Subgroups — C₂×M₃₂×D₁₀₉ #28376126

Order 13952.

400 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812639	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812638	C₂	1	2	yes	maximal, complemented
4	M₃₂×D₁₀₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₀₉	C₄	1	1	yes	characteristic
8	C₂²×#11724784	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₄	1	4	yes	—
10	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
14	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812639	C₂²	1	4	yes	complemented
16	#16812638	C₂²	1	8	yes	complemented
17	C₂×#13182004	C₂²	1	2	yes	complemented
18	C₁₆×D₁₀₉	C₂²	1	8	yes	complemented
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	M₃₂×C₁₀₉	C₂²	1	4	yes	complemented
21	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	complemented
22	C₂³×D₁₀₉	C₈	1	1	yes	characteristic
23	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	C₈	1	4	yes	—
26	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
31	C₂×#11724784	C₂×C₄	1	4	yes	—
32	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
33	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
34	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
35	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
36	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
37	C₈×D₁₀₉	C₂³	1	2	yes	—
38	C₈×D₁₀₉	C₂³	1	2	yes	—
39	#13182004	C₂³	1	4	yes	complemented
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₁₆×C₁₀₉	C₂³	1	4	yes	complemented
46	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
47	C₂²×D₁₀₉	—	2	4	no	—
48	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₁₀₉	C₂×C₈	1	4	yes	—
50	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
56	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
57	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
58	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
59	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
60	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
61	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
62	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
63	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
64	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
65	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
67	#11724784	C₂²×C₄	1	2	yes	—
68	#11724784	C₂²×C₄	1	2	yes	—
69	C₂²×C₁₀₉	—	2	2	no	—
70	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
71	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×D₁₀₉	—	2	2	no	—
73	C₂×D₁₀₉	—	2	8	no	—
74	C₂×D₁₀₉	M₃₂	1	4	yes	direct product factor
75	C₂×D₁₀₉	C₂²×C₈	1	2	yes	—
76	C₂×D₁₀₉	C₂²×C₈	1	2	yes	—
77	C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
78	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
80	Dic₁₀₉	C₂²×C₈	1	2	yes	—
81	Dic₁₀₉	C₂²×C₈	1	2	yes	—
82	C₂×C₁₀₉	—	2	2	no	—
83	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	complemented
84	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
85	D₁₀₉	—	2	2	no	—
86	D₁₀₉	C₂×M₃₂	1	4	yes	direct product factor
87	C₂²×M₃₂	—	109	1	no	maximal, Sylow
88	C₁₀₉	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₈	—	109	1	no	—
90	C₂×M₃₂	—	109	1	no	—
91	C₂×M₃₂	—	109	8	no	—
92	C₂×M₃₂	—	109	2	no	—
93	C₂×M₃₂	D₁₀₉	1	1	yes	characteristic, direct product factor
94	C₂²×C₁₆	—	109	2	no	—
95	C₂³×C₄	—	109	1	no	—
96	C₂²×C₈	—	109	1	no	—
97	C₂²×C₈	—	109	4	no	—
98	C₂²×C₈	—	109	1	no	—
99	C₂²×C₈	—	109	1	no	—
100	C₂²×C₈	—	109	4	no	—
