Subgroups — C₂×#2848×C₁₃₁ #45722496

Order 33536.

7502 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	C₂⁵×C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Q₈×C₁₃₁	C₂	1	3	yes	maximal, complemented
3	#2848×C₁₃₁	C₂	1	16	yes	maximal, direct product factor
4	C₂×#385×C₁₃₁	C₂	1	6	yes	maximal, complemented
5	C₂×#380×C₁₃₁	C₂	1	3	yes	maximal, complemented
6	C₂×#379×C₁₃₁	C₂	1	2	yes	maximal
7	C₂⁶×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂³×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
9	C₂⁴×C₄×C₁₃₁	—	2	24	no	—
10	C₂⁴×C₄×C₁₃₁	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₁₃₁	C₂²	1	8	yes	—
12	C₂⁴×C₄×C₁₃₁	C₂²	1	3	yes	—
13	C₂×C₂²⋊Q₈×C₁₃₁	—	2	24	no	—
14	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	3	yes	complemented
16	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	6	yes	complemented
17	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	6	yes	—
19	#385×C₁₃₁	C₂²	1	48	yes	complemented
20	#380×C₁₃₁	C₂²	1	24	yes	complemented
21	#379×C₁₃₁	C₂²	1	16	yes	—
22	C₂×#95×C₁₃₁	C₂²	1	6	yes	complemented
23	C₂⁵×C₁₃₁	—	2	12	no	—
24	C₂⁵×C₁₃₁	—	2	12	no	—
25	C₂⁵×C₁₃₁	Q₈	1	8	yes	complemented
26	C₂⁵×C₁₃₁	C₂³	1	4	yes	—
27	C₂⁵×C₁₃₁	C₂³	1	3	yes	—
28	C₂²×Q₈×C₁₃₁	—	2	12	no	—
29	C₂²×Q₈×C₁₃₁	C₂³	1	4	yes	complemented
30	C₂³×C₄×C₁₃₁	—	2	128	no	—
31	C₂³×C₄×C₁₃₁	—	2	12	no	—
32	C₂³×C₄×C₁₃₁	—	2	24	no	—
33	C₂³×C₄×C₁₃₁	—	2	96	no	—
34	C₂³×C₄×C₁₃₁	—	2	24	no	—
35	C₂³×C₄×C₁₃₁	D₄	1	24	yes	—
36	C₂³×C₄×C₁₃₁	C₂³	1	12	yes	—
37	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃₁	C₂³	1	6	yes	—
39	C₂³×C₄×C₁₃₁	C₂³	1	12	yes	—
40	C₂³×C₄×C₁₃₁	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂²⋊Q₈×C₁₃₁	—	2	96	no	—
43	C₂×C₄⋊C₄×C₁₃₁	—	2	12	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	24	no	—
45	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	12	yes	complemented
46	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	24	yes	complemented
47	C₂×C₂²⋊C₄×C₁₃₁	—	2	24	no	—
48	C₂×C₂²⋊C₄×C₁₃₁	—	2	24	no	—
49	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	24	yes	—
50	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	24	yes	—
51	#95×C₁₃₁	C₂³	1	24	yes	complemented
52	C₂⁴×C₁₃₁	—	2	32	no	—
53	C₂⁴×C₁₃₁	—	2	96	no	—
54	C₂⁴×C₁₃₁	—	2	96	no	—
55	C₂⁴×C₁₃₁	—	2	24	no	—
56	C₂⁴×C₁₃₁	—	2	12	no	—
57	C₂⁴×C₁₃₁	—	2	24	no	—
58	C₂⁴×C₁₃₁	—	2	12	no	—
