Subgroups — C₂×#2860×C₁₃₁ #45722513

Order 33536.

2838 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	#2860×C₁₃₁	C₂	1	16	yes	maximal, direct product factor
3	C₂×#390×C₁₃₁	C₂	1	3	yes	maximal
4	C₂×#389×C₁₃₁	C₂	1	3	yes	maximal, complemented
5	C₂×#388×C₁₃₁	C₂	1	6	yes	maximal, complemented
6	C₂×C₄³×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×#126×C₁₃₁	—	2	16	no	—
9	C₂×#126×C₁₃₁	C₂²	1	16	yes	complemented
10	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	6	yes	—
11	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	6	yes	—
13	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	6	yes	—
14	C₂²×C₄²×C₁₃₁	C₂²	1	3	yes	—
15	C₂²×C₄²×C₁₃₁	C₂²	1	3	yes	—
16	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
17	#390×C₁₃₁	C₂²	1	24	yes	—
18	#389×C₁₃₁	C₂²	1	24	yes	complemented
19	#388×C₁₃₁	C₂²	1	48	yes	complemented
20	C₂×#95×C₁₃₁	C₂²	1	6	yes	—
21	C₄³×C₁₃₁	C₂²	1	8	yes	complemented
22	C₂⁵×C₁₃₁	C₂³	1	2	yes	—
23	C₂²×D₄×C₁₃₁	—	2	12	no	—
24	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
25	C₂³×C₄×C₁₃₁	C₂³	1	6	yes	—
26	C₂³×C₄×C₁₃₁	C₂³	1	3	yes	—
27	C₂³×C₄×C₁₃₁	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
29	#126×C₁₃₁	—	2	64	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	24	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	12	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	24	yes	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	12	yes	—
34	C₂×C₂²⋊C₄×C₁₃₁	—	2	24	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	2	24	no	—
36	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	24	yes	—
38	C₂×C₄²×C₁₃₁	—	2	12	no	—
39	C₂×C₄²×C₁₃₁	—	2	12	no	—
40	C₂×C₄²×C₁₃₁	D₄	1	8	yes	complemented
41	C₂×C₄²×C₁₃₁	C₂³	1	12	yes	—
42	C₂×C₄²×C₁₃₁	C₂³	1	12	yes	—
43	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
44	#95×C₁₃₁	C₂³	1	24	yes	—
45	C₂⁴×C₁₃₁	—	2	4	no	—
46	C₂⁴×C₁₃₁	—	2	12	no	—
47	C₂⁴×C₁₃₁	—	2	12	no	—
48	C₂⁴×C₁₃₁	C₂⁴	1	4	yes	—
49	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₄×C₁₃₁	—	2	24	no	—
51	C₂×D₄×C₁₃₁	—	4	16	no	—
52	C₂²×C₄×C₁₃₁	—	2	6	no	—
53	C₂²×C₄×C₁₃₁	—	2	6	no	—
54	C₂²×C₄×C₁₃₁	—	2	6	no	—
55	C₂²×C₄×C₁₃₁	—	2	6	no	—
56	C₂²×C₄×C₁₃₁	—	2	12	no	—
57	C₂²×C₄×C₁₃₁	—	2	12	no	—
58	C₂²×C₄×C₁₃₁	—	2	12	no	—
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₁₃₁	#41	1	12	yes	—
61	C₂²×C₄×C₁₃₁	#41	1	12	yes	—
62	C₂²×C₄×C₁₃₁	C₂⁴	1	12	yes	—
63	C₂²×C₄×C₁₃₁	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₁₃₁	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
66	C₄⋊C₄×C₁₃₁	—	2	48	no	—
67	C₄⋊C₄×C₁₃₁	—	2	24	no	—
68	C₂²⋊C₄×C₁₃₁	—	2	48	no	—
69	C₂²⋊C₄×C₁₃₁	—	2	48	no	—
70	C₄²×C₁₃₁	—	2	24	no	—
71	C₄²×C₁₃₁	—	2	24	no	—
72	C₄²×C₁₃₁	C₂×D₄	1	16	yes	complemented
73	C₂³×C₁₃₁	—	2	4	no	—
74	C₂³×C₁₃₁	—	2	12	no	—
75	C₂³×C₁₃₁	—	2	12	no	—
76	C₂³×C₁₃₁	—	4	8	no	—
77	C₂³×C₁₃₁	—	4	24	no	—
78	C₂³×C₁₃₁	—	4	24	no	—
79	C₂³×C₁₃₁	C₂²×D₄	1	2	yes	—
80	C₂³×C₁₃₁	C₂×#41	1	6	yes	—
81	C₂³×C₁₃₁	C₂×#41	1	6	yes	—
82	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
83	D₄×C₁₃₁	—	4	16	no	—
84	C₂×C₄×C₁₃₁	—	2	24	no	—
85	C₂×C₄×C₁₃₁	—	2	24	no	—
