Subgroups — C₂×#2232×C₁₃₁ #45719523

Order 33536.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2232×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#366×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂×#357×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#121×C₁₃₁	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₁₃₁	—	2	1	no	—
11	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
12	#466×C₁₃₁	—	2	2	no	—
13	#466×C₁₃₁	C₂²	1	4	yes	—
14	C₂×#107×C₁₃₁	—	2	2	no	—
15	C₂×#107×C₁₃₁	—	2	1	no	—
16	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
18	C₂×#107×C₁₃₁	C₂²	1	4	yes	complemented
19	C₂×#102×C₁₃₁	—	2	1	no	—
20	#366×C₁₃₁	C₂²	1	8	yes	complemented
21	#357×C₁₃₁	C₂²	1	8	yes	complemented
22	C₂²⋊C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₁₃₁	—	2	1	no	—
24	C₂²×D₄×C₁₃₁	—	4	1	no	—
25	C₂³×C₄×C₁₃₁	—	4	1	no	—
26	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂×D₈×C₁₃₁	—	2	1	no	—
28	C₂×D₈×C₁₃₁	—	2	4	no	—
29	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃₁	—	2	4	no	—
31	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	#121×C₁₃₁	—	2	4	no	—
36	#121×C₁₃₁	—	4	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
43	C₂×C₁₆×C₁₃₁	—	2	4	no	—
44	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
45	#107×C₁₃₁	—	2	2	no	—
46	#107×C₁₃₁	—	2	4	no	—
47	#107×C₁₃₁	—	2	1	no	—
48	#107×C₁₃₁	C₂³	1	4	yes	—
49	#107×C₁₃₁	C₂³	1	2	yes	—
50	#102×C₁₃₁	—	2	4	no	—
51	C₂⁴×C₁₃₁	—	4	1	no	—
52	C₂⁴×C₁₃₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₁₃₁	—	2	2	no	—
54	C₂×D₄×C₁₃₁	—	4	4	no	—
55	C₂×D₄×C₁₃₁	—	4	2	no	—
56	C₂×D₄×C₁₃₁	—	4	1	no	—
57	C₂×D₄×C₁₃₁	—	4	2	no	—
58	C₂×D₄×C₁₃₁	—	8	4	no	—
59	C₂²×C₄×C₁₃₁	—	2	2	no	—
60	C₂²×C₄×C₁₃₁	—	2	2	no	—
61	C₂²×C₄×C₁₃₁	—	4	4	no	—
62	C₂²×C₄×C₁₃₁	—	4	1	no	—
63	C₂²×C₄×C₁₃₁	—	4	2	no	—
64	C₂²×C₄×C₁₃₁	—	8	4	no	—
65	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
66	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₁₃₁	—	2	4	no	—
70	D₈×C₁₃₁	—	4	2	no	—
71	C₂×C₈×C₁₃₁	—	2	2	no	—
72	C₂×C₈×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	2	4	no	—
74	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₁	#41	1	4	yes	—
77	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₁₃₁	—	2	2	no	—
79	C₄⋊C₄×C₁₃₁	—	2	4	no	—
80	C₄⋊C₄×C₁₃₁	—	4	4	no	—
81	C₄⋊C₄×C₁₃₁	—	4	2	no	—
82	C₄⋊C₄×C₁₃₁	—	4	1	no	—
83	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
84	C₁₆×C₁₃₁	—	2	4	no	—
85	C₂³×C₁₃₁	—	2	2	no	—
86	C₂³×C₁₃₁	—	2	2	no	—
87	C₂³×C₁₃₁	—	4	2	no	—
88	C₂³×C₁₃₁	—	8	2	no	—
89	C₂³×C₁₃₁	—	8	2	no	—
90	C₂³×C₁₃₁	—	8	2	no	—
91	C₂³×C₁₃₁	D₁₆	1	4	yes	—
92	C₂³×C₁₃₁	C₂×D₈	1	2	yes	—
93	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
94	D₄×C₁₃₁	—	4	2	no	—
95	D₄×C₁₃₁	—	4	2	no	—
96	D₄×C₁₃₁	—	8	8	no	—
97	D₄×C₁₃₁	—	8	2	no	—
98	C₂×C₄×C₁₃₁	—	2	2	no	—
99	C₂×C₄×C₁₃₁	—	2	2	no	—
100	C₂×C₄×C₁₃₁	—	2	2	no	—
101	C₂×C₄×C₁₃₁	—	2	4	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	4	no	—
104	C₂×C₄×C₁₃₁	—	8	8	no	—
105	C₂×C₄×C₁₃₁	—	8	4	no	—
106	C₂×C₄×C₁₃₁	—	8	2	no	—
107	C₂×C₄×C₁₃₁	#123	1	2	yes	—
108	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	#137	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃₁	—	2	2	no	—
113	C₈×C₁₃₁	#123	1	2	yes	—
114	C₈×C₁₃₁	C₂×#41	1	2	yes	—
115	C₂²×C₁₃₁	—	2	2	no	—
116	C₂²×C₁₃₁	—	2	2	no	—
117	C₂²×C₁₃₁	—	2	8	no	—
118	C₂²×C₁₃₁	—	8	2	no	—
119	C₂²×C₁₃₁	—	8	2	no	—
120	C₂²×C₁₃₁	—	8	2	no	—
121	C₂²×C₁₃₁	—	16	4	no	—
122	C₂²×C₁₃₁	#480	1	2	yes	—
123	C₂²×C₁₃₁	C₂×D₁₆	1	4	yes	—
124	C₂²×C₁₃₁	C₂×D₁₆	1	2	yes	—
125	C₂²×C₁₃₁	#510	1	2	yes	—
126	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₁₃₁	—	2	2	no	—
128	C₄×C₁₃₁	—	8	4	no	—
129	C₄×C₁₃₁	—	8	2	no	—
130	C₄×C₁₃₁	#480	1	2	yes	—
131	C₄×C₁₃₁	C₂×#123	1	1	yes	characteristic
132	C₄×C₁₃₁	C₂×#137	1	1	yes	characteristic
133	C₂×C₁₃₁	—	2	4	no	—
134	C₂×C₁₃₁	—	16	2	no	—
135	C₂×C₁₃₁	#2232	1	4	yes	direct product factor
136	C₂×C₁₃₁	C₂×#480	1	1	yes	characteristic