59	C₂⁴×C₁₃₁	C₂×D₄	1	12	yes	—
60	C₂⁴×C₁₃₁	C₂×Q₈	1	12	yes	—
61	C₂⁴×C₁₃₁	C₂×Q₈	1	16	yes	complemented
62	C₂⁴×C₁₃₁	#41	1	12	yes	—
63	C₂⁴×C₁₃₁	C₂⁴	1	6	yes	—
64	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
65	C₂×Q₈×C₁₃₁	—	2	24	no	—
66	C₂×Q₈×C₁₃₁	—	4	16	no	—
67	C₂²×C₄×C₁₃₁	—	2	96	no	—
68	C₂²×C₄×C₁₃₁	—	2	256	no	—
69	C₂²×C₄×C₁₃₁	—	2	24	no	—
70	C₂²×C₄×C₁₃₁	—	2	12	no	—
71	C₂²×C₄×C₁₃₁	—	2	96	no	—
72	C₂²×C₄×C₁₃₁	—	2	48	no	—
73	C₂²×C₄×C₁₃₁	—	2	48	no	—
74	C₂²×C₄×C₁₃₁	—	2	12	no	—
75	C₂²×C₄×C₁₃₁	—	2	12	no	—
76	C₂²×C₄×C₁₃₁	—	2	24	no	—
77	C₂²×C₄×C₁₃₁	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₁₃₁	C₂×D₄	1	48	yes	—
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	12	yes	—
80	C₂²×C₄×C₁₃₁	C₂⁴	1	12	yes	—
81	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₁₃₁	C₂⁴	1	6	yes	—
83	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
84	C₄⋊C₄×C₁₃₁	—	2	24	no	—
85	C₄⋊C₄×C₁₃₁	—	2	48	no	—
86	C₂²⋊C₄×C₁₃₁	—	2	48	no	—
87	C₂²⋊C₄×C₁₃₁	—	2	48	no	—
88	C₂³×C₁₃₁	—	2	32	no	—
89	C₂³×C₁₃₁	—	2	256	no	—
90	C₂³×C₁₃₁	—	2	96	no	—
91	C₂³×C₁₃₁	—	2	24	no	—
92	C₂³×C₁₃₁	—	2	48	no	—
93	C₂³×C₁₃₁	—	2	96	no	—
94	C₂³×C₁₃₁	—	2	24	no	—
95	C₂³×C₁₃₁	—	2	48	no	—
96	C₂³×C₁₃₁	—	2	12	no	—
97	C₂³×C₁₃₁	—	2	12	no	—
98	C₂³×C₁₃₁	C₂²⋊Q₈	1	48	yes	complemented
99	C₂³×C₁₃₁	C₂²×D₄	1	12	yes	—
100	C₂³×C₁₃₁	C₂²×D₄	1	6	yes	—
101	C₂³×C₁₃₁	C₂²×Q₈	1	12	yes	—
102	C₂³×C₁₃₁	C₂²×Q₈	1	2	yes	—
103	C₂³×C₁₃₁	C₂×#41	1	12	yes	—
104	C₂³×C₁₃₁	C₂×#41	1	6	yes	—
105	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
106	Q₈×C₁₃₁	—	4	16	no	—
107	C₂×C₄×C₁₃₁	—	2	96	no	—
108	C₂×C₄×C₁₃₁	—	2	24	no	—
109	C₂×C₄×C₁₃₁	—	2	12	no	—
110	C₂×C₄×C₁₃₁	—	2	96	no	—
111	C₂×C₄×C₁₃₁	—	2	12	no	—
112	C₂×C₄×C₁₃₁	—	2	12	no	—
113	C₂×C₄×C₁₃₁	—	2	24	no	—
114	C₂×C₄×C₁₃₁	—	4	8	no	—
115	C₂×C₄×C₁₃₁	—	4	24	no	—
116	C₂×C₄×C₁₃₁	#120	1	16	yes	—
117	C₂×C₄×C₁₃₁	C₂²×D₄	1	12	yes	—
118	C₂×C₄×C₁₃₁	C₂²×D₄	1	12	yes	—
119	C₂²×C₁₃₁	—	2	128	no	—
120	C₂²×C₁₃₁	—	2	96	no	—
121	C₂²×C₁₃₁	—	2	12	no	—
122	C₂²×C₁₃₁	—	2	24	no	—
123	C₂²×C₁₃₁	—	2	12	no	—
124	C₂²×C₁₃₁	—	2	24	no	—
125	C₂²×C₁₃₁	C₂×#120	1	4	yes	—
126	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	12	yes	—
127	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	24	yes	complemented
128	C₂²×C₁₃₁	#525	1	12	yes	—
129	C₂²×C₁₃₁	C₂³×D₄	1	3	yes	—
130	C₂²×C₁₃₁	C₂³×Q₈	1	1	yes	characteristic
131	C₂²×C₁₃₁	C₂²×#41	1	3	yes	—