86	C₂×C₄×C₁₃₁	—	2	6	no	—
87	C₂×C₄×C₁₃₁	—	2	6	no	—
88	C₂×C₄×C₁₃₁	—	2	6	no	—
89	C₂×C₄×C₁₃₁	—	2	6	no	—
90	C₂×C₄×C₁₃₁	—	2	12	no	—
91	C₂×C₄×C₁₃₁	—	2	12	no	—
92	C₂×C₄×C₁₃₁	—	2	12	no	—
93	C₂×C₄×C₁₃₁	—	4	24	no	—
94	C₂×C₄×C₁₃₁	#126	1	16	yes	complemented
95	C₂×C₄×C₁₃₁	C₂²×D₄	1	12	yes	—
96	C₂×C₄×C₁₃₁	C₂×#41	1	12	yes	—
97	C₂×C₄×C₁₃₁	C₂×#41	1	12	yes	—
98	C₂²×C₁₃₁	—	4	4	no	—
99	C₂²×C₁₃₁	—	4	12	no	—
100	C₂²×C₁₃₁	—	4	12	no	—
101	C₂²×C₁₃₁	—	8	16	no	—
102	C₂²×C₁₃₁	C₂×#126	1	4	yes	—
103	C₂²×C₁₃₁	#529	1	12	yes	—
104	C₂²×C₁₃₁	#532	1	12	yes	—
105	C₂²×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
106	C₂²×C₁₃₁	C₂²×#41	1	3	yes	—
107	C₂²×C₁₃₁	C₂²×#41	1	3	yes	—
108	C₄×C₁₃₁	—	2	12	no	—
109	C₄×C₁₃₁	—	2	12	no	—
110	C₄×C₁₃₁	—	4	12	no	—
111	C₄×C₁₃₁	C₂×#126	1	8	yes	complemented
112	C₂×C₁₃₁	—	8	4	no	—
113	C₂×C₁₃₁	#2860	1	8	yes	direct product factor
114	C₂×C₁₃₁	C₂²×#126	1	1	yes	characteristic
115	C₂×C₁₃₁	C₂×#529	1	3	yes	—
116	C₂×C₁₃₁	C₂×#532	1	3	yes	—
117	C₂×#2860	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₁₃₁	C₂×#2860	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×#126	C₂×C₁₃₁	1	2	yes	complemented
120	#2860	C₂×C₁₃₁	1	16	yes	direct product factor
121	C₂×#390	C₂×C₁₃₁	1	3	yes	—
122	C₂×#389	C₂×C₁₃₁	1	3	yes	complemented
123	C₂×#388	C₂×C₁₃₁	1	6	yes	complemented
124	C₂×C₄³	C₂×C₁₃₁	1	1	yes	characteristic, complemented
125	C₂³×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
126	C₂×#126	—	2	16	no	—
127	C₂×#126	C₂²×C₁₃₁	1	16	yes	complemented
128	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	6	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	3	yes	—
130	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	6	yes	—
131	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	6	yes	—
132	C₂²×C₄²	C₂²×C₁₃₁	1	3	yes	—
133	C₂²×C₄²	C₂²×C₁₃₁	1	3	yes	—
134	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
135	#390	C₂²×C₁₃₁	1	24	yes	—
136	#389	C₂²×C₁₃₁	1	24	yes	complemented
137	#388	C₂²×C₁₃₁	1	48	yes	complemented
138	C₂×#95	C₂²×C₁₃₁	1	6	yes	—
139	C₄³	C₂²×C₁₃₁	1	8	yes	complemented
140	C₂⁵	C₂³×C₁₃₁	1	2	yes	—
141	C₂²×D₄	—	2	12	no	—
142	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
143	C₂³×C₄	C₂³×C₁₃₁	1	6	yes	—
144	C₂³×C₄	C₂³×C₁₃₁	1	3	yes	—
145	C₂³×C₄	C₂³×C₁₃₁	1	3	yes	—
146	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
147	#126	—	2	64	no	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₄⋊C₄	—	2	12	no	—
150	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	24	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	12	yes	—
152	C₂×C₂²⋊C₄	—	2	24	no	—
153	C₂×C₂²⋊C₄	—	2	24	no	—
154	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	24	yes	—
155	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	24	yes	—
156	C₂×C₄²	—	2	12	no	—
157	C₂×C₄²	—	2	12	no	—
158	C₂×C₄²	D₄×C₁₃₁	1	8	yes	complemented
159	C₂×C₄²	C₂³×C₁₃₁	1	12	yes	—
160	C₂×C₄²	C₂³×C₁₃₁	1	12	yes	—
161	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
162	#95	C₂³×C₁₃₁	1	24	yes	—
163	C₂⁴	—	2	4	no	—
164	C₂⁴	—	2	12	no	—
165	C₂⁴	—	2	12	no	—
