Subgroups — C₂×D₁₀₉⋊#131 #28380021

Order 13952.

1504 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812787	C₂	1	2	yes	maximal, complemented
2	D₁₀₉⋊#131	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16812786	C₂	1	6	yes	maximal, complemented
4	C₂×D₁₀₉⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	C₂²×D₁₀₉⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#16812787	C₂²	1	16	yes	complemented
8	#16812786	C₂²	1	48	yes	complemented
9	C₂×#13182039	C₂²	1	6	yes	complemented
10	D₁₀₉⋊M₁₆	C₂²	1	24	yes	complemented
11	C₂×#13182038	C₂²	1	6	yes	complemented
12	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₀₉⋊C₈	C₂²	1	3	yes	—
14	C₂×D₁₀₉⋊C₈	C₂²	1	6	yes	complemented
15	C₂×D₁₀₉⋊C₈	C₂²	1	24	yes	complemented
16	C₂²×#11724786	C₂²	1	6	yes	complemented
17	C₂×#13182030	C₂²	1	1	yes	characteristic
18	C₂×#13182029	C₄	1	3	yes	—
19	C₂×#13182028	C₂²	1	3	yes	—
20	#41×D₁₀₉	C₄	1	8	yes	—
21	#41×D₁₀₉	C₂²	1	8	yes	—
22	C₂×Q₈×D₁₀₉	C₄	1	1	yes	characteristic
23	C₂×D₄×D₁₀₉	C₄	1	3	yes	—
24	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
25	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
26	#13182039	C₂³	1	24	yes	complemented
27	#13182038	C₂³	1	24	yes	complemented
28	D₁₀₉⋊C₈	C₂³	1	4	yes	—
29	D₁₀₉⋊C₈	C₂³	1	12	yes	—
30	D₁₀₉⋊C₈	C₂³	1	24	yes	complemented
31	C₂×#11724786	C₂³	1	2	yes	—
32	C₂×#11724786	C₂³	1	6	yes	—
33	C₂×#11724786	C₂³	1	24	yes	complemented
34	C₂³×D₁₀₉	C₂×C₄	1	3	yes	—
35	#13182030	C₂×C₄	1	4	yes	—
36	#13182030	C₂³	1	4	yes	—
37	#13182029	C₂×C₄	1	24	yes	—
38	#13182028	C₂×C₄	1	12	yes	—
39	#13182028	C₂³	1	12	yes	—
40	C₂×Dic₂₁₈	C₂×C₄	1	3	yes	—
41	Q₈×D₁₀₉	C₂×C₄	1	8	yes	—
42	C₂×D₄₃₆	C₂×C₄	1	3	yes	—
43	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	—
44	D₄×D₁₀₉	C₂×C₄	1	24	yes	—
45	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
46	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₁₀₉	C₂×C₄	1	12	yes	—
48	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
49	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
51	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
52	#41×C₁₀₉	C₂×C₄	1	8	yes	—
53	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
55	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
56	C₂³×C₁₀₉	C₂²×C₄	1	3	yes	—
57	C₂²×D₁₀₉	—	2	12	no	—
58	C₂²×D₁₀₉	C₂²×C₄	1	12	yes	—
59	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
60	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
62	D₄×C₁₀₉	C₂²×C₄	1	12	yes	—
63	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	—
67	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
68	C₂×Dic₁₀₉	C₂²×C₄	1	6	yes	—
69	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
71	D₄₃₆	C₂²×C₄	1	12	yes	—
72	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
73	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
74	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
75	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
76	C₄×D₁₀₉	C₂⁴	1	2	yes	—
77	C₄×D₁₀₉	C₂⁴	1	6	yes	—
78	Dic₂₁₈	C₂²×C₄	1	12	yes	—
79	#11724786	C₂⁴	1	4	yes	—
80	#11724786	C₂⁴	1	12	yes	—
81	C₂²×C₁₀₉	—	2	6	no	—
82	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
83	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×D₁₀₉	—	2	6	no	—
85	C₂×D₁₀₉	—	2	24	no	—
86	C₂×D₁₀₉	#131	1	4	yes	complemented
87	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
88	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
89	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
90	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
91	Dic₁₀₉	C₂³×C₄	1	6	yes	—
92	Dic₁₀₉	C₂³×C₄	1	1	yes	characteristic
93	Dic₁₀₉	C₂⁵	1	1	yes	characteristic
94	C₂×C₁₀₉	—	2	6	no	—
95	C₂×C₁₀₉	C₂×#131	1	2	yes	complemented
96	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
97	D₁₀₉	—	2	6	no	—
98	D₁₀₉	C₂×#131	1	4	yes	complemented
