Subgroups — C₁₃₁×#7622 #45693506

Order 33536.

462 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1839×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1271×C₁₃₁	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₁	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
8	C₂×#97×C₁₃₁	—	2	1	no	—
9	#391×C₁₃₁	—	2	1	no	—
10	#389×C₁₃₁	—	2	1	no	—
11	#389×C₁₃₁	C₂²	1	2	yes	—
12	#384×C₁₃₁	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₃₁	C₄	1	2	yes	—
15	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
16	#336×C₁₃₁	—	2	1	no	—
17	#336×C₁₃₁	—	2	1	no	—
18	#336×C₁₃₁	C₂²	1	2	yes	—
19	#336×C₁₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃₁	—	2	1	no	—
21	C₂²×C₈×C₁₃₁	—	2	2	no	—
22	C₂²×C₈×C₁₃₁	—	2	2	no	—
23	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
24	C₄×Q₈×C₁₃₁	—	4	4	no	—
25	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃₁	—	2	2	no	—
32	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₁	—	2	4	no	—
40	#97×C₁₃₁	—	4	2	no	—
41	#95×C₁₃₁	—	2	2	no	—
42	#95×C₁₃₁	—	2	2	no	—
43	C₂×Q₈×C₁₃₁	—	2	4	no	—
44	C₂×Q₈×C₁₃₁	—	4	2	no	—
45	C₂×Q₈×C₁₃₁	—	4	2	no	—
46	C₂²×C₄×C₁₃₁	—	2	2	no	—
47	C₂²×C₄×C₁₃₁	—	2	2	no	—
48	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃₁	—	2	4	no	—
56	C₂×C₈×C₁₃₁	—	4	2	no	—
57	C₂×C₈×C₁₃₁	—	4	2	no	—
58	C₂×C₈×C₁₃₁	—	4	2	no	—
59	C₂×C₈×C₁₃₁	—	4	2	no	—
60	C₂×C₈×C₁₃₁	—	4	2	no	—
61	C₄⋊C₄×C₁₃₁	—	2	4	no	—
62	C₄⋊C₄×C₁₃₁	—	2	2	no	—
63	C₄⋊C₄×C₁₃₁	—	4	2	no	—
64	C₄⋊C₄×C₁₃₁	—	4	2	no	—
65	C₄⋊C₄×C₁₃₁	—	4	2	no	—
66	C₄⋊C₄×C₁₃₁	—	4	4	no	—
67	C₄⋊C₄×C₁₃₁	M₁₆	1	4	yes	—
68	C₄²×C₁₃₁	—	2	2	no	—
69	C₄²×C₁₃₁	—	2	2	no	—
70	C₄²×C₁₃₁	—	2	2	no	—
71	C₄²×C₁₃₁	—	2	2	no	—
72	C₄²×C₁₃₁	—	4	1	no	—
73	C₄²×C₁₃₁	—	4	1	no	—
74	C₄²×C₁₃₁	—	4	4	no	—
75	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
76	C₄²×C₁₃₁	Q₁₆	1	2	yes	—
77	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₁₃₁	—	4	8	no	—
79	C₂×C₄×C₁₃₁	—	2	1	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
82	C₂×C₄×C₁₃₁	—	2	1	no	—
83	C₂×C₄×C₁₃₁	—	2	1	no	—
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	1	no	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	1	no	—
92	C₂×C₄×C₁₃₁	—	2	1	no	—
93	C₂×C₄×C₁₃₁	—	2	4	no	—
94	C₂×C₄×C₁₃₁	—	4	2	no	—
95	C₂×C₄×C₁₃₁	—	4	2	no	—
96	C₂×C₄×C₁₃₁	—	4	2	no	—
97	C₂×C₄×C₁₃₁	—	4	2	no	—
98	C₂×C₄×C₁₃₁	—	4	2	no	—
99	C₂×C₄×C₁₃₁	#101	1	2	yes	—
100	C₂×C₄×C₁₃₁	#103	1	2	yes	—
101	C₂×C₄×C₁₃₁	#103	1	2	yes	—
102	C₂×C₄×C₁₃₁	#104	1	2	yes	—
103	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₈×C₁₃₁	—	4	4	no	—
108	C₈×C₁₃₁	—	8	2	no	—
109	C₂²×C₁₃₁	#407	1	1	yes	characteristic
110	C₂²×C₁₃₁	#410	1	1	yes	characteristic
111	C₂²×C₁₃₁	C₂×#101	1	1	yes	characteristic
112	C₂²×C₁₃₁	C₂×#103	1	1	yes	characteristic
113	C₂²×C₁₃₁	#418	1	1	yes	characteristic
114	C₂²×C₁₃₁	C₂×#104	1	1	yes	characteristic
115	C₂²×C₁₃₁	#421	1	1	yes	characteristic
116	C₄×C₁₃₁	—	2	2	no	—
117	C₄×C₁₃₁	—	2	2	no	—
118	C₄×C₁₃₁	—	2	2	no	—
119	C₄×C₁₃₁	—	2	2	no	—
120	C₄×C₁₃₁	—	2	1	no	—
121	C₄×C₁₃₁	—	2	1	no	—
