Subgroups — C₁₃₁×#8147 #45694031

Order 33536.

618 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×#322×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1273×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1273×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×M₁₆×C₁₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	4	yes	—
9	C₄×M₁₆×C₁₃₁	—	2	4	no	—
10	C₂×#97×C₁₃₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#336×C₁₃₁	C₄	1	4	yes	—
16	#322×C₁₃₁	—	2	4	no	—
17	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₃₁	—	2	4	no	—
20	C₂×M₁₆×C₁₃₁	—	2	4	no	—
21	C₂×M₁₆×C₁₃₁	—	2	4	no	—
22	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃₁	—	2	2	no	—
27	C₂×C₄²×C₁₃₁	—	2	1	no	—
28	C₂×C₄²×C₁₃₁	—	2	1	no	—
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₂²⋊C₈×C₁₃₁	—	2	8	no	—
33	#97×C₁₃₁	—	2	4	no	—
34	C₄×C₈×C₁₃₁	—	2	2	no	—
35	C₄×C₈×C₁₃₁	—	2	2	no	—
36	C₄×C₈×C₁₃₁	—	2	2	no	—
37	C₄×C₈×C₁₃₁	—	2	2	no	—
38	C₄×C₈×C₁₃₁	—	2	2	no	—
39	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
40	C₂⁴×C₁₃₁	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	—	2	1	no	—
42	C₂²×C₄×C₁₃₁	—	2	1	no	—
43	C₂²×C₄×C₁₃₁	—	2	2	no	—
44	C₂²×C₄×C₁₃₁	—	2	2	no	—
45	C₂²×C₄×C₁₃₁	—	2	2	no	—
46	C₂²×C₄×C₁₃₁	—	2	4	no	—
47	C₂²×C₄×C₁₃₁	—	2	4	no	—
48	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	M₁₆	1	4	yes	—
51	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
53	M₁₆×C₁₃₁	—	2	16	no	—
54	C₂×C₈×C₁₃₁	—	2	2	no	—
55	C₂×C₈×C₁₃₁	—	2	4	no	—
56	C₂×C₈×C₁₃₁	—	2	2	no	—
57	C₂×C₈×C₁₃₁	—	2	2	no	—
58	C₂×C₈×C₁₃₁	—	2	2	no	—
59	C₂×C₈×C₁₃₁	—	2	4	no	—
60	C₂×C₈×C₁₃₁	—	2	4	no	—
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
64	C₂×C₈×C₁₃₁	M₁₆	1	4	yes	—
65	C₂×C₈×C₁₃₁	M₁₆	1	8	yes	complemented
66	C₄²×C₁₃₁	—	2	1	no	—
67	C₄²×C₁₃₁	—	2	1	no	—
68	C₄²×C₁₃₁	—	2	2	no	—
69	C₄²×C₁₃₁	—	2	2	no	—
70	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
71	C₄²×C₁₃₁	#41	1	1	yes	characteristic
72	C₄²×C₁₃₁	#41	1	1	yes	characteristic
73	C₂³×C₁₃₁	—	2	1	no	—
74	C₂³×C₁₃₁	—	2	2	no	—
75	C₂³×C₁₃₁	—	2	1	no	—
76	C₂³×C₁₃₁	—	2	2	no	—
77	C₂³×C₁₃₁	#97	1	2	yes	—
78	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	—	2	1	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	4	no	—
82	C₂×C₄×C₁₃₁	—	2	4	no	—
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	4	4	no	—
87	C₂×C₄×C₁₃₁	—	4	4	no	—
88	C₂×C₄×C₁₃₁	#97	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂×M₁₆	1	4	yes	—
96	C₂×C₄×C₁₃₁	#131	1	4	yes	—
97	C₈×C₁₃₁	—	2	4	no	—
98	C₈×C₁₃₁	—	2	4	no	—
99	C₈×C₁₃₁	—	2	8	no	—
100	C₈×C₁₃₁	—	4	4	no	—
101	C₂²×C₁₃₁	—	2	4	no	—
102	C₂²×C₁₃₁	—	2	2	no	—
103	C₂²×C₁₃₁	—	4	1	no	—
104	C₂²×C₁₃₁	—	4	1	no	—
105	C₂²×C₁₃₁	—	4	2	no	—
106	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₃₁	C₂×#97	1	1	yes	characteristic
108	C₂²×C₁₃₁	C₄×M₁₆	1	2	yes	—
109	C₂²×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₁₃₁	#405	1	2	yes	—
111	C₄×C₁₃₁	—	2	2	no	—
112	C₄×C₁₃₁	—	4	2	no	—
113	C₄×C₁₃₁	#407	1	2	yes	—
114	C₄×C₁₃₁	#432	1	2	yes	—
115	C₄×C₁₃₁	#435	1	4	yes	—
116	C₄×C₁₃₁	#436	1	4	yes	—
117	C₂×C₁₃₁	—	4	2	no	—
118	C₂×C₁₃₁	#1451	1	2	yes	—
119	C₂×C₁₃₁	#1453	1	1	yes	characteristic
120	C₂×C₁₃₁	#1455	1	1	yes	characteristic
121	C₂×C₁₃₁	#1749	1	1	yes	characteristic