132	C₄×C₁₃₁	—	2	12	no	—
133	C₄×C₁₃₁	—	4	4	no	—
134	C₄×C₁₃₁	—	4	12	no	—
135	C₄×C₁₃₁	C₂×#120	1	8	yes	—
136	C₂×C₁₃₁	—	2	24	no	—
137	C₂×C₁₃₁	#2848	1	8	yes	direct product factor
138	C₂×C₁₃₁	C₂²×#120	1	1	yes	characteristic
139	C₂×C₁₃₁	C₂²×C₂²⋊Q₈	1	3	yes	—
140	C₂×C₁₃₁	C₂×#525	1	3	yes	—
141	C₂×#2848	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₁	C₂×#2848	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁵×C₄	C₂×C₁₃₁	1	1	yes	characteristic
144	C₂²×C₂²⋊Q₈	C₂×C₁₃₁	1	3	yes	complemented
145	#2848	C₂×C₁₃₁	1	16	yes	direct product factor
146	C₂×#385	C₂×C₁₃₁	1	6	yes	complemented
147	C₂×#380	C₂×C₁₃₁	1	3	yes	complemented
148	C₂×#379	C₂×C₁₃₁	1	2	yes	—
149	C₂⁶	C₂²×C₁₃₁	1	1	yes	characteristic
150	C₂³×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
151	C₂⁴×C₄	—	2	24	no	—
152	C₂⁴×C₄	C₂²×C₁₃₁	1	8	yes	—
153	C₂⁴×C₄	C₂²×C₁₃₁	1	3	yes	—
154	C₂⁴×C₄	C₂²×C₁₃₁	1	3	yes	—
155	C₂×C₂²⋊Q₈	—	2	24	no	—
156	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	24	yes	complemented
157	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	3	yes	complemented
158	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	6	yes	complemented
159	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	6	yes	—
160	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	6	yes	—
161	#385	C₂²×C₁₃₁	1	48	yes	complemented
162	#380	C₂²×C₁₃₁	1	24	yes	complemented
163	#379	C₂²×C₁₃₁	1	16	yes	—
164	C₂×#95	C₂²×C₁₃₁	1	6	yes	complemented
165	C₂⁵	—	2	12	no	—
166	C₂⁵	—	2	12	no	—
167	C₂⁵	Q₈×C₁₃₁	1	8	yes	complemented
168	C₂⁵	C₂³×C₁₃₁	1	4	yes	—
169	C₂⁵	C₂³×C₁₃₁	1	3	yes	—
170	C₂²×Q₈	—	2	12	no	—
171	C₂²×Q₈	C₂³×C₁₃₁	1	4	yes	complemented
172	C₂³×C₄	—	2	128	no	—
173	C₂³×C₄	—	2	12	no	—
174	C₂³×C₄	—	2	24	no	—
175	C₂³×C₄	—	2	96	no	—
176	C₂³×C₄	—	2	24	no	—
177	C₂³×C₄	D₄×C₁₃₁	1	24	yes	—
178	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
179	C₂³×C₄	C₂³×C₁₃₁	1	6	yes	—
180	C₂³×C₄	C₂³×C₁₃₁	1	12	yes	—
181	C₂³×C₄	C₂³×C₁₃₁	1	3	yes	—
182	C₂³×C₄	C₂³×C₁₃₁	1	12	yes	—
183	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
184	C₂²⋊Q₈	—	2	96	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	24	no	—
187	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	12	yes	complemented
188	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	24	yes	complemented
189	C₂×C₂²⋊C₄	—	2	24	no	—
190	C₂×C₂²⋊C₄	—	2	24	no	—
