Subgroups — C₁₃₁×#33248 #45719132

Order 33536.

1006 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#490×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2641×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1972×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1970×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂²×D₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×#127×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
12	#499×C₁₃₁	—	2	4	no	—
13	#493×C₁₃₁	—	2	4	no	—
14	#490×C₁₃₁	—	2	8	no	—
15	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#102×C₁₃₁	C₂²	1	8	yes	—
18	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	#394×C₁₃₁	—	2	8	no	—
23	#390×C₁₃₁	C₂²	1	8	yes	—
24	#376×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂²×D₄×C₁₃₁	—	2	4	no	—
26	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
27	C₂×D₈×C₁₃₁	—	2	4	no	—
28	C₂×D₈×C₁₃₁	—	2	4	no	—
29	C₂×D₈×C₁₃₁	—	2	8	no	—
30	C₂×D₈×C₁₃₁	—	2	8	no	—
31	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
34	#127×C₁₃₁	—	2	4	no	—
35	#127×C₁₃₁	—	2	4	no	—
36	#125×C₁₃₁	—	2	4	no	—
37	#125×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₁₃₁	—	2	16	no	—
41	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	#107×C₁₃₁	—	2	4	no	—
44	#106×C₁₃₁	—	2	4	no	—
45	#102×C₁₃₁	—	2	16	no	—
46	#97×C₁₃₁	—	2	4	no	—
47	#97×C₁₃₁	D₄	1	4	yes	—
48	C₄×C₈×C₁₃₁	—	2	4	no	—
49	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
50	#95×C₁₃₁	D₄	1	4	yes	—
51	C₂⁴×C₁₃₁	—	2	4	no	—
52	C₂×D₄×C₁₃₁	—	2	4	no	—
53	C₂×D₄×C₁₃₁	—	2	4	no	—
54	C₂×D₄×C₁₃₁	—	2	8	no	—
55	C₂×D₄×C₁₃₁	—	2	8	no	—
56	C₂×D₄×C₁₃₁	—	4	8	no	—
57	C₂×D₄×C₁₃₁	—	8	8	no	—
58	C₂²×C₄×C₁₃₁	—	2	4	no	—
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
62	D₈×C₁₃₁	—	4	8	no	—
63	D₈×C₁₃₁	—	4	8	no	—
64	C₂×C₈×C₁₃₁	—	2	2	no	—
65	C₂×C₈×C₁₃₁	—	2	2	no	—
66	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃₁	#41	1	4	yes	—
70	C₂×C₈×C₁₃₁	#41	1	4	yes	—
71	C₄⋊C₄×C₁₃₁	—	2	4	no	—
72	C₄⋊C₄×C₁₃₁	—	2	4	no	—
73	C₄⋊C₄×C₁₃₁	—	4	8	no	—
74	C₂²⋊C₄×C₁₃₁	—	8	16	no	—
75	C₄²×C₁₃₁	—	2	2	no	—
76	C₄²×C₁₃₁	—	2	2	no	—
77	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
79	C₂³×C₁₃₁	—	4	4	no	—
80	C₂³×C₁₃₁	—	4	8	no	—
81	C₂³×C₁₃₁	—	4	16	no	—
82	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
83	D₄×C₁₃₁	—	4	4	no	—
84	D₄×C₁₃₁	—	4	4	no	—
85	D₄×C₁₃₁	—	4	8	no	—
86	D₄×C₁₃₁	—	8	16	no	—
87	C₂×C₄×C₁₃₁	—	2	4	no	—
88	C₂×C₄×C₁₃₁	—	2	4	no	—
89	C₂×C₄×C₁₃₁	—	2	8	no	—
90	C₂×C₄×C₁₃₁	—	4	4	no	—
91	C₂×C₄×C₁₃₁	—	4	8	no	—
92	C₂×C₄×C₁₃₁	#127	1	4	yes	—
93	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
97	C₈×C₁₃₁	—	2	4	no	—
98	C₈×C₁₃₁	#124	1	4	yes	—
99	C₈×C₁₃₁	#124	1	4	yes	—
100	C₂²×C₁₃₁	—	8	4	no	—
101	C₂²×C₁₃₁	—	8	8	no	—
102	C₂²×C₁₃₁	—	8	16	no	—
103	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
104	C₂²×C₁₃₁	#532	1	2	yes	—
105	C₂²×C₁₃₁	#576	1	4	yes	—
106	C₄×C₁₃₁	—	2	4	no	—
107	C₄×C₁₃₁	—	4	4	no	—
108	C₄×C₁₃₁	—	8	4	no	—
109	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
110	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
111	C₄×C₁₃₁	#532	1	2	yes	—
112	C₂×C₁₃₁	—	16	4	no	—
113	C₂×C₁₃₁	#2858	1	1	yes	characteristic
114	C₂×C₁₃₁	#3155	1	2	yes	—
115	C₂×C₁₃₁	#3176	1	4	yes	—
116	#33248	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₁	#33248	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic, complemented
119	C₂×#493	C₂×C₁₃₁	1	1	yes	characteristic
120	C₂×#490	C₂×C₁₃₁	1	2	yes	complemented
121	#2641	C₂×C₁₃₁	1	2	yes	complemented
122	#1972	C₂×C₁₃₁	1	4	yes	complemented
123	#1970	C₂×C₁₃₁	1	4	yes	complemented
