Subgroups — C₂×#2169×C₁₃₁ #45719315

Order 33536.

742 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2169×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₁₆×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₁₃₁	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₃₁	—	2	8	no	—
10	C₂×M₃₂×C₁₃₁	C₂²	1	8	yes	complemented
11	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₈×D₄×C₁₃₁	C₄	1	4	yes	—
15	C₈×D₄×C₁₃₁	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
20	C₂²⋊C₁₆×C₁₃₁	C₂²	1	8	yes	complemented
21	C₄×C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
22	C₂²×D₄×C₁₃₁	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	—	2	4	no	—
25	C₂²×C₈×C₁₃₁	—	2	4	no	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₁₃₁	C₈	1	4	yes	—
32	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₃₁	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₁₃₁	—	2	16	no	—
37	C₂×C₁₆×C₁₃₁	—	2	4	no	—
38	C₂×C₁₆×C₁₃₁	—	2	2	no	—
39	C₂×C₁₆×C₁₃₁	D₄	1	4	yes	complemented
40	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
41	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
43	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
44	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
47	C₂⁴×C₁₃₁	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₁₃₁	—	2	4	no	—
49	C₂×D₄×C₁₃₁	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₁₃₁	—	2	4	no	—
51	C₂²×C₄×C₁₃₁	—	2	4	no	—
52	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	—	2	4	no	—
59	C₂×C₈×C₁₃₁	—	2	2	no	—
60	C₂×C₈×C₁₃₁	—	2	8	no	—
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃₁	#41	1	2	yes	—
68	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₁₃₁	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	8	yes	—
71	C₄²×C₁₃₁	C₂×C₈	1	2	yes	—
72	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₁₆×C₁₃₁	—	2	4	no	—
74	C₁₆×C₁₃₁	—	2	2	no	—
75	C₁₆×C₁₃₁	C₂×D₄	1	4	yes	complemented
76	C₂³×C₁₃₁	—	2	4	no	—
77	C₂³×C₁₃₁	—	2	4	no	—
78	C₂³×C₁₃₁	—	2	4	no	—
79	C₂³×C₁₃₁	C₂²×C₈	1	4	yes	—
80	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
81	D₄×C₁₃₁	—	2	8	no	—
82	C₂×C₄×C₁₃₁	—	2	4	no	—
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	4	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	8	no	—
87	C₂×C₄×C₁₃₁	M₃₂	1	4	yes	complemented
88	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₃₁	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₁₃₁	#131	1	2	yes	—
94	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₁₃₁	—	2	4	no	—
96	C₈×C₁₃₁	—	2	2	no	—
97	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
98	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₁₃₁	—	2	4	no	—
101	C₂²×C₁₃₁	—	2	4	no	—
102	C₂²×C₁₃₁	—	2	4	no	—
103	C₂²×C₁₃₁	—	4	8	no	—
104	C₂²×C₁₃₁	C₈×D₄	1	2	yes	—
105	C₂²×C₁₃₁	C₂×M₃₂	1	2	yes	—
106	C₂²×C₁₃₁	#504	1	2	yes	—
107	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
108	C₄×C₁₃₁	—	2	2	no	—
109	C₄×C₁₃₁	—	2	4	no	—
110	C₄×C₁₃₁	C₈×D₄	1	2	yes	—
111	C₄×C₁₃₁	C₂×M₃₂	1	4	yes	complemented
112	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₁₃₁	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₃₁	—	4	4	no	—
115	C₂×C₁₃₁	#2169	1	4	yes	direct product factor
116	C₂×C₁₃₁	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₁₃₁	C₂²×M₃₂	1	1	yes	characteristic
118	C₂×C₁₃₁	C₂×#504	1	1	yes	characteristic
119	C₂×#2169	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₃₁	C₂×#2169	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×M₃₂	C₂×C₁₃₁	1	2	yes	complemented
122	C₂×C₈×D₄	C₂×C₁₃₁	1	1	yes	characteristic
123	#2169	C₂×C₁₃₁	1	8	yes	direct product factor
124	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
125	C₂×C₂²⋊C₁₆	C₂×C₁₃₁	1	2	yes	complemented
126	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
127	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
128	C₂×C₄×D₄	C₄×C₁₃₁	1	1	yes	characteristic
129	C₂×M₃₂	—	2	8	no	—
130	C₂×M₃₂	C₂²×C₁₃₁	1	8	yes	complemented
131	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
133	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
134	C₈×D₄	C₄×C₁₃₁	1	4	yes	—
135	C₈×D₄	C₂²×C₁₃₁	1	4	yes	—
136	C₂×C₄⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
137	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
140	C₂²⋊C₁₆	C₂²×C₁₃₁	1	8	yes	complemented
141	C₄×C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
