Subgroups — C₁₃₁×#23723 #45709607

Order 33536.

972 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₃₁	C₂	1	3	yes	maximal, complemented
2	#2879×C₁₃₁	C₂	1	3	yes	maximal
3	#2091×C₁₃₁	C₂	1	8	yes	maximal, complemented
4	C₂×#338×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#41×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₃₁	C₂²	1	3	yes	—
7	C₂²×M₁₆×C₁₃₁	C₂²	1	3	yes	—
8	#430×C₁₃₁	—	2	6	no	—
9	#430×C₁₃₁	C₂²	1	12	yes	—
10	C₂×#108×C₁₃₁	—	2	6	no	—
11	#408×C₁₃₁	—	2	6	no	—
12	#408×C₁₃₁	—	2	6	no	—
13	#407×C₁₃₁	C₂²	1	12	yes	—
14	#338×C₁₃₁	C₂²	1	4	yes	—
15	C₂×#41×C₁₃₁	—	2	3	no	—
16	C₂×#41×C₁₃₁	—	2	3	no	—
17	C₂×#41×C₁₃₁	—	2	3	no	—
18	C₂×#41×C₁₃₁	—	2	3	no	—
19	C₂²×Q₈×C₁₃₁	Q₈	1	1	yes	characteristic
20	C₂²×D₄×C₁₃₁	Q₈	1	2	yes	—
21	C₂²×D₄×C₁₃₁	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₃₁	—	2	6	no	—
25	C₂×M₁₆×C₁₃₁	—	2	6	no	—
26	C₂×M₁₆×C₁₃₁	—	2	3	no	—
27	C₂×M₁₆×C₁₃₁	—	2	6	no	—
28	C₂×M₁₆×C₁₃₁	—	2	6	no	—
29	C₂×M₁₆×C₁₃₁	D₄	1	6	yes	—
30	C₂×M₁₆×C₁₃₁	C₂³	1	6	yes	—
31	C₂×M₁₆×C₁₃₁	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₃₁	—	2	3	no	—
33	C₂²×C₈×C₁₃₁	D₄	1	6	yes	—
34	#108×C₁₃₁	—	2	24	no	—
35	C₂²⋊C₈×C₁₃₁	—	2	24	no	—
36	C₂⁴×C₁₃₁	C₂×Q₈	1	2	yes	—
37	C₂⁴×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
38	#41×C₁₃₁	—	4	3	no	—
39	#41×C₁₃₁	—	4	3	no	—
40	#41×C₁₃₁	—	4	6	no	—
41	#41×C₁₃₁	—	4	3	no	—
42	#41×C₁₃₁	—	4	1	no	—
43	C₂×Q₈×C₁₃₁	—	2	3	no	—
44	C₂×Q₈×C₁₃₁	—	2	3	no	—
45	C₂×D₄×C₁₃₁	—	2	6	no	—
46	C₂×D₄×C₁₃₁	—	2	6	no	—
47	C₂×D₄×C₁₃₁	—	2	3	no	—
48	C₂×D₄×C₁₃₁	—	2	3	no	—
49	C₂²×C₄×C₁₃₁	—	2	6	no	—
50	C₂²×C₄×C₁₃₁	—	2	6	no	—
51	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
52	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	#41	1	3	yes	—
56	C₂²×C₄×C₁₃₁	#41	1	3	yes	—
57	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₁₃₁	—	2	12	no	—
59	M₁₆×C₁₃₁	—	2	12	no	—
60	M₁₆×C₁₃₁	—	2	12	no	—
61	M₁₆×C₁₃₁	—	4	3	no	—
62	M₁₆×C₁₃₁	—	4	3	no	—
63	C₂×C₈×C₁₃₁	—	2	6	no	—
64	C₂×C₈×C₁₃₁	—	2	6	no	—
65	C₂×C₈×C₁₃₁	—	2	6	no	—
66	C₂×C₈×C₁₃₁	—	4	6	no	—
67	C₂×C₈×C₁₃₁	C₂×D₄	1	12	yes	—
68	C₂³×C₁₃₁	—	2	6	no	—
69	C₂³×C₁₃₁	—	8	1	no	—
70	C₂³×C₁₃₁	—	8	2	no	—
71	C₂³×C₁₃₁	C₂²⋊Q₈	1	3	yes	—
72	C₂³×C₁₃₁	C₂²⋊Q₈	1	3	yes	—
73	C₂³×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
74	Q₈×C₁₃₁	—	4	1	no	—
75	Q₈×C₁₃₁	—	4	3	no	—
76	D₄×C₁₃₁	—	4	2	no	—
77	D₄×C₁₃₁	—	4	6	no	—
78	D₄×C₁₃₁	—	4	1	no	—
79	D₄×C₁₃₁	—	4	3	no	—
80	C₂×C₄×C₁₃₁	—	2	6	no	—
81	C₂×C₄×C₁₃₁	—	4	6	no	—
82	C₂×C₄×C₁₃₁	—	4	2	no	—
