₀₉	C₂²	1	14	yes	—
13	C₂²×Dic₁₀₉	—	2	56	no	—
14	C₂²×Dic₁₀₉	C₂×C₄	1	112	yes	complemented
15	C₂²×Dic₁₀₉	C₂³	1	28	yes	—
16	C₂²×Dic₁₀₉	C₂³	1	28	yes	—
17	#13182008	C₂³	1	64	yes	direct product factor
18	C₂⁴×C₁₀₉	C₂×C₄	1	8	yes	—
19	C₂⁴×C₁₀₉	D₄	1	8	yes	—
20	C₂⁴×C₁₀₉	Q₈	1	8	yes	—
21	C₂⁴×C₁₀₉	C₂³	1	7	yes	—
22	C₂²×C₄×C₁₀₉	—	2	56	no	—
23	C₂²×C₄×C₁₀₉	C₂³	1	28	yes	—
24	C₂³×C₁₀₉	C₄⋊C₄	1	64	yes	complemented
25	C₂³×C₁₀₉	C₂²×C₄	1	28	yes	—
26	C₂³×C₁₀₉	C₂×D₄	1	28	yes	—
27	C₂³×C₁₀₉	C₂×Q₈	1	28	yes	—
28	C₂³×C₁₀₉	C₂⁴	1	7	yes	—
29	C₂×C₄×C₁₀₉	—	2	56	no	—
30	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
31	C₂×Dic₁₀₉	—	2	56	no	—
32	C₂×Dic₁₀₉	C₂²×C₄	1	112	yes	complemented
33	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
34	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
35	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	112	yes	complemented
36	C₂²×C₁₀₉	C₂³×C₄	1	14	yes	—
37	C₂²×C₁₀₉	C₂²×D₄	1	14	yes	—
38	C₂²×C₁₀₉	C₂²×Q₈	1	14	yes	—
39	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
40	C₄×C₁₀₉	—	2	8	no	—
41	Dic₁₀₉	—	2	8	no	—
42	Dic₁₀₉	C₂³×C₄	1	16	yes	complemented
43	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	28	yes	complemented
44	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
45	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
47	C₂³×C₄⋊C₄	—	109	1	no	maximal, Sylow
48	C₁₀₉	C₂³×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴×C₄	—	109	1	no	—
50	C₂⁴×C₄	—	109	1	no	—
51	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic
52	C₂²×C₄⋊C₄	—	109	28	no	—
53	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
54	C₂³×C₄	—	2	8	no	—
55	C₂³×C₄	—	109	16	no	—
56	C₂³×C₄	—	109	14	no	—
57	C₂³×C₄	—	109	14	no	—
58	C₂³×C₄	—	218	8	no	—
59	C₂³×C₄	C₂×D₁₀₉	1	14	yes	—
60	C₂×C₄⋊C₄	—	109	112	no	—
61	C₂⁴	Dic₂₁₈	1	8	yes	—
62	C₂⁴	C₄×D₁₀₉	1	8	yes	—
63	C₂⁴	C₂²⋊D₁₀₉	1	8	yes	—
64	C₂⁴	C₂²×D₁₀₉	1	7	yes	—
65	C₂²×C₄	—	2	56	no	—
66	C₂²×C₄	—	109	112	no	—
67	C₂²×C₄	—	109	28	no	—
68	C₂²×C₄	—	109	28	no	—
69	C₂²×C₄	—	218	56	no	—
70	C₂²×C₄	C₂²×D₁₀₉	1	28	yes	—
71	C₄⋊C₄	—	109	64	no	—
72	C₂³	#13182008	1	64	yes	direct product factor
73	C₂³	C₂×C₄×D₁₀₉	1	28	yes	—
74	C₂³	C₂×C₂²⋊D₁₀₉	1	28	yes	—
75	C₂³	C₂×Dic₂₁₈	1	28	yes	—
76	C₂³	C₂³×D₁₀₉	1	7	yes	—
77	C₂×C₄	—	2	56	no	—
78	C₂×C₄	—	109	112	no	—
79	C₂×C₄	—	109	8	no	—
80	C₂×C₄	—	109	8	no	—
81	C₂×C₄	—	218	56	no	—
82	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
83	C₂²	C₂×#13182008	1	112	yes	direct product factor
84	C₂²	C₂²×C₄×D₁₀₉	1	14	yes	—
85	C₂²	C₂²×C₂²⋊D₁₀₉	1	14	yes	—
86	C₂²	C₂²×Dic₂₁₈	1	14	yes	—
87	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
88	C₄	—	2	8	no	—
89	C₄	—	109	16	no	—
90	C₄	—	218	8	no	—
91	C₂	C₂²×#13182008	1	28	yes	direct product factor
92	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
93	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₂	C₂³×Dic₂₁₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂⁴×C₄×C₁₀₉	C₂
socle, prime radical	C₂⁵×C₁₀₉	C₂²
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂⁴×C₄	D₁₀₉
hypocenter	C₁₀₉	C₂³×C₄⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#13182008	C₂	direct (N × H)
6. C₂³×Dic₁₀₉	C₄	semidirect
9. C₂×#13182008	C₂²	direct (N × H)
14. C₂²×Dic₁₀₉	C₂×C₄	semidirect
17. #13182008	C₂³	direct (N × H)
24. C₂³×C₁₀₉	C₄⋊C₄	semidirect
32. C₂×Dic₁₀₉	C₂²×C₄	semidirect
35. C₂²×C₁₀₉	C₂×C₄⋊C₄	semidirect
42. Dic₁₀₉	C₂³×C₄	semidirect
43. C₂×C₁₀₉	C₂²×C₄⋊C₄	semidirect
48. C₁₀₉	C₂³×C₄⋊C₄	semidirect
72. C₂³	#13182008	direct (N × H)
83. C₂²	C₂×#13182008	direct (N × H)
91. C₂	C₂²×#13182008	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₀₉	2
C₂²×#13182008	2
C₂⁴×C₄×C₁₀₉	2
C₂³×C₄⋊C₄	109

Subgroups — C23×#13182008 #28376192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13182008 #28376192