Subgroups — C₁₃₁×#33263 #45719147

Order 33536.

694 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	#2859×C₁₃₁	C₂	1	2	yes	maximal
2	#1944×C₁₃₁	C₂	1	4	yes	maximal
3	#1943×C₁₃₁	C₂	1	4	yes	maximal
4	#1845×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1844×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1834×C₁₃₁	C₂	1	2	yes	maximal
7	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₁₃₁	—	2	2	no	—
9	C₂×#107×C₁₃₁	C₂²	1	4	yes	—
10	C₂×#106×C₁₃₁	C₂²	1	4	yes	—
11	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
12	C₂×#97×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	#389×C₁₃₁	—	2	2	no	—
16	#389×C₁₃₁	—	2	2	no	—
17	#389×C₁₃₁	C₂²	1	4	yes	—
18	#389×C₁₃₁	C₂²	1	4	yes	—
19	#384×C₁₃₁	C₂²	1	2	yes	—
20	#384×C₁₃₁	C₂²	1	2	yes	—
21	#382×C₁₃₁	—	2	2	no	—
22	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
23	#340×C₁₃₁	C₂²	1	8	yes	—
24	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
25	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
26	#125×C₁₃₁	—	4	8	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
40	#107×C₁₃₁	—	2	8	no	—
41	#106×C₁₃₁	—	2	8	no	—
42	#97×C₁₃₁	—	2	4	no	—
43	#97×C₁₃₁	—	2	4	no	—
44	C₄×C₈×C₁₃₁	Q₈	1	4	yes	—
45	C₄×C₈×C₁₃₁	Q₈	1	4	yes	—
46	#95×C₁₃₁	—	2	4	no	—
47	#95×C₁₃₁	—	2	4	no	—
48	C₂²×C₄×C₁₃₁	—	2	4	no	—
49	C₂²×C₄×C₁₃₁	—	2	4	no	—
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
55	C₂×C₈×C₁₃₁	—	2	2	no	—
56	C₂×C₈×C₁₃₁	—	2	2	no	—
57	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₁₃₁	#41	1	4	yes	—
60	C₂×C₈×C₁₃₁	#41	1	4	yes	—
61	C₄⋊C₄×C₁₃₁	—	2	8	no	—
62	C₄⋊C₄×C₁₃₁	—	2	8	no	—
63	C₄⋊C₄×C₁₃₁	—	4	8	no	—
64	C₄⋊C₄×C₁₃₁	—	4	8	no	—
65	C₄⋊C₄×C₁₃₁	—	4	4	no	—
66	C₄⋊C₄×C₁₃₁	—	4	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	4	no	—
68	C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄²×C₁₃₁	—	2	4	no	—
70	C₄²×C₁₃₁	—	2	4	no	—
71	C₄²×C₁₃₁	—	2	2	no	—
72	C₄²×C₁₃₁	—	2	2	no	—
73	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
74	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
75	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	2	1	no	—
78	C₂×C₄×C₁₃₁	—	2	1	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	—	2	2	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	—	4	4	no	—
84	C₂×C₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	—	4	4	no	—
86	C₂×C₄×C₁₃₁	—	4	4	no	—
87	C₂×C₄×C₁₃₁	—	4	4	no	—
88	C₂×C₄×C₁₃₁	—	4	4	no	—
89	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	—
90	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
91	C₂×C₄×C₁₃₁	#137	1	4	yes	—
92	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
96	C₈×C₁₃₁	—	2	4	no	—
97	C₈×C₁₃₁	#125	1	8	yes	complemented
98	C₂²×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₁₃₁	#532	1	1	yes	characteristic
100	C₂²×C₁₃₁	#532	1	1	yes	characteristic
101	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
102	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
103	C₄×C₁₃₁	—	4	2	no	—
104	C₄×C₁₃₁	—	4	1	no	—
105	C₄×C₁₃₁	—	4	1	no	—
106	C₄×C₁₃₁	—	8	4	no	—
107	C₄×C₁₃₁	—	8	4	no	—
108	C₄×C₁₃₁	#501	1	8	yes	—
109	C₄×C₁₃₁	C₂×#125	1	2	yes	—
110	C₄×C₁₃₁	#533	1	2	yes	—
