Finite Group Databasebeta

← C₂×Q₃₂×D₁₀₉ #28376143

Subgroups — C₂×Q₃₂×D₁₀₉ #28376143

Order 13952.

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×D₁₀₉	C₂	1	2	yes	maximal
2	C₂×Dic₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812645	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₁₀₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₁₀₉	—	2	2	no	—
8	C₂×Dic₄₃₆	C₂²	1	2	yes	—
9	C₂×#13182019	—	2	2	no	—
10	Q₁₆×D₁₀₉	—	2	4	no	—
11	Q₁₆×D₁₀₉	C₂²	1	8	yes	—
12	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	Dic₈₇₂	C₂²	1	4	yes	complemented
14	#16812645	C₂²	1	8	yes	complemented
15	C₂×#13182004	C₂²	1	1	yes	characteristic
16	C₁₆×D₁₀₉	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
18	Q₃₂×C₁₀₉	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₈	—	2	2	no	—
21	C₂×Dic₂₁₈	—	4	2	no	—
22	Q₈×D₁₀₉	—	2	8	no	—
23	Q₈×D₁₀₉	—	4	4	no	—
24	C₂×C₄×D₁₀₉	—	4	2	no	—
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	Dic₄₃₆	—	2	2	no	—
27	Dic₄₃₆	C₂³	1	4	yes	—
28	#13182019	—	2	8	no	—
29	C₂×#11724784	D₄	1	1	yes	characteristic
30	C₈×D₁₀₉	D₄	1	2	yes	—
31	C₈×D₁₀₉	C₂³	1	2	yes	—
32	#13182004	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₀₉	—	2	2	no	—
34	Q₁₆×C₁₀₉	—	2	2	no	—
35	Q₁₆×C₁₀₉	C₂³	1	4	yes	—
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
39	Q₈×C₁₀₉	—	2	4	no	—
40	Q₈×C₁₀₉	—	4	2	no	—
41	C₂×C₄×C₁₀₉	—	4	2	no	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	4	2	no	—
44	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
45	C₄×D₁₀₉	—	4	8	no	—
46	C₄×D₁₀₉	D₈	1	2	yes	—
47	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
48	Dic₂₁₈	—	2	4	no	—
49	Dic₂₁₈	—	4	2	no	—
50	Dic₂₁₈	—	4	8	no	—
51	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
53	#11724784	C₂×D₄	1	2	yes	—
54	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
55	C₂×D₁₀₉	Q₃₂	1	4	yes	direct product factor
56	C₂×D₁₀₉	C₂×D₈	1	2	yes	—
57	C₄×C₁₀₉	—	4	4	no	—
58	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
59	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
60	Dic₁₀₉	—	4	4	no	—
61	Dic₁₀₉	C₂×D₈	1	2	yes	—
62	C₂×C₁₀₉	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
64	D₁₀₉	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	109	1	no	maximal, Sylow
66	C₁₀₉	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	109	2	no	—
68	C₂×Q₃₂	—	109	2	no	—
69	C₂×Q₃₂	—	109	8	no	—
70	C₂×Q₃₂	—	109	1	no	—
71	C₂×Q₃₂	D₁₀₉	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	109	1	no	—
73	C₂²×Q₈	—	218	2	no	—
74	C₂×Q₁₆	—	109	8	no	—
75	C₂×Q₁₆	—	109	2	no	—
76	C₂×Q₁₆	—	218	4	no	—
77	C₂×Q₁₆	—	218	2	no	—
78	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
79	C₂²×C₈	—	109	1	no	—
80	Q₃₂	—	109	8	no	—
81	Q₃₂	—	109	4	no	—
82	Q₃₂	C₂×D₁₀₉	1	4	yes	direct product factor
83	C₂×C₁₆	—	109	1	no	—
84	C₂×C₁₆	—	109	4	no	—
85	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	436	2	no	—
88	C₂×Q₈	—	436	4	no	—
89	C₂×Q₈	—	218	8	no	—
90	C₂×Q₈	—	218	2	no	—
91	C₂²×C₄	—	109	1	no	—
92	C₂²×C₄	—	436	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	109	4	no	—
95	Q₁₆	—	218	8	no	—
96	Q₁₆	—	218	2	no	—
97	Q₁₆	C₂²×D₁₀₉	1	4	yes	—
98	C₂×C₈	—	109	1	no	—
99	C₂×C₈	—	109	2	no	—
100	C₂×C₈	—	109	2	no	—
101	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
102	C₁₆	—	109	2	no	—
103	C₁₆	C₂²×D₁₀₉	1	2	yes	—
104	C₂³	—	109	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	436	8	no	—
108	Q₈	—	436	2	no	—
109	Q₈	—	218	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	109	1	no	—
112	C₂×C₄	—	109	2	no	—
113	C₂×C₄	—	109	2	no	—
114	C₂×C₄	—	436	8	no	—
115	C₂×C₄	—	436	2	no	—
116	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
117	C₈	—	109	2	no	—
118	C₈	D₄×D₁₀₉	1	1	yes	characteristic
119	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
120	C₂²	—	109	4	no	—
121	C₂²	—	109	2	no	—
122	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	109	2	no	—
125	C₄	—	436	4	no	—
126	C₄	D₈×D₁₀₉	1	1	yes	characteristic
127	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
128	C₂	—	109	4	no	—
129	C₂	Q₃₂×D₁₀₉	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₈×C₁₀₉	C₂⁴
Frattini	C₈	C₂³×D₁₀₉
Fitting	C₂×Q₃₂×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂²×D₁₀₉	D₈
hypercenter	C₂×Q₃₂	D₁₀₉
hypocenter	C₁₀₉	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
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×Dic₈₇₂	C₂	semidirect
3. C₂×#16812645	C₂	semidirect
4. Q₃₂×D₁₀₉	C₂	direct (N × H)
6. C₂×Q₃₂×C₁₀₉	C₂	semidirect
13. Dic₈₇₂	C₂²	semidirect
14. #16812645	C₂²	semidirect
18. Q₃₂×C₁₀₉	C₂²	semidirect
55. C₂×D₁₀₉	Q₃₂	direct (N × H)
62. C₂×C₁₀₉	C₂×Q₃₂	semidirect
64. D₁₀₉	C₂×Q₃₂	direct (N × H)
66. C₁₀₉	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₁₀₉	direct (N × H)
82. Q₃₂	C₂×D₁₀₉	direct (N × H)
129. C₂	Q₃₂×D₁₀₉	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×D₁₀₉	2
C₂×Dic₈₇₂	2
C₂×#16812645	2
Q₃₂×D₁₀₉	2
C₂×C₁₆×D₁₀₉	2
C₂×Q₃₂×C₁₀₉	2
C₂²×Q₃₂	109