99	C₂²×#131	—	109	1	no	maximal, Sylow
100	C₁₀₉	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
101	C₂²×#41	—	109	1	no	—
102	C₂×#131	—	109	6	no	—
103	C₂×#131	—	109	16	no	—
104	C₂×#131	—	109	2	no	—
105	C₂²×M₁₆	—	109	3	no	—
106	C₂³×C₈	—	109	3	no	—
107	C₂×#41	—	109	8	no	—
108	C₂×#41	—	109	8	no	—
109	C₂×#41	—	109	1	no	—
110	C₂×#41	—	109	3	no	—
111	C₂×#41	—	109	3	no	—
112	C₂×#41	#89137	1	1	yes	characteristic
113	C₂²×Q₈	—	109	1	no	—
114	C₂²×D₄	—	109	3	no	—
115	C₂³×C₄	—	109	3	no	—
116	#131	—	109	48	no	—
117	#131	—	109	16	no	—
118	C₂×M₁₆	—	109	6	no	—
119	C₂×M₁₆	—	109	24	no	—
120	C₂×M₁₆	—	109	6	no	—
121	C₂²×C₈	—	109	1	no	—
122	C₂²×C₈	—	109	3	no	—
123	C₂²×C₈	—	109	6	no	—
124	C₂²×C₈	—	109	6	no	—
125	C₂²×C₈	—	109	24	no	—
126	C₂⁴	—	109	3	no	—
127	#41	—	109	4	no	—
128	#41	—	109	4	no	—
129	#41	—	109	12	no	—
130	#41	—	109	12	no	—
131	#41	—	109	24	no	—
132	#41	C₂×#89137	1	8	yes	—
133	C₂×Q₈	—	109	8	no	—
134	C₂×Q₈	—	109	3	no	—
135	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
136	C₂×D₄	—	109	6	no	—
137	C₂×D₄	—	109	24	no	—
138	C₂×D₄	—	109	3	no	—
139	C₂×D₄	C₂×#89137	1	3	yes	—
140	C₂²×C₄	—	109	3	no	—
141	C₂²×C₄	—	109	3	no	—
142	C₂²×C₄	—	109	12	no	—
143	C₂²×C₄	—	109	3	no	—
144	C₂²×C₄	—	109	12	no	—
145	C₂²×C₄	—	109	3	no	—
146	C₂²×C₄	—	109	1	no	—
147	C₂²×C₄	C₂×#89137	1	3	yes	—
148	M₁₆	—	109	24	no	—
149	M₁₆	—	109	24	no	—
150	C₂×C₈	—	109	2	no	—
151	C₂×C₈	—	109	6	no	—
152	C₂×C₈	—	109	24	no	—
153	C₂×C₈	—	109	4	no	—
154	C₂×C₈	—	109	12	no	—
155	C₂×C₈	—	109	24	no	—
156	C₂³	—	109	1	no	—
157	C₂³	—	109	12	no	—
158	C₂³	—	109	3	no	—
159	C₂³	—	218	12	no	—
160	C₂³	C₂²×#89137	1	3	yes	—
161	Q₈	—	109	12	no	—
162	Q₈	C₂²×#89137	1	4	yes	—
163	D₄	—	109	24	no	—
164	D₄	—	109	12	no	—
165	D₄	C₂²×#89137	1	12	yes	—
166	C₂×C₄	—	109	1	no	—
167	C₂×C₄	—	109	6	no	—
168	C₂×C₄	—	109	3	no	—
169	C₂×C₄	—	109	6	no	—
170	C₂×C₄	—	109	2	no	—
171	C₂×C₄	—	109	6	no	—
172	C₂×C₄	—	109	2	no	—
173	C₂×C₄	—	109	12	no	—
174	C₂×C₄	—	109	6	no	—
175	C₂×C₄	—	109	12	no	—
176	C₂×C₄	C₂²×#89137	1	3	yes	—
177	C₂×C₄	C₂²×#89137	1	12	yes	—
178	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
179	C₈	—	109	4	no	—
180	C₈	—	109	12	no	—
181	C₂²	—	2	6	no	—
182	C₂²	—	109	4	no	—
183	C₂²	—	109	2	no	—
184	C₂²	—	109	6	no	—
185	C₂²	—	218	24	no	—
186	C₂²	—	218	6	no	—
187	C₂²	C₂³×#89137	1	6	yes	—
188	C₂²	C₂³×#89137	1	1	yes	characteristic
189	C₄	—	109	1	no	—
190	C₄	—	109	1	no	—
191	C₄	—	109	6	no	—
192	C₄	C₂³×#89137	1	2	yes	—
193	C₄	C₂³×#89137	1	6	yes	—
194	C₂	—	2	6	no	—
195	C₂	—	109	4	no	—
196	C₂	—	218	6	no	—
197	C₂	D₁₀₉⋊#131	1	2	yes	direct product factor
198	C₂	C₂⁴×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂	C₂⁴×#89137
Fitting	C₂×#41×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#41×D₁₀₉	C₂
cosocle	Dic₁₀₉	C₂⁵
hypercenter	C₂×#41	#89137
hypocenter	C₁₀₉	C₂²×#131
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16812787	C₂	semidirect
2. D₁₀₉⋊#131	C₂	direct (N × H)
3. C₂×#16812786	C₂	semidirect
4. C₂×D₁₀₉⋊M₁₆	C₂	semidirect
5. C₂²×D₁₀₉⋊C₈	C₂	semidirect
7. #16812787	C₂²	semidirect
8. #16812786	C₂²	semidirect
9. C₂×#13182039	C₂²	semidirect
10. D₁₀₉⋊M₁₆	C₂²	semidirect
11. C₂×#13182038	C₂²	semidirect
14. C₂×D₁₀₉⋊C₈	C₂²	semidirect
15. C₂×D₁₀₉⋊C₈	C₂²	semidirect
16. C₂²×#11724786	C₂²	semidirect
26. #13182039	C₂³	semidirect
27. #13182038	C₂³	semidirect
30. D₁₀₉⋊C₈	C₂³	semidirect
33. C₂×#11724786	C₂³	semidirect
86. C₂×D₁₀₉	#131	semidirect
95. C₂×C₁₀₉	C₂×#131	semidirect
98. D₁₀₉	C₂×#131	semidirect
100. C₁₀₉	C₂²×#131	semidirect
197. C₂	D₁₀₉⋊#131	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×#16812787	2
D₁₀₉⋊#131	2
C₂×#16812786	2
C₂×D₁₀₉⋊M₁₆	2
C₂²×D₁₀₉⋊C₈	2
C₂×#41×D₁₀₉	2
C₂²×#131	109

Subgroups — C2×D109⋊#131 #28380021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₀₉⋊#131 #28380021