83	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	3	yes	—
84	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	3	yes	—
85	C₂×C₄×C₁₃₁	#123	1	3	yes	—
86	C₂×C₄×C₁₃₁	#123	1	3	yes	—
87	C₂×C₄×C₁₃₁	#123	1	6	yes	—
88	C₂×C₄×C₁₃₁	#127	1	3	yes	—
89	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₁₃₁	C₂×#41	1	3	yes	—
92	C₈×C₁₃₁	—	2	12	no	—
93	C₈×C₁₃₁	—	4	6	no	—
94	C₂²×C₁₃₁	—	4	1	no	—
95	C₂²×C₁₃₁	—	4	3	no	—
96	C₂²×C₁₃₁	—	4	3	no	—
97	C₂²×C₁₃₁	—	4	12	no	—
98	C₂²×C₁₃₁	—	4	2	no	—
99	C₂²×C₁₃₁	#399	1	1	yes	characteristic
100	C₂²×C₁₃₁	#399	1	3	yes	—
101	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	3	yes	—
102	C₄×C₁₃₁	—	4	2	no	—
103	C₄×C₁₃₁	#399	1	1	yes	characteristic
104	C₄×C₁₃₁	#399	1	3	yes	—
105	C₄×C₁₃₁	C₂×#123	1	3	yes	—
106	C₄×C₁₃₁	C₂×#127	1	1	yes	characteristic
107	C₂×C₁₃₁	—	2	3	no	—
108	C₂×C₁₃₁	—	2	1	no	—
109	C₂×C₁₃₁	—	2	3	no	—
110	C₂×C₁₃₁	—	8	2	no	—
111	C₂×C₁₃₁	C₂×#399	1	1	yes	characteristic
112	#23723	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₁	#23723	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#430	C₂×C₁₃₁	1	3	yes	complemented
115	#2879	C₂×C₁₃₁	1	3	yes	—
116	#2091	C₂×C₁₃₁	1	8	yes	complemented
117	C₂×#338	C₂×C₁₃₁	1	1	yes	characteristic, complemented
118	C₂²×#41	C₂²×C₁₃₁	1	1	yes	characteristic
119	C₂²×M₁₆	C₂²×C₁₃₁	1	3	yes	—
120	C₂²×M₁₆	C₂²×C₁₃₁	1	3	yes	—
121	#430	—	2	6	no	—
122	#430	C₂²×C₁₃₁	1	12	yes	—
123	C₂×#108	—	2	6	no	—
124	#408	—	2	6	no	—
125	#408	—	2	6	no	—
126	#407	C₂²×C₁₃₁	1	12	yes	—
127	#338	C₂²×C₁₃₁	1	4	yes	—
128	C₂×#41	—	2	3	no	—
129	C₂×#41	—	2	3	no	—
130	C₂×#41	—	2	3	no	—
131	C₂×#41	—	2	3	no	—
132	C₂²×Q₈	Q₈×C₁₃₁	1	1	yes	characteristic
133	C₂²×D₄	Q₈×C₁₃₁	1	2	yes	—
134	C₂²×D₄	Q₈×C₁₃₁	1	1	yes	characteristic
135	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
136	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
137	C₂×M₁₆	—	2	6	no	—
138	C₂×M₁₆	—	2	6	no	—
139	C₂×M₁₆	—	2	3	no	—
140	C₂×M₁₆	—	2	6	no	—
141	C₂×M₁₆	—	2	6	no	—
142	C₂×M₁₆	D₄×C₁₃₁	1	6	yes	—
143	C₂×M₁₆	C₂³×C₁₃₁	1	6	yes	—
144	C₂×M₁₆	C₂³×C₁₃₁	1	6	yes	—
145	C₂²×C₈	—	2	3	no	—
146	C₂²×C₈	D₄×C₁₃₁	1	6	yes	—
147	#108	—	2	24	no	—
148	C₂²⋊C₈	—	2	24	no	—
149	C₂⁴	C₂×Q₈×C₁₃₁	1	2	yes	—
150	C₂⁴	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
151	#41	—	4	3	no	—
152	#41	—	4	3	no	—
153	#41	—	4	6	no	—
154	#41	—	4	3	no	—
155	#41	—	4	1	no	—
156	C₂×Q₈	—	2	3	no	—
157	C₂×Q₈	—	2	3	no	—
158	C₂×D₄	—	2	6	no	—
159	C₂×D₄	—	2	6	no	—
160	C₂×D₄	—	2	3	no	—
161	C₂×D₄	—	2	3	no	—
