Subgroups — C₂×#1335×C₁₃₁ #45697485

Order 33536.

414 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈²×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1335×C₁₃₁	C₂	1	4	yes	maximal, direct product factor
5	C₂×#135×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×#131×C₁₃₁	—	2	1	no	—
7	C₂×M₃₂×C₁₃₁	—	2	1	no	—
8	#443×C₁₃₁	—	2	1	no	—
9	#443×C₁₃₁	—	2	1	no	—
10	#443×C₁₃₁	C₄	1	2	yes	—
11	#443×C₁₃₁	C₂²	1	2	yes	—
12	C₂×#108×C₁₃₁	C₄	1	1	yes	characteristic
13	C₂×#104×C₁₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#364×C₁₃₁	—	2	1	no	—
17	#364×C₁₃₁	C₂²	1	2	yes	complemented
18	C₈²×C₁₃₁	—	2	1	no	—
19	C₈²×C₁₃₁	C₂²	1	2	yes	complemented
20	C₂×#41×C₁₃₁	—	2	1	no	—
21	#135×C₁₃₁	—	2	1	no	—
22	#135×C₁₃₁	—	2	1	no	—
23	#135×C₁₃₁	C₈	1	2	yes	complemented
24	#135×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₁₃₁	C₈	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₁₃₁	—	2	1	no	—
27	C₂×D₈×C₁₃₁	C₈	1	1	yes	characteristic, complemented
28	#131×C₁₃₁	—	2	2	no	—
29	#131×C₁₃₁	—	2	2	no	—
30	#131×C₁₃₁	—	4	1	no	—
31	#131×C₁₃₁	—	4	1	no	—
32	C₂×M₁₆×C₁₃₁	—	2	1	no	—
33	C₂×M₁₆×C₁₃₁	—	4	1	no	—
34	C₂²×C₈×C₁₃₁	—	2	2	no	—
35	C₂²×C₈×C₁₃₁	—	4	1	no	—
36	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
39	M₃₂×C₁₃₁	—	2	2	no	—
40	M₃₂×C₁₃₁	—	4	1	no	—
41	C₂×C₁₆×C₁₃₁	—	4	1	no	—
42	#108×C₁₃₁	—	2	1	no	—
43	#108×C₁₃₁	C₂×C₄	1	2	yes	—
44	#104×C₁₃₁	—	2	2	no	—
45	#104×C₁₃₁	—	2	2	no	—
46	C₄×C₈×C₁₃₁	—	2	2	no	—
47	C₄×C₈×C₁₃₁	—	2	2	no	—
48	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
52	#41×C₁₃₁	—	2	2	no	—
53	#41×C₁₃₁	—	2	2	no	—
54	#41×C₁₃₁	—	4	1	no	—
55	#41×C₁₃₁	—	4	1	no	—
56	C₂×Q₈×C₁₃₁	—	2	1	no	—
57	C₂×D₄×C₁₃₁	—	2	1	no	—
58	C₂×D₄×C₁₃₁	—	4	1	no	—
59	C₂²×C₄×C₁₃₁	—	2	1	no	—
60	C₂²×C₄×C₁₃₁	—	4	1	no	—
61	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	Q₁₆×C₁₃₁	—	2	1	no	—
63	Q₁₆×C₁₃₁	C₂×C₈	1	2	yes	complemented
64	SD₁₆×C₁₃₁	—	2	2	no	—
65	SD₁₆×C₁₃₁	—	2	2	no	—
66	D₈×C₁₃₁	—	2	1	no	—
67	D₈×C₁₃₁	C₂×C₈	1	2	yes	complemented
68	M₁₆×C₁₃₁	—	2	2	no	—
69	M₁₆×C₁₃₁	—	4	1	no	—
70	M₁₆×C₁₃₁	—	4	2	no	—
71	M₁₆×C₁₃₁	—	4	2	no	—
72	C₂×C₈×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	2	8	no	—
74	C₂×C₈×C₁₃₁	—	4	1	no	—
75	C₂×C₈×C₁₃₁	—	4	2	no	—
76	C₂×C₈×C₁₃₁	—	4	2	no	—
77	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	M₁₆	1	1	yes	characteristic
82	C₂×C₈×C₁₃₁	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
84	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
85	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃₁	—	2	1	no	—
90	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₁₃₁	—	4	2	no	—
93	C₂³×C₁₃₁	—	4	1	no	—
94	C₂³×C₁₃₁	#104	1	1	yes	characteristic
95	Q₈×C₁₃₁	—	2	2	no	—
96	Q₈×C₁₃₁	—	4	1	no	—
97	D₄×C₁₃₁	—	2	2	no	—
98	D₄×C₁₃₁	—	4	2	no	—
99	D₄×C₁₃₁	—	4	2	no	—
100	D₄×C₁₃₁	—	4	1	no	—
101	C₂×C₄×C₁₃₁	—	2	4	no	—
102	C₂×C₄×C₁₃₁	—	2	1	no	—
103	C₂×C₄×C₁₃₁	—	2	1	no	—
104	C₂×C₄×C₁₃₁	—	4	1	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	—	4	2	no	—
107	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#104	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₁₃₁	—	2	8	no	—
114	C₈×C₁₃₁	—	4	2	no	—
115	C₈×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
117	C₈×C₁₃₁	#102	1	1	yes	characteristic
118	C₈×C₁₃₁	#102	1	1	yes	characteristic
119	C₈×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
120	C₈×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
121	C₈×C₁₃₁	C₂×D₈	1	1	yes	characteristic
122	C₈×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×C₁₃₁	—	2	2	no	—
124	C₂²×C₁₃₁	—	4	2	no	—
125	C₂²×C₁₃₁	—	8	2	no	—
126	C₂²×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₁₃₁	C₂×#104	1	1	yes	characteristic
129	C₄×C₁₃₁	—	2	2	no	—
130	C₄×C₁₃₁	—	2	2	no	—
131	C₄×C₁₃₁	—	4	2	no	—
132	C₄×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
134	C₄×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
136	C₂×C₁₃₁	—	2	1	no	—
137	C₂×C₁₃₁	—	2	1	no	—
138	C₂×C₁₃₁	—	8	2	no	—
139	C₂×C₁₃₁	#1335	1	2	yes	direct product factor
140	C₂×C₁₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
141	C₂×#1335	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₁	C₂×#1335	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#443	C₂×C₁₃₁	1	1	yes	characteristic
144	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic, complemented