142	C₂²×D₄	C₈×C₁₃₁	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
147	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
148	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
150	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
151	C₄×D₄	C₈×C₁₃₁	1	4	yes	—
152	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
153	C₂×C₄⋊C₄	C₈×C₁₃₁	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₈×C₁₃₁	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
156	M₃₂	—	2	16	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	D₄×C₁₃₁	1	4	yes	complemented
160	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
161	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
162	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
163	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
164	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
165	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
166	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
167	C₂⁴	C₂×C₈×C₁₃₁	1	2	yes	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	C₂×C₈×C₁₃₁	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
174	C₂²×C₄	C₂×C₈×C₁₃₁	1	4	yes	—
175	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
184	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
185	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
187	C₂×C₈	#41×C₁₃₁	1	2	yes	—
188	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
189	C₄⋊C₄	C₂×C₈×C₁₃₁	1	4	yes	—
190	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	8	yes	—
191	C₄²	C₂×C₈×C₁₃₁	1	2	yes	—
192	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂×D₄×C₁₃₁	1	4	yes	complemented
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	C₂²×C₈×C₁₃₁	1	4	yes	—
200	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	M₃₂×C₁₃₁	1	4	yes	complemented
208	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
209	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₁₃₁	1	4	yes	—
211	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
213	C₂×C₄	#131×C₁₃₁	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
218	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
219	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₈×D₄×C₁₃₁	1	2	yes	—
225	C₂²	C₂×M₃₂×C₁₃₁	1	2	yes	—
226	C₂²	#504×C₁₃₁	1	2	yes	—
227	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	C₈×D₄×C₁₃₁	1	2	yes	—
231	C₄	C₂×M₃₂×C₁₃₁	1	4	yes	complemented
232	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
233	C₄	C₂×#131×C₁₃₁	1	1	yes	characteristic
234	C₂	—	4	4	no	—
235	C₂	#2169×C₁₃₁	1	4	yes	direct product factor
236	C₂	C₂×C₈×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂	C₂²×M₃₂×C₁₃₁	1	1	yes	characteristic
238	C₂	C₂×#504×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₃₁	C₂³
commutator	C₂²	C₂³×C₈×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂²×C₈
prime radical	C₂²×D₄×C₁₃₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₁₃₁	C₂	semidirect
3. #2169×C₁₃₁	C₂	direct (N × H)
4. C₂×C₄⋊C₁₆×C₁₃₁	C₂	semidirect
5. C₂×C₂²⋊C₁₆×C₁₃₁	C₂	semidirect
6. C₂×C₄×C₁₆×C₁₃₁	C₂	semidirect
10. C₂×M₃₂×C₁₃₁	C₂²	semidirect
19. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
20. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
21. C₄×C₁₆×C₁₃₁	C₂²	semidirect
39. C₂×C₁₆×C₁₃₁	D₄	semidirect
75. C₁₆×C₁₃₁	C₂×D₄	semidirect
87. C₂×C₄×C₁₃₁	M₃₂	semidirect
111. C₄×C₁₃₁	C₂×M₃₂	semidirect
115. C₂×C₁₃₁	#2169	direct (N × H)
119. C₂×#2169	C₁₃₁	direct (N × H)
120. C₁₃₁	C₂×#2169	direct (N × H)
121. C₂²×M₃₂	C₂×C₁₃₁	semidirect
123. #2169	C₂×C₁₃₁	direct (N × H)
124. C₂×C₄⋊C₁₆	C₂×C₁₃₁	semidirect
125. C₂×C₂²⋊C₁₆	C₂×C₁₃₁	semidirect
126. C₂×C₄×C₁₆	C₂×C₁₃₁	semidirect
130. C₂×M₃₂	C₂²×C₁₃₁	semidirect
139. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
140. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
141. C₄×C₁₆	C₂²×C₁₃₁	semidirect
159. C₂×C₁₆	D₄×C₁₃₁	semidirect
195. C₁₆	C₂×D₄×C₁₃₁	semidirect
207. C₂×C₄	M₃₂×C₁₃₁	semidirect
231. C₄	C₂×M₃₂×C₁₃₁	semidirect
235. C₂	#2169×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×M₃₂×C₁₃₁	2
C₂×C₈×D₄×C₁₃₁	2
#2169×C₁₃₁	2
C₂×C₄⋊C₁₆×C₁₃₁	2
C₂×C₂²⋊C₁₆×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
C₂×#2169	131

Subgroups — C2×#2169×C131 #45719315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2169×C₁₃₁ #45719315