Finite Group Databasebeta

← C₂×Q₃₂×D₁₀₇ #28349572

Subgroups — C₂×Q₃₂×D₁₀₇ #28349572

Order 13696.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₁₀₇	C₂	1	2	yes	maximal
2	C₂×Dic₈₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812055	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₁₀₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₁₀₇	—	2	2	no	—
8	C₂×Dic₄₂₈	C₂²	1	2	yes	—
9	C₂×#13133808	—	2	2	no	—
10	Q₁₆×D₁₀₇	—	2	4	no	—
11	Q₁₆×D₁₀₇	C₂²	1	8	yes	—
12	C₂×C₈×D₁₀₇	C₂²	1	1	yes	characteristic
13	Dic₈₅₆	C₂²	1	4	yes	complemented
14	#16812055	C₂²	1	8	yes	complemented
15	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₁₀₇	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
18	Q₃₂×C₁₀₇	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₄	—	2	2	no	—
21	C₂×Dic₂₁₄	—	4	2	no	—
22	Q₈×D₁₀₇	—	2	8	no	—
23	Q₈×D₁₀₇	—	4	4	no	—
24	C₂×C₄×D₁₀₇	—	4	2	no	—
25	C₂×C₄×D₁₀₇	D₄	1	1	yes	characteristic
26	Dic₄₂₈	—	2	2	no	—
27	Dic₄₂₈	C₂³	1	4	yes	—
28	#13133808	—	2	8	no	—
29	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₁₀₇	D₄	1	2	yes	—
31	C₈×D₁₀₇	C₂³	1	2	yes	—
32	C₁₀₇⋊C₁₆	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₀₇	—	2	2	no	—
34	Q₁₆×C₁₀₇	—	2	2	no	—
35	Q₁₆×C₁₀₇	C₂³	1	4	yes	—
36	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
37	C₁₆×C₁₀₇	C₂³	1	2	yes	—
38	C₂²×D₁₀₇	D₈	1	1	yes	characteristic
39	Q₈×C₁₀₇	—	2	4	no	—
40	Q₈×C₁₀₇	—	4	2	no	—
41	C₂×C₄×C₁₀₇	—	4	2	no	—
42	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₀₇	—	4	2	no	—
44	C₂×Dic₁₀₇	D₈	1	1	yes	characteristic
45	C₄×D₁₀₇	—	4	8	no	—
46	C₄×D₁₀₇	D₈	1	2	yes	—
47	C₄×D₁₀₇	C₂×D₄	1	2	yes	—
48	Dic₂₁₄	—	2	4	no	—
49	Dic₂₁₄	—	4	2	no	—
50	Dic₂₁₄	—	4	8	no	—
51	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
53	C₁₀₇⋊C₈	C₂×D₄	1	2	yes	—
54	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
55	C₂×D₁₀₇	Q₃₂	1	4	yes	direct product factor
56	C₂×D₁₀₇	C₂×D₈	1	2	yes	—
57	C₄×C₁₀₇	—	4	4	no	—
58	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
59	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
60	Dic₁₀₇	—	4	4	no	—
61	Dic₁₀₇	C₂×D₈	1	2	yes	—
62	C₂×C₁₀₇	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
64	D₁₀₇	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	107	1	no	maximal, Sylow
66	C₁₀₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	107	2	no	—
68	C₂×Q₃₂	—	107	2	no	—
69	C₂×Q₃₂	—	107	8	no	—
70	C₂×Q₃₂	—	107	1	no	—
71	C₂×Q₃₂	D₁₀₇	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	107	1	no	—
73	C₂²×Q₈	—	214	2	no	—
74	C₂×Q₁₆	—	107	8	no	—
75	C₂×Q₁₆	—	107	2	no	—
76	C₂×Q₁₆	—	214	4	no	—
77	C₂×Q₁₆	—	214	2	no	—
78	C₂×Q₁₆	C₂×D₁₀₇	1	2	yes	—
79	C₂²×C₈	—	107	1	no	—
80	Q₃₂	—	107	8	no	—
81	Q₃₂	—	107	4	no	—
82	Q₃₂	C₂×D₁₀₇	1	4	yes	direct product factor
83	C₂×C₁₆	—	107	1	no	—
84	C₂×C₁₆	—	107	4	no	—
85	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	428	2	no	—
88	C₂×Q₈	—	428	4	no	—
89	C₂×Q₈	—	214	8	no	—
90	C₂×Q₈	—	214	2	no	—
91	C₂²×C₄	—	107	1	no	—
92	C₂²×C₄	—	428	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	107	4	no	—
95	Q₁₆	—	214	8	no	—
96	Q₁₆	—	214	2	no	—
97	Q₁₆	C₂²×D₁₀₇	1	4	yes	—
98	C₂×C₈	—	107	1	no	—
99	C₂×C₈	—	107	2	no	—
100	C₂×C₈	—	107	2	no	—
101	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
102	C₁₆	—	107	2	no	—
103	C₁₆	C₂²×D₁₀₇	1	2	yes	—
104	C₂³	—	107	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	428	8	no	—
108	Q₈	—	428	2	no	—
109	Q₈	—	214	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	107	1	no	—
112	C₂×C₄	—	107	2	no	—
113	C₂×C₄	—	107	2	no	—
114	C₂×C₄	—	428	8	no	—
115	C₂×C₄	—	428	2	no	—
116	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
117	C₈	—	107	2	no	—
118	C₈	D₄×D₁₀₇	1	1	yes	characteristic
119	C₈	C₂³×D₁₀₇	1	1	yes	characteristic
120	C₂²	—	107	2	no	—
121	C₂²	—	107	4	no	—
122	C₂²	D₈×D₁₀₇	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	107	2	no	—
125	C₄	—	428	4	no	—
126	C₄	D₈×D₁₀₇	1	1	yes	characteristic
127	C₄	C₂×D₄×D₁₀₇	1	1	yes	characteristic
128	C₂	—	107	4	no	—
129	C₂	Q₃₂×D₁₀₇	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₇
commutator, cosocle	C₈×C₁₀₇	C₂⁴
Frattini	C₈	C₂³×D₁₀₇
Fitting	C₂×Q₃₂×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₈
prime radical	C₂²×D₁₀₇	D₈
hypercenter	C₂×Q₃₂	D₁₀₇
hypocenter	C₁₀₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₈₅₆	C₂	semidirect
3. C₂×#16812055	C₂	semidirect
4. Q₃₂×D₁₀₇	C₂	direct (N × H)
6. C₂×Q₃₂×C₁₀₇	C₂	semidirect
13. Dic₈₅₆	C₂²	semidirect
14. #16812055	C₂²	semidirect
18. Q₃₂×C₁₀₇	C₂²	semidirect
55. C₂×D₁₀₇	Q₃₂	direct (N × H)
62. C₂×C₁₀₇	C₂×Q₃₂	semidirect
64. D₁₀₇	C₂×Q₃₂	direct (N × H)
66. C₁₀₇	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₁₀₇	direct (N × H)
82. Q₃₂	C₂×D₁₀₇	direct (N × H)
129. C₂	Q₃₂×D₁₀₇	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×Q₁₆×D₁₀₇	2
C₂×Dic₈₅₆	2
C₂×#16812055	2
Q₃₂×D₁₀₇	2
C₂×C₁₆×D₁₀₇	2
C₂×Q₃₂×C₁₀₇	2
C₂²×Q₃₂	107