145	C₂×C₈²	C₂×C₁₃₁	1	1	yes	characteristic, complemented
146	#1335	C₂×C₁₃₁	1	4	yes	direct product factor
147	C₂×#135	C₄×C₁₃₁	1	1	yes	characteristic
148	C₂×#131	—	2	1	no	—
149	C₂×M₃₂	—	2	1	no	—
150	#443	—	2	1	no	—
151	#443	—	2	1	no	—
152	#443	C₄×C₁₃₁	1	2	yes	—
153	#443	C₂²×C₁₃₁	1	2	yes	—
154	C₂×#108	C₄×C₁₃₁	1	1	yes	characteristic
155	C₂×#104	—	2	1	no	—
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
158	#364	—	2	1	no	—
159	#364	C₂²×C₁₃₁	1	2	yes	complemented
160	C₈²	—	2	1	no	—
161	C₈²	C₂²×C₁₃₁	1	2	yes	complemented
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	C₈×C₁₃₁	1	2	yes	complemented
166	#135	C₂×C₄×C₁₃₁	1	2	yes	—
167	C₂×Q₁₆	C₈×C₁₃₁	1	1	yes	characteristic, complemented
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	C₈×C₁₃₁	1	1	yes	characteristic, complemented
170	#131	—	2	2	no	—
171	#131	—	2	2	no	—
172	#131	—	4	1	no	—
173	#131	—	4	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	4	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	4	1	no	—
178	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	4	1	no	—
183	C₂×C₁₆	—	4	1	no	—
184	#108	—	2	1	no	—
185	#108	C₂×C₄×C₁₃₁	1	2	yes	—
186	#104	—	2	2	no	—
187	#104	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
191	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
192	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×C₈×C₁₃₁	1	2	yes	complemented
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×C₈×C₁₃₁	1	2	yes	complemented
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂×C₈×C₁₃₁	1	1	yes	characteristic
222	C₂×C₈	C₂×C₈×C₁₃₁	1	1	yes	characteristic
223	C₂×C₈	M₁₆×C₁₃₁	1	1	yes	characteristic
224	C₂×C₈	M₁₆×C₁₃₁	1	1	yes	characteristic
225	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
226	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
227	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
228	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
234	C₁₆	—	4	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	#104×C₁₃₁	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	Q₈	—	4	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄	#104×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	4	2	no	—
257	C₈	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
258	C₈	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
259	C₈	#102×C₁₃₁	1	1	yes	characteristic
260	C₈	#102×C₁₃₁	1	1	yes	characteristic
261	C₈	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
262	C₈	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
263	C₈	C₂×D₈×C₁₃₁	1	1	yes	characteristic
264	C₈	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
269	C₂²	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
270	C₂²	C₂×#104×C₁₃₁	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
275	C₄	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
276	C₄	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
277	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	8	2	no	—
281	C₂	#1335×C₁₃₁	1	2	yes	direct product factor
282	C₂	C₂×D₄⋊C₈×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₁	D₈
commutator	C₈	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	D₄⋊C₈
prime radical	C₂×#135×C₁₃₁	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₃₁	C₂	semidirect
3. C₂×C₈²×C₁₃₁	C₂	semidirect
4. #1335×C₁₃₁	C₂	direct (N × H)
17. #364×C₁₃₁	C₂²	semidirect
19. C₈²×C₁₃₁	C₂²	semidirect
23. #135×C₁₃₁	C₈	semidirect
25. C₂×Q₁₆×C₁₃₁	C₈	semidirect
27. C₂×D₈×C₁₃₁	C₈	semidirect
63. Q₁₆×C₁₃₁	C₂×C₈	semidirect
67. D₈×C₁₃₁	C₂×C₈	semidirect
139. C₂×C₁₃₁	#1335	direct (N × H)
141. C₂×#1335	C₁₃₁	direct (N × H)
142. C₁₃₁	C₂×#1335	direct (N × H)
144. C₂×#364	C₂×C₁₃₁	semidirect
145. C₂×C₈²	C₂×C₁₃₁	semidirect
146. #1335	C₂×C₁₃₁	direct (N × H)
159. #364	C₂²×C₁₃₁	semidirect
161. C₈²	C₂²×C₁₃₁	semidirect
165. #135	C₈×C₁₃₁	semidirect
167. C₂×Q₁₆	C₈×C₁₃₁	semidirect
169. C₂×D₈	C₈×C₁₃₁	semidirect
205. Q₁₆	C₂×C₈×C₁₃₁	semidirect
209. D₈	C₂×C₈×C₁₃₁	semidirect
281. C₂	#1335×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×#443×C₁₃₁	2
C₂×#364×C₁₃₁	2
C₂×C₈²×C₁₃₁	2
#1335×C₁₃₁	2
C₂×#1335	131

Subgroups — C2×#1335×C131 #45697485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1335×C₁₃₁ #45697485