122	C₄×C₁₃₁	—	2	2	no	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	4	4	no	—
125	C₄×C₁₃₁	—	8	2	no	—
126	C₂×C₁₃₁	#1467	1	1	yes	characteristic
127	C₂×C₁₃₁	#1487	1	1	yes	characteristic
128	C₂×C₁₃₁	#1490	1	1	yes	characteristic
129	C₂×C₁₃₁	#1503	1	1	yes	characteristic
130	C₂×C₁₃₁	#1514	1	1	yes	characteristic
131	C₂×C₁₃₁	#1531	1	1	yes	characteristic
132	C₂×C₁₃₁	#1549	1	1	yes	characteristic
133	#7622	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₁	#7622	1	1	yes	characteristic, direct product factor, Sylow
135	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	1	yes	characteristic
136	#1839	C₂×C₁₃₁	1	1	yes	characteristic
137	#1767	C₂×C₁₃₁	1	1	yes	characteristic
138	#1271	C₂×C₁₃₁	1	4	yes	—
139	C₂×C₄⋊Q₈	C₄×C₁₃₁	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	2	1	no	—
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×#97	—	2	1	no	—
143	#391	—	2	1	no	—
144	#389	—	2	1	no	—
145	#389	C₂²×C₁₃₁	1	2	yes	—
146	#384	C₄×C₁₃₁	1	1	yes	characteristic
147	C₄×C₄⋊C₄	—	2	1	no	—
148	C₄×C₄⋊C₄	C₄×C₁₃₁	1	2	yes	—
149	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
150	#336	—	2	1	no	—
151	#336	—	2	1	no	—
152	#336	C₂²×C₁₃₁	1	2	yes	—
153	#336	C₂²×C₁₃₁	1	2	yes	—
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄×Q₈	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
173	C₄⋊C₈	—	2	4	no	—
174	#97	—	4	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	M₁₆×C₁₃₁	1	4	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	4	1	no	—
208	C₄²	—	4	4	no	—
209	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
210	C₄²	Q₁₆×C₁₃₁	1	2	yes	—
211	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
212	Q₈	—	4	8	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	#101×C₁₃₁	1	2	yes	—
234	C₂×C₄	#103×C₁₃₁	1	2	yes	—
235	C₂×C₄	#103×C₁₃₁	1	2	yes	—
236	C₂×C₄	#104×C₁₃₁	1	2	yes	—
237	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
238	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
240	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	8	2	no	—
243	C₂²	#407×C₁₃₁	1	1	yes	characteristic
244	C₂²	#410×C₁₃₁	1	1	yes	characteristic
245	C₂²	C₂×#101×C₁₃₁	1	1	yes	characteristic
246	C₂²	C₂×#103×C₁₃₁	1	1	yes	characteristic
247	C₂²	#418×C₁₃₁	1	1	yes	characteristic
248	C₂²	C₂×#104×C₁₃₁	1	1	yes	characteristic
249	C₂²	#421×C₁₃₁	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	4	no	—
259	C₄	—	8	2	no	—
260	C₂	#1467×C₁₃₁	1	1	yes	characteristic
261	C₂	#1487×C₁₃₁	1	1	yes	characteristic
262	C₂	#1490×C₁₃₁	1	1	yes	characteristic
263	C₂	#1503×C₁₃₁	1	1	yes	characteristic
264	C₂	#1514×C₁₃₁	1	1	yes	characteristic
265	C₂	#1531×C₁₃₁	1	1	yes	characteristic
266	C₂	#1549×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
133. #7622	C₁₃₁	direct (N × H)
134. C₁₃₁	#7622	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₄⋊Q₈×C₁₃₁	2
#1839×C₁₃₁	2
#1767×C₁₃₁	2
#1271×C₁₃₁	2
#7622	131

Subgroups — C131×#7622 #45693506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7622 #45693506