101	C₂²×C₈	—	109	1	no	—
102	C₂²×C₈	—	109	1	no	—
103	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
104	M₃₂	—	109	4	no	—
105	M₃₂	—	109	8	no	—
106	M₃₂	C₂×D₁₀₉	1	4	yes	direct product factor
107	C₂×C₁₆	—	109	2	no	—
108	C₂×C₁₆	—	109	8	no	—
109	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	complemented
110	C₂⁴	—	109	1	no	—
111	C₂²×C₄	—	109	1	no	—
112	C₂²×C₄	—	109	1	no	—
113	C₂²×C₄	—	109	4	no	—
114	C₂²×C₄	—	109	1	no	—
115	C₂²×C₄	—	109	4	no	—
116	C₂²×C₄	—	109	1	no	—
117	C₂²×C₄	—	109	1	no	—
118	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
119	C₂×C₈	—	109	4	no	—
120	C₂×C₈	—	109	1	no	—
121	C₂×C₈	—	109	1	no	—
122	C₂×C₈	—	109	4	no	—
123	C₂×C₈	—	109	4	no	—
124	C₂×C₈	—	109	2	no	—
125	C₂×C₈	—	109	2	no	—
126	C₂×C₈	—	109	2	no	—
127	C₂×C₈	—	109	2	no	—
128	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
129	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
130	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
131	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
132	C₁₆	—	109	4	no	—
133	C₁₆	C₂²×D₁₀₉	1	4	yes	complemented
134	C₂³	—	109	1	no	—
135	C₂³	—	109	4	no	—
136	C₂³	—	109	1	no	—
137	C₂³	—	218	4	no	—
138	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
139	C₂×C₄	—	109	4	no	—
140	C₂×C₄	—	109	1	no	—
141	C₂×C₄	—	109	1	no	—
142	C₂×C₄	—	109	4	no	—
143	C₂×C₄	—	109	2	no	—
144	C₂×C₄	—	109	4	no	—
145	C₂×C₄	—	109	2	no	—
146	C₂×C₄	—	109	2	no	—
147	C₂×C₄	—	109	2	no	—
148	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
149	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
152	C₈	—	109	2	no	—
153	C₈	—	109	2	no	—
154	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
155	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
156	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	—	109	4	no	—
159	C₂²	—	109	2	no	—
160	C₂²	—	109	2	no	—
161	C₂²	—	218	8	no	—
162	C₂²	—	218	2	no	—
163	C₂²	C₂×C₈×D₁₀₉	1	2	yes	—
164	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
165	C₄	—	109	2	no	—
166	C₄	—	109	2	no	—
167	C₄	C₂×C₈×D₁₀₉	1	2	yes	—
168	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
169	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	109	4	no	—
172	C₂	—	218	2	no	—
173	C₂	M₃₂×D₁₀₉	1	2	yes	direct product factor
174	C₂	C₂²×C₈×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₈
Frattini	C₈	C₂³×D₁₀₉
Fitting	C₂×M₃₂×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂³×D₁₀₉	C₈
cosocle	C₈×C₁₀₉	C₂⁴
hypercenter	C₂×M₃₂	D₁₀₉
hypocenter	C₁₀₉	C₂²×M₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#16812639	C₂	semidirect
3. C₂×#16812638	C₂	semidirect
4. M₃₂×D₁₀₉	C₂	direct (N × H)
5. C₂×C₁₆×D₁₀₉	C₂	semidirect
6. C₂×M₃₂×C₁₀₉	C₂	semidirect
15. #16812639	C₂²	semidirect
16. #16812638	C₂²	semidirect
17. C₂×#13182004	C₂²	semidirect
18. C₁₆×D₁₀₉	C₂²	semidirect
20. M₃₂×C₁₀₉	C₂²	semidirect
21. C₂×C₁₆×C₁₀₉	C₂²	semidirect
39. #13182004	C₂³	semidirect
45. C₁₆×C₁₀₉	C₂³	semidirect
74. C₂×D₁₀₉	M₃₂	direct (N × H)
83. C₂×C₁₀₉	C₂×M₃₂	semidirect
86. D₁₀₉	C₂×M₃₂	direct (N × H)
88. C₁₀₉	C₂²×M₃₂	semidirect
93. C₂×M₃₂	D₁₀₉	direct (N × H)
106. M₃₂	C₂×D₁₀₉	direct (N × H)
109. C₂×C₁₆	C₂×D₁₀₉	semidirect
133. C₁₆	C₂²×D₁₀₉	semidirect
173. C₂	M₃₂×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
C₂²×C₈×D₁₀₉	2
C₂×#16812639	2
C₂×#16812638	2
M₃₂×D₁₀₉	2
C₂×C₁₆×D₁₀₉	2
C₂×M₃₂×C₁₀₉	2
C₂²×M₃₂	109

Subgroups — C2×M32×D109 #28376126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₃₂×D₁₀₉ #28376126