166	C₂⁴	C₂⁴×C₁₃₁	1	4	yes	—
167	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
168	C₂×D₄	—	2	24	no	—
169	C₂×D₄	—	4	16	no	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	C₂×D₄×C₁₃₁	1	12	yes	—
178	C₂²×C₄	#41×C₁₃₁	1	12	yes	—
179	C₂²×C₄	#41×C₁₃₁	1	12	yes	—
180	C₂²×C₄	C₂⁴×C₁₃₁	1	12	yes	—
181	C₂²×C₄	C₂⁴×C₁₃₁	1	6	yes	—
182	C₂²×C₄	C₂⁴×C₁₃₁	1	6	yes	—
183	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
184	C₄⋊C₄	—	2	48	no	—
185	C₄⋊C₄	—	2	24	no	—
186	C₂²⋊C₄	—	2	48	no	—
187	C₂²⋊C₄	—	2	48	no	—
188	C₄²	—	2	24	no	—
189	C₄²	—	2	24	no	—
190	C₄²	C₂×D₄×C₁₃₁	1	16	yes	complemented
191	C₂³	—	2	4	no	—
192	C₂³	—	2	12	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	4	8	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	4	24	no	—
197	C₂³	C₂²×D₄×C₁₃₁	1	2	yes	—
198	C₂³	C₂×#41×C₁₃₁	1	6	yes	—
199	C₂³	C₂×#41×C₁₃₁	1	6	yes	—
200	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
201	D₄	—	4	16	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	2	24	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	4	24	no	—
212	C₂×C₄	#126×C₁₃₁	1	16	yes	complemented
213	C₂×C₄	C₂²×D₄×C₁₃₁	1	12	yes	—
214	C₂×C₄	C₂×#41×C₁₃₁	1	12	yes	—
215	C₂×C₄	C₂×#41×C₁₃₁	1	12	yes	—
216	C₂²	—	4	4	no	—
217	C₂²	—	4	12	no	—
218	C₂²	—	4	12	no	—
219	C₂²	—	8	16	no	—
220	C₂²	C₂×#126×C₁₃₁	1	4	yes	—
221	C₂²	#529×C₁₃₁	1	12	yes	—
222	C₂²	#532×C₁₃₁	1	12	yes	—
223	C₂²	C₂³×D₄×C₁₃₁	1	1	yes	characteristic
224	C₂²	C₂²×#41×C₁₃₁	1	3	yes	—
225	C₂²	C₂²×#41×C₁₃₁	1	3	yes	—
226	C₄	—	2	12	no	—
227	C₄	—	2	12	no	—
228	C₄	—	4	12	no	—
229	C₄	C₂×#126×C₁₃₁	1	8	yes	complemented
230	C₂	—	8	4	no	—
231	C₂	#2860×C₁₃₁	1	8	yes	direct product factor
232	C₂	C₂²×#126×C₁₃₁	1	1	yes	characteristic
233	C₂	C₂×#529×C₁₃₁	1	3	yes	—
234	C₂	C₂×#532×C₁₃₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂³×D₄×C₁₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#126×C₁₃₁	C₂	semidirect
2. #2860×C₁₃₁	C₂	direct (N × H)
4. C₂×#389×C₁₃₁	C₂	semidirect
5. C₂×#388×C₁₃₁	C₂	semidirect
6. C₂×C₄³×C₁₃₁	C₂	semidirect
9. C₂×#126×C₁₃₁	C₂²	semidirect
18. #389×C₁₃₁	C₂²	semidirect
19. #388×C₁₃₁	C₂²	semidirect
21. C₄³×C₁₃₁	C₂²	semidirect
40. C₂×C₄²×C₁₃₁	D₄	semidirect
72. C₄²×C₁₃₁	C₂×D₄	semidirect
94. C₂×C₄×C₁₃₁	#126	semidirect
111. C₄×C₁₃₁	C₂×#126	semidirect
113. C₂×C₁₃₁	#2860	direct (N × H)
117. C₂×#2860	C₁₃₁	direct (N × H)
118. C₁₃₁	C₂×#2860	direct (N × H)
119. C₂²×#126	C₂×C₁₃₁	semidirect
120. #2860	C₂×C₁₃₁	direct (N × H)
122. C₂×#389	C₂×C₁₃₁	semidirect
123. C₂×#388	C₂×C₁₃₁	semidirect
124. C₂×C₄³	C₂×C₁₃₁	semidirect
127. C₂×#126	C₂²×C₁₃₁	semidirect
136. #389	C₂²×C₁₃₁	semidirect
137. #388	C₂²×C₁₃₁	semidirect
139. C₄³	C₂²×C₁₃₁	semidirect
158. C₂×C₄²	D₄×C₁₃₁	semidirect
190. C₄²	C₂×D₄×C₁₃₁	semidirect
212. C₂×C₄	#126×C₁₃₁	semidirect
229. C₄	C₂×#126×C₁₃₁	semidirect
231. C₂	#2860×C₁₃₁	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×#126×C₁₃₁	2
#2860×C₁₃₁	2
C₂×#390×C₁₃₁	2
C₂×#389×C₁₃₁	2
C₂×#388×C₁₃₁	2
C₂×C₄³×C₁₃₁	2
C₂×#2860	131

Subgroups — C2×#2860×C131 #45722513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2860×C₁₃₁ #45722513