122	C₂×C₁₃₁	#1753	1	2	yes	—
123	#8147	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₁₃₁	#8147	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₄×M₁₆	C₂×C₁₃₁	1	1	yes	characteristic
126	C₄×C₂²⋊C₈	C₂×C₁₃₁	1	2	yes	—
127	C₂×#322	C₂×C₁₃₁	1	2	yes	complemented
128	#1273	C₂×C₁₃₁	1	1	yes	characteristic, complemented
129	#1273	C₂×C₁₃₁	1	1	yes	characteristic, complemented
130	C₂²×M₁₆	C₄×C₁₃₁	1	2	yes	—
131	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
132	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	4	yes	—
133	C₄×M₁₆	—	2	4	no	—
134	C₂×#97	C₄×C₁₃₁	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	#336	C₄×C₁₃₁	1	4	yes	—
140	#322	—	2	4	no	—
141	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	4	no	—
146	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
147	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
148	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
149	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
156	C₂²⋊C₈	—	2	8	no	—
157	#97	—	2	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
164	C₂⁴	C₄²×C₁₃₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
173	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
174	C₂²×C₄	M₁₆×C₁₃₁	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
177	M₁₆	—	2	16	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
188	C₂×C₈	M₁₆×C₁₃₁	1	4	yes	—
189	C₂×C₈	M₁₆×C₁₃₁	1	8	yes	complemented
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
195	C₄²	#41×C₁₃₁	1	1	yes	characteristic
196	C₄²	#41×C₁₃₁	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	#97×C₁₃₁	1	2	yes	—
202	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	#97×C₁₃₁	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
216	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
217	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
218	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
219	C₂×C₄	C₂×M₁₆×C₁₃₁	1	4	yes	—
220	C₂×C₄	#131×C₁₃₁	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	2	4	no	—
223	C₈	—	2	8	no	—
224	C₈	—	4	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
231	C₂²	C₂×#97×C₁₃₁	1	1	yes	characteristic
232	C₂²	C₄×M₁₆×C₁₃₁	1	2	yes	—
233	C₂²	C₄×M₁₆×C₁₃₁	1	1	yes	characteristic
234	C₂²	#405×C₁₃₁	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	4	2	no	—
237	C₄	#407×C₁₃₁	1	2	yes	—
238	C₄	#432×C₁₃₁	1	2	yes	—
239	C₄	#435×C₁₃₁	1	4	yes	—
240	C₄	#436×C₁₃₁	1	4	yes	—
241	C₂	—	4	2	no	—
242	C₂	#1451×C₁₃₁	1	2	yes	—
243	C₂	#1453×C₁₃₁	1	1	yes	characteristic
244	C₂	#1455×C₁₃₁	1	1	yes	characteristic
245	C₂	#1749×C₁₃₁	1	1	yes	characteristic
246	C₂	#1753×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×#322×C₁₃₁	C₂	semidirect
4. #1273×C₁₃₁	C₂	semidirect
5. #1273×C₁₃₁	C₂	semidirect
65. C₂×C₈×C₁₃₁	M₁₆	semidirect
123. #8147	C₁₃₁	direct (N × H)
124. C₁₃₁	#8147	direct (N × H)
127. C₂×#322	C₂×C₁₃₁	semidirect
128. #1273	C₂×C₁₃₁	semidirect
129. #1273	C₂×C₁₃₁	semidirect
189. C₂×C₈	M₁₆×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₁₃₁	2
C₄×C₂²⋊C₈×C₁₃₁	2
C₂×#322×C₁₃₁	2
#1273×C₁₃₁	2
#8147	131

Subgroups — C131×#8147 #45694031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8147 #45694031