137	C₂×C₁₃₁	C₂²×D₁₆	1	1	yes	characteristic
138	C₂×C₁₃₁	C₂×#510	1	1	yes	characteristic
139	C₂×#2232	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	C₂×#2232	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#466	C₂×C₁₃₁	1	1	yes	characteristic
142	C₂²×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
143	#2232	C₂×C₁₃₁	1	8	yes	direct product factor
144	C₂×#366	C₂×C₁₃₁	1	2	yes	complemented
145	C₂×#357	C₂×C₁₃₁	1	2	yes	complemented
146	C₂×C₂²⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
147	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
149	C₂×#121	—	2	1	no	—
150	C₂²×C₄⋊C₄	—	2	1	no	—
151	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
152	#466	—	2	2	no	—
153	#466	C₂²×C₁₃₁	1	4	yes	—
154	C₂×#107	—	2	2	no	—
155	C₂×#107	—	2	1	no	—
156	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
157	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
158	C₂×#107	C₂²×C₁₃₁	1	4	yes	complemented
159	C₂×#102	—	2	1	no	—
160	#366	C₂²×C₁₃₁	1	8	yes	complemented
161	#357	C₂²×C₁₃₁	1	8	yes	complemented
162	C₂²⋊C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂³×C₄	—	4	1	no	—
166	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
175	#121	—	2	4	no	—
176	#121	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	4	1	no	—
181	C₂×C₄⋊C₄	—	4	2	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
185	#107	—	2	2	no	—
186	#107	—	2	4	no	—
187	#107	—	2	1	no	—
188	#107	C₂³×C₁₃₁	1	4	yes	—
189	#107	C₂³×C₁₃₁	1	2	yes	—
190	#102	—	2	4	no	—
191	C₂⁴	—	4	1	no	—
192	C₂⁴	D₈×C₁₃₁	1	1	yes	characteristic
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	8	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
206	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₁₃₁	1	4	yes	—
217	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₁₆	—	2	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	D₁₆×C₁₃₁	1	4	yes	—
232	C₂³	C₂×D₈×C₁₃₁	1	2	yes	—
233	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	#123×C₁₃₁	1	2	yes	—
248	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
249	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	#137×C₁₃₁	1	2	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	#123×C₁₃₁	1	2	yes	—
254	C₈	C₂×#41×C₁₃₁	1	2	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	#480×C₁₃₁	1	2	yes	—
263	C₂²	C₂×D₁₆×C₁₃₁	1	4	yes	—
264	C₂²	C₂×D₁₆×C₁₃₁	1	2	yes	—
265	C₂²	#510×C₁₃₁	1	2	yes	—
266	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	#480×C₁₃₁	1	2	yes	—
271	C₄	C₂×#123×C₁₃₁	1	1	yes	characteristic
272	C₄	C₂×#137×C₁₃₁	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	#2232×C₁₃₁	1	4	yes	direct product factor
276	C₂	C₂×#480×C₁₃₁	1	1	yes	characteristic
277	C₂	C₂²×D₁₆×C₁₃₁	1	1	yes	characteristic
278	C₂	C₂×#510×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂×#466×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#107×C₁₃₁	C₂	semidirect
3. #2232×C₁₃₁	C₂	direct (N × H)
4. C₂×#366×C₁₃₁	C₂	semidirect
5. C₂×#357×C₁₃₁	C₂	semidirect
6. C₂×C₂²⋊C₁₆×C₁₃₁	C₂	semidirect
18. C₂×#107×C₁₃₁	C₂²	semidirect
20. #366×C₁₃₁	C₂²	semidirect
21. #357×C₁₃₁	C₂²	semidirect
22. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
135. C₂×C₁₃₁	#2232	direct (N × H)
139. C₂×#2232	C₁₃₁	direct (N × H)
140. C₁₃₁	C₂×#2232	direct (N × H)
142. C₂²×#107	C₂×C₁₃₁	semidirect
143. #2232	C₂×C₁₃₁	direct (N × H)
144. C₂×#366	C₂×C₁₃₁	semidirect
145. C₂×#357	C₂×C₁₃₁	semidirect
146. C₂×C₂²⋊C₁₆	C₂×C₁₃₁	semidirect
158. C₂×#107	C₂²×C₁₃₁	semidirect
160. #366	C₂²×C₁₃₁	semidirect
161. #357	C₂²×C₁₃₁	semidirect
162. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
275. C₂	#2232×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
C₂×#466×C₁₃₁	2
C₂²×#107×C₁₃₁	2
#2232×C₁₃₁	2
C₂×#366×C₁₃₁	2
C₂×#357×C₁₃₁	2
C₂×C₂²⋊C₁₆×C₁₃₁	2
C₂×#2232	131

Subgroups — C2×#2232×C131 #45719523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2232×C₁₃₁ #45719523