191	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	24	yes	—
192	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	24	yes	—
193	#95	C₂³×C₁₃₁	1	24	yes	complemented
194	C₂⁴	—	2	32	no	—
195	C₂⁴	—	2	96	no	—
196	C₂⁴	—	2	96	no	—
197	C₂⁴	—	2	24	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	24	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	C₂×D₄×C₁₃₁	1	12	yes	—
202	C₂⁴	C₂×Q₈×C₁₃₁	1	12	yes	—
203	C₂⁴	C₂×Q₈×C₁₃₁	1	16	yes	complemented
204	C₂⁴	#41×C₁₃₁	1	12	yes	—
205	C₂⁴	C₂⁴×C₁₃₁	1	6	yes	—
206	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
207	C₂×Q₈	—	2	24	no	—
208	C₂×Q₈	—	4	16	no	—
209	C₂²×C₄	—	2	96	no	—
210	C₂²×C₄	—	2	256	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	96	no	—
214	C₂²×C₄	—	2	48	no	—
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	2	12	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	24	no	—
219	C₂²×C₄	C₂×D₄×C₁₃₁	1	12	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃₁	1	48	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃₁	1	12	yes	—
222	C₂²×C₄	C₂⁴×C₁₃₁	1	12	yes	—
223	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
224	C₂²×C₄	C₂⁴×C₁₃₁	1	6	yes	—
225	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
226	C₄⋊C₄	—	2	24	no	—
227	C₄⋊C₄	—	2	48	no	—
228	C₂²⋊C₄	—	2	48	no	—
229	C₂²⋊C₄	—	2	48	no	—
230	C₂³	—	2	32	no	—
231	C₂³	—	2	256	no	—
232	C₂³	—	2	96	no	—
233	C₂³	—	2	24	no	—
234	C₂³	—	2	48	no	—
235	C₂³	—	2	96	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	2	48	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	12	no	—
240	C₂³	C₂²⋊Q₈×C₁₃₁	1	48	yes	complemented
241	C₂³	C₂²×D₄×C₁₃₁	1	12	yes	—
242	C₂³	C₂²×D₄×C₁₃₁	1	6	yes	—
243	C₂³	C₂²×Q₈×C₁₃₁	1	12	yes	—
244	C₂³	C₂²×Q₈×C₁₃₁	1	2	yes	—
245	C₂³	C₂×#41×C₁₃₁	1	12	yes	—
246	C₂³	C₂×#41×C₁₃₁	1	6	yes	—
247	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
248	Q₈	—	4	16	no	—
249	C₂×C₄	—	2	96	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	12	no	—
252	C₂×C₄	—	2	96	no	—
253	C₂×C₄	—	2	12	no	—
254	C₂×C₄	—	2	12	no	—
255	C₂×C₄	—	2	24	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	24	no	—
258	C₂×C₄	#120×C₁₃₁	1	16	yes	—
259	C₂×C₄	C₂²×D₄×C₁₃₁	1	12	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃₁	1	12	yes	—
261	C₂²	—	2	128	no	—
262	C₂²	—	2	96	no	—
263	C₂²	—	2	12	no	—
264	C₂²	—	2	24	no	—
265	C₂²	—	2	12	no	—
266	C₂²	—	2	24	no	—
267	C₂²	C₂×#120×C₁₃₁	1	4	yes	—