162	C₂²×C₄	—	2	6	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
165	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
166	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
167	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
168	C₂²×C₄	#41×C₁₃₁	1	3	yes	—
169	C₂²×C₄	#41×C₁₃₁	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
171	M₁₆	—	2	12	no	—
172	M₁₆	—	2	12	no	—
173	M₁₆	—	2	12	no	—
174	M₁₆	—	4	3	no	—
175	M₁₆	—	4	3	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	—	2	6	no	—
178	C₂×C₈	—	2	6	no	—
179	C₂×C₈	—	4	6	no	—
180	C₂×C₈	C₂×D₄×C₁₃₁	1	12	yes	—
181	C₂³	—	2	6	no	—
182	C₂³	—	8	1	no	—
183	C₂³	—	8	2	no	—
184	C₂³	C₂²⋊Q₈×C₁₃₁	1	3	yes	—
185	C₂³	C₂²⋊Q₈×C₁₃₁	1	3	yes	—
186	C₂³	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
187	Q₈	—	4	1	no	—
188	Q₈	—	4	3	no	—
189	D₄	—	4	2	no	—
190	D₄	—	4	6	no	—
191	D₄	—	4	1	no	—
192	D₄	—	4	3	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	4	6	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	3	yes	—
197	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	3	yes	—
198	C₂×C₄	#123×C₁₃₁	1	3	yes	—
199	C₂×C₄	#123×C₁₃₁	1	3	yes	—
200	C₂×C₄	#123×C₁₃₁	1	6	yes	—
201	C₂×C₄	#127×C₁₃₁	1	3	yes	—
202	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄×C₁₃₁	1	3	yes	—
204	C₂×C₄	C₂×#41×C₁₃₁	1	3	yes	—
205	C₈	—	2	12	no	—
206	C₈	—	4	6	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	3	no	—
209	C₂²	—	4	3	no	—
210	C₂²	—	4	12	no	—
211	C₂²	—	4	2	no	—
212	C₂²	#399×C₁₃₁	1	1	yes	characteristic
213	C₂²	#399×C₁₃₁	1	3	yes	—
214	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	3	yes	—
215	C₄	—	4	2	no	—
216	C₄	#399×C₁₃₁	1	1	yes	characteristic
217	C₄	#399×C₁₃₁	1	3	yes	—
218	C₄	C₂×#123×C₁₃₁	1	3	yes	—
219	C₄	C₂×#127×C₁₃₁	1	1	yes	characteristic
220	C₂	—	2	3	no	—
221	C₂	—	2	1	no	—
222	C₂	—	2	3	no	—
223	C₂	—	8	2	no	—
224	C₂	C₂×#399×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	C₂×#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂×C₁₃₁	C₂×#399
prime radical	C₂²×#41×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#430×C₁₃₁	C₂	semidirect
3. #2091×C₁₃₁	C₂	semidirect
4. C₂×#338×C₁₃₁	C₂	semidirect
112. #23723	C₁₃₁	direct (N × H)
113. C₁₃₁	#23723	direct (N × H)
114. C₂×#430	C₂×C₁₃₁	semidirect
116. #2091	C₂×C₁₃₁	semidirect
117. C₂×#338	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#430×C₁₃₁	2
#2879×C₁₃₁	2
#2091×C₁₃₁	2
C₂×#338×C₁₃₁	2
#23723	131

Subgroups — C131×#23723 #45709607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23723 #45709607