Subgroups — C₂×#1674×C₁₃₁ #45705000

Order 33536.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1674×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#332×C₁₃₁	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₁₃₁	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	#500×C₁₃₁	—	2	2	no	—
11	#500×C₁₃₁	C₂²	1	4	yes	—
12	#462×C₁₃₁	—	2	4	no	—
13	#462×C₁₃₁	C₂²	1	8	yes	—
14	C₈×Q₈×C₁₃₁	—	2	2	no	—
15	C₈×Q₈×C₁₃₁	C₂²	1	4	yes	—
16	C₂×#107×C₁₃₁	—	2	1	no	—
17	C₂×#107×C₁₃₁	—	2	1	no	—
18	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#103×C₁₃₁	—	2	2	no	—
22	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	#335×C₁₃₁	C₂²	1	4	yes	—
25	#332×C₁₃₁	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₁	—	2	2	no	—
27	C₂²×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₁₃₁	—	2	8	no	—
29	C₂×Q₁₆×C₁₃₁	—	4	8	no	—
30	C₂²×C₈×C₁₃₁	—	2	1	no	—
31	C₂²×C₈×C₁₃₁	—	2	2	no	—
32	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
35	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
36	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
37	C₄⋊Q₈×C₁₃₁	C₂³	1	4	yes	—
38	C₄×Q₈×C₁₃₁	—	2	1	no	—
39	C₄×Q₈×C₁₃₁	—	2	1	no	—
40	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
41	C₄×Q₈×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₁	—	2	1	no	—
47	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
48	#107×C₁₃₁	—	2	4	no	—
49	#107×C₁₃₁	—	2	4	no	—
50	C₄⋊C₈×C₁₃₁	—	2	4	no	—
51	C₄⋊C₈×C₁₃₁	—	2	1	no	—
52	C₄⋊C₈×C₁₃₁	—	2	2	no	—
53	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
54	C₄⋊C₈×C₁₃₁	C₂³	1	4	yes	—
55	#103×C₁₃₁	—	2	8	no	—
56	C₄×C₈×C₁₃₁	—	2	4	no	—
57	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
58	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
59	C₂×Q₈×C₁₃₁	—	2	1	no	—
60	C₂×Q₈×C₁₃₁	—	2	2	no	—
61	C₂×Q₈×C₁₃₁	—	2	4	no	—
62	C₂×Q₈×C₁₃₁	—	4	2	no	—
63	C₂×Q₈×C₁₃₁	—	4	4	no	—
64	C₂×Q₈×C₁₃₁	—	4	4	no	—
65	C₂×Q₈×C₁₃₁	D₈	1	4	yes	—
66	C₂×Q₈×C₁₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₁	—	2	1	no	—
68	C₂²×C₄×C₁₃₁	—	2	1	no	—
69	C₂²×C₄×C₁₃₁	—	4	2	no	—
70	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
73	Q₁₆×C₁₃₁	—	4	16	no	—
74	C₂×C₈×C₁₃₁	—	2	2	no	—
75	C₂×C₈×C₁₃₁	—	2	2	no	—
76	C₂×C₈×C₁₃₁	—	2	4	no	—
77	C₂×C₈×C₁₃₁	—	4	4	no	—
78	C₂×C₈×C₁₃₁	—	4	2	no	—
79	C₂×C₈×C₁₃₁	Q₁₆	1	4	yes	complemented
80	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₄⋊C₄×C₁₃₁	—	2	2	no	—
83	C₄⋊C₄×C₁₃₁	—	2	1	no	—
84	C₄⋊C₄×C₁₃₁	—	2	2	no	—
85	C₄⋊C₄×C₁₃₁	—	2	4	no	—
86	C₄⋊C₄×C₁₃₁	—	2	4	no	—
87	C₄⋊C₄×C₁₃₁	—	4	2	no	—
88	C₄⋊C₄×C₁₃₁	—	4	2	no	—
89	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
90	C₄²×C₁₃₁	—	2	2	no	—
91	C₄²×C₁₃₁	—	2	2	no	—
92	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃₁	#41	1	1	yes	characteristic
95	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
96	C₂³×C₁₃₁	#121	1	1	yes	characteristic
97	Q₈×C₁₃₁	—	2	2	no	—
98	Q₈×C₁₃₁	—	2	4	no	—
99	Q₈×C₁₃₁	—	4	4	no	—
100	Q₈×C₁₃₁	—	4	4	no	—
101	Q₈×C₁₃₁	—	8	4	no	—
102	Q₈×C₁₃₁	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₃₁	—	2	2	no	—
104	C₂×C₄×C₁₃₁	—	2	4	no	—
105	C₂×C₄×C₁₃₁	—	2	2	no	—
106	C₂×C₄×C₁₃₁	—	2	2	no	—
107	C₂×C₄×C₁₃₁	—	4	2	no	—
108	C₂×C₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	8	4	no	—
110	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₁₃₁	#135	1	2	yes	—
116	C₂×C₄×C₁₃₁	#137	1	2	yes	—
117	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
120	C₈×C₁₃₁	—	2	2	no	—
121	C₈×C₁₃₁	—	4	4	no	—
122	C₈×C₁₃₁	—	4	2	no	—
123	C₈×C₁₃₁	C₂×Q₁₆	1	4	yes	complemented
124	C₂²×C₁₃₁	#462	1	2	yes	—
125	C₂²×C₁₃₁	#466	1	2	yes	—
126	C₂²×C₁₃₁	#473	1	2	yes	—
127	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
128	C₄×C₁₃₁	—	2	4	no	—
129	C₄×C₁₃₁	—	2	1	no	—
130	C₄×C₁₃₁	—	2	1	no	—
131	C₄×C₁₃₁	—	4	2	no	—
132	C₄×C₁₃₁	—	8	4	no	—
133	C₄×C₁₃₁	#462	1	2	yes	—
134	C₄×C₁₃₁	#466	1	2	yes	—