268	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	12	yes	—
269	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	24	yes	complemented
270	C₂²	#525×C₁₃₁	1	12	yes	—
271	C₂²	C₂³×D₄×C₁₃₁	1	3	yes	—
272	C₂²	C₂³×Q₈×C₁₃₁	1	1	yes	characteristic
273	C₂²	C₂²×#41×C₁₃₁	1	3	yes	—
274	C₄	—	2	12	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	12	no	—
277	C₄	C₂×#120×C₁₃₁	1	8	yes	—
278	C₂	—	2	24	no	—
279	C₂	#2848×C₁₃₁	1	8	yes	direct product factor
280	C₂	C₂²×#120×C₁₃₁	1	1	yes	characteristic
281	C₂	C₂²×C₂²⋊Q₈×C₁₃₁	1	3	yes	—
282	C₂	C₂×#525×C₁₃₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂⁶×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
2. C₂²×C₂²⋊Q₈×C₁₃₁	C₂	semidirect
3. #2848×C₁₃₁	C₂	direct (N × H)
4. C₂×#385×C₁₃₁	C₂	semidirect
5. C₂×#380×C₁₃₁	C₂	semidirect
8. C₂³×Q₈×C₁₃₁	C₂²	semidirect
14. C₂×C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
15. C₂²×C₄⋊C₄×C₁₃₁	C₂²	semidirect
16. C₂²×C₄⋊C₄×C₁₃₁	C₂²	semidirect
19. #385×C₁₃₁	C₂²	semidirect
20. #380×C₁₃₁	C₂²	semidirect
22. C₂×#95×C₁₃₁	C₂²	semidirect
25. C₂⁵×C₁₃₁	Q₈	semidirect
29. C₂²×Q₈×C₁₃₁	C₂³	semidirect
45. C₂×C₄⋊C₄×C₁₃₁	C₂³	semidirect
46. C₂×C₄⋊C₄×C₁₃₁	C₂³	semidirect
51. #95×C₁₃₁	C₂³	semidirect
61. C₂⁴×C₁₃₁	C₂×Q₈	semidirect
98. C₂³×C₁₃₁	C₂²⋊Q₈	semidirect
127. C₂²×C₁₃₁	C₂×C₂²⋊Q₈	semidirect
137. C₂×C₁₃₁	#2848	direct (N × H)
141. C₂×#2848	C₁₃₁	direct (N × H)
142. C₁₃₁	C₂×#2848	direct (N × H)
144. C₂²×C₂²⋊Q₈	C₂×C₁₃₁	semidirect
145. #2848	C₂×C₁₃₁	direct (N × H)
146. C₂×#385	C₂×C₁₃₁	semidirect
147. C₂×#380	C₂×C₁₃₁	semidirect
150. C₂³×Q₈	C₂²×C₁₃₁	semidirect
156. C₂×C₂²⋊Q₈	C₂²×C₁₃₁	semidirect
157. C₂²×C₄⋊C₄	C₂²×C₁₃₁	semidirect
158. C₂²×C₄⋊C₄	C₂²×C₁₃₁	semidirect
161. #385	C₂²×C₁₃₁	semidirect
162. #380	C₂²×C₁₃₁	semidirect
164. C₂×#95	C₂²×C₁₃₁	semidirect
167. C₂⁵	Q₈×C₁₃₁	semidirect
171. C₂²×Q₈	C₂³×C₁₃₁	semidirect
187. C₂×C₄⋊C₄	C₂³×C₁₃₁	semidirect
188. C₂×C₄⋊C₄	C₂³×C₁₃₁	semidirect
193. #95	C₂³×C₁₃₁	semidirect
203. C₂⁴	C₂×Q₈×C₁₃₁	semidirect
240. C₂³	C₂²⋊Q₈×C₁₃₁	semidirect
269. C₂²	C₂×C₂²⋊Q₈×C₁₃₁	semidirect
279. C₂	#2848×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
C₂⁵×C₄×C₁₃₁	2
C₂²×C₂²⋊Q₈×C₁₃₁	2
#2848×C₁₃₁	2
C₂×#385×C₁₃₁	2
C₂×#380×C₁₃₁	2
C₂×#379×C₁₃₁	2
C₂×#2848	131

Subgroups — C2×#2848×C131 #45722496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2848×C₁₃₁ #45722496