124	#1726	C₂×C₁₃₁	1	1	yes	characteristic, complemented
125	C₂²×D₈	C₂²×C₁₃₁	1	2	yes	—
126	C₂²×D₈	C₂²×C₁₃₁	1	2	yes	—
127	C₂×#127	C₂²×C₁₃₁	1	2	yes	—
128	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
129	#499	—	2	4	no	—
130	#493	—	2	4	no	—
131	#490	—	2	8	no	—
132	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
133	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
134	C₂×#102	C₂²×C₁₃₁	1	8	yes	—
135	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
136	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	#394	—	2	8	no	—
140	#390	C₂²×C₁₃₁	1	8	yes	—
141	#376	C₂²×C₁₃₁	1	1	yes	characteristic
142	C₂²×D₄	—	2	4	no	—
143	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	8	no	—
147	C₂×D₈	—	2	8	no	—
148	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
149	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	2	4	no	—
153	#125	—	2	4	no	—
154	#125	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	16	no	—
158	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
159	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
160	#107	—	2	4	no	—
161	#106	—	2	4	no	—
162	#102	—	2	16	no	—
163	#97	—	2	4	no	—
164	#97	D₄×C₁₃₁	1	4	yes	—
165	C₄×C₈	—	2	4	no	—
166	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
167	#95	D₄×C₁₃₁	1	4	yes	—
168	C₂⁴	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	2	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	8	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
177	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
178	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
179	D₈	—	4	8	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
184	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
186	C₂×C₈	#41×C₁₃₁	1	4	yes	—
187	C₂×C₈	#41×C₁₃₁	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	8	16	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	8	no	—
198	C₂³	—	4	16	no	—
199	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	8	no	—
203	D₄	—	8	16	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	#127×C₁₃₁	1	4	yes	—
210	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
213	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	#124×C₁₃₁	1	4	yes	—
216	C₈	#124×C₁₃₁	1	4	yes	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	8	no	—
219	C₂²	—	8	16	no	—
220	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
221	C₂²	#532×C₁₃₁	1	2	yes	—
222	C₂²	#576×C₁₃₁	1	4	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	4	no	—
226	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
227	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
228	C₄	#532×C₁₃₁	1	2	yes	—
229	C₂	—	16	4	no	—
230	C₂	#2858×C₁₃₁	1	1	yes	characteristic
231	C₂	#3155×C₁₃₁	1	2	yes	—
232	C₂	#3176×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#493×C₁₃₁	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₁	C₂	semidirect
3. C₂×#490×C₁₃₁	C₂	semidirect
4. #2641×C₁₃₁	C₂	semidirect
5. #1972×C₁₃₁	C₂	semidirect
6. #1970×C₁₃₁	C₂	semidirect
7. #1726×C₁₃₁	C₂	semidirect
116. #33248	C₁₃₁	direct (N × H)
117. C₁₃₁	#33248	direct (N × H)
118. C₂×#499	C₂×C₁₃₁	semidirect
120. C₂×#490	C₂×C₁₃₁	semidirect
121. #2641	C₂×C₁₃₁	semidirect
122. #1972	C₂×C₁₃₁	semidirect
123. #1970	C₂×C₁₃₁	semidirect
124. #1726	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×#499×C₁₃₁	2
C₂×#493×C₁₃₁	2
C₂×#490×C₁₃₁	2
#2641×C₁₃₁	2
#1972×C₁₃₁	2
#1970×C₁₃₁	2
#1726×C₁₃₁	2
#33248	131

Subgroups — C131×#33248 #45719132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33248 #45719132