135	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
137	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
138	C₄×C₁₃₁	C₂×#137	1	1	yes	characteristic
139	C₂×C₁₃₁	#1674	1	4	yes	direct product factor
140	C₂×C₁₃₁	C₂×#462	1	1	yes	characteristic
141	C₂×C₁₃₁	C₂×#466	1	1	yes	characteristic
142	C₂×C₁₃₁	C₂×#473	1	1	yes	characteristic
143	C₂×#1674	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₁	C₂×#1674	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
146	C₂×#462	C₂×C₁₃₁	1	2	yes	—
147	C₂×C₈×Q₈	C₂×C₁₃₁	1	1	yes	characteristic
148	#1674	C₂×C₁₃₁	1	8	yes	direct product factor
149	C₂×#335	C₂×C₁₃₁	1	1	yes	characteristic
150	C₂×#332	C₂×C₁₃₁	1	2	yes	—
151	C₂²×Q₁₆	—	2	2	no	—
152	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
153	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
154	#500	—	2	2	no	—
155	#500	C₂²×C₁₃₁	1	4	yes	—
156	#462	—	2	4	no	—
157	#462	C₂²×C₁₃₁	1	8	yes	—
158	C₈×Q₈	—	2	2	no	—
159	C₈×Q₈	C₂²×C₁₃₁	1	4	yes	—
160	C₂×#107	—	2	1	no	—
161	C₂×#107	—	2	1	no	—
162	C₂×C₄⋊C₈	—	2	1	no	—
163	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
165	C₂×#103	—	2	2	no	—
166	C₂×C₄×C₈	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
168	#335	C₂²×C₁₃₁	1	4	yes	—
169	#332	C₂²×C₁₃₁	1	8	yes	—
170	C₂²×Q₈	—	2	2	no	—
171	C₂²×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
172	C₂×Q₁₆	—	2	8	no	—
173	C₂×Q₁₆	—	4	8	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂³×C₁₃₁	1	4	yes	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
185	C₄×Q₈	C₂³×C₁₃₁	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
198	C₄⋊C₈	C₂³×C₁₃₁	1	4	yes	—
199	#103	—	2	8	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
202	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂×Q₈	—	4	4	no	—
209	C₂×Q₈	D₈×C₁₃₁	1	4	yes	—
210	C₂×Q₈	C₂×D₄×C₁₃₁	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
217	Q₁₆	—	4	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	Q₁₆×C₁₃₁	1	4	yes	complemented
224	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
238	C₄²	#41×C₁₃₁	1	1	yes	characteristic
239	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
240	C₂³	#121×C₁₃₁	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	8	4	no	—
246	Q₈	C₂×D₈×C₁₃₁	1	4	yes	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
255	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
256	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
258	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
259	C₂×C₄	#135×C₁₃₁	1	2	yes	—
260	C₂×C₄	#137×C₁₃₁	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	4	no	—
266	C₈	—	4	2	no	—
267	C₈	C₂×Q₁₆×C₁₃₁	1	4	yes	complemented
268	C₂²	#462×C₁₃₁	1	2	yes	—
269	C₂²	#466×C₁₃₁	1	2	yes	—
270	C₂²	#473×C₁₃₁	1	2	yes	—
271	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	4	no	—
277	C₄	#462×C₁₃₁	1	2	yes	—
278	C₄	#466×C₁₃₁	1	2	yes	—
279	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
280	C₄	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
281	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
282	C₄	C₂×#137×C₁₃₁	1	1	yes	characteristic
283	C₂	#1674×C₁₃₁	1	4	yes	direct product factor
284	C₂	C₂×#462×C₁₃₁	1	1	yes	characteristic
285	C₂	C₂×#466×C₁₃₁	1	1	yes	characteristic
286	C₂	C₂×#473×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1674×C₁₃₁	C₂	direct (N × H)
79. C₂×C₈×C₁₃₁	Q₁₆	semidirect
123. C₈×C₁₃₁	C₂×Q₁₆	semidirect
139. C₂×C₁₃₁	#1674	direct (N × H)
143. C₂×#1674	C₁₃₁	direct (N × H)
144. C₁₃₁	C₂×#1674	direct (N × H)
148. #1674	C₂×C₁₃₁	direct (N × H)
223. C₂×C₈	Q₁₆×C₁₃₁	semidirect
267. C₈	C₂×Q₁₆×C₁₃₁	semidirect
283. C₂	#1674×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×#500×C₁₃₁	2
C₂×#462×C₁₃₁	2
C₂×C₈×Q₈×C₁₃₁	2
#1674×C₁₃₁	2
C₂×#335×C₁₃₁	2
C₂×#332×C₁₃₁	2
C₂×#1674	131

Subgroups — C2×#1674×C131 #45705000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1674×C₁₃₁ #45705000