111	C₂×C₁₃₁	#2859	1	1	yes	characteristic
112	C₂×C₁₃₁	C₂×#501	1	2	yes	—
113	C₂×C₁₃₁	#3182	1	2	yes	—
114	C₂×C₁₃₁	#3184	1	1	yes	characteristic
115	C₂×C₁₃₁	#3185	1	1	yes	characteristic
116	#33263	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₁	#33263	1	1	yes	characteristic, direct product factor, Sylow
118	#2859	C₂×C₁₃₁	1	2	yes	—
119	#1944	C₂×C₁₃₁	1	4	yes	—
120	#1943	C₂×C₁₃₁	1	4	yes	—
121	#1845	C₂×C₁₃₁	1	1	yes	characteristic
122	#1844	C₂×C₁₃₁	1	1	yes	characteristic
123	#1834	C₂×C₁₃₁	1	2	yes	—
124	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic
125	C₂×#125	—	2	2	no	—
126	C₂×#107	C₂²×C₁₃₁	1	4	yes	—
127	C₂×#106	C₂²×C₁₃₁	1	4	yes	—
128	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
129	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
132	#389	—	2	2	no	—
133	#389	—	2	2	no	—
134	#389	C₂²×C₁₃₁	1	4	yes	—
135	#389	C₂²×C₁₃₁	1	4	yes	—
136	#384	C₂²×C₁₃₁	1	2	yes	—
137	#384	C₂²×C₁₃₁	1	2	yes	—
138	#382	—	2	2	no	—
139	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
140	#340	C₂²×C₁₃₁	1	8	yes	—
141	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
142	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
143	#125	—	4	8	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
152	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
153	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
154	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
157	#107	—	2	8	no	—
158	#106	—	2	8	no	—
159	#97	—	2	4	no	—
160	#97	—	2	4	no	—
161	C₄×C₈	Q₈×C₁₃₁	1	4	yes	—
162	C₄×C₈	Q₈×C₁₃₁	1	4	yes	—
163	#95	—	2	4	no	—
164	#95	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
171	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
175	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
176	C₂×C₈	#41×C₁₃₁	1	4	yes	—
177	C₂×C₈	#41×C₁₃₁	1	4	yes	—
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
191	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
192	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	4	yes	—
207	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
208	C₂×C₄	#137×C₁₃₁	1	4	yes	—
209	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
212	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	#125×C₁₃₁	1	8	yes	complemented
215	C₂²	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
216	C₂²	#532×C₁₃₁	1	1	yes	characteristic
217	C₂²	#532×C₁₃₁	1	1	yes	characteristic
218	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
219	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	4	no	—
225	C₄	#501×C₁₃₁	1	8	yes	—
226	C₄	C₂×#125×C₁₃₁	1	2	yes	—
227	C₄	#533×C₁₃₁	1	2	yes	—
228	C₂	#2859×C₁₃₁	1	1	yes	characteristic
229	C₂	C₂×#501×C₁₃₁	1	2	yes	—
230	C₂	#3182×C₁₃₁	1	2	yes	—
231	C₂	#3184×C₁₃₁	1	1	yes	characteristic
232	C₂	#3185×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
97. C₈×C₁₃₁	#125	semidirect
116. #33263	C₁₃₁	direct (N × H)
117. C₁₃₁	#33263	direct (N × H)
214. C₈	#125×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2859×C₁₃₁	2
#1944×C₁₃₁	2
#1943×C₁₃₁	2
#1845×C₁₃₁	2
#1844×C₁₃₁	2
#1834×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33263	131

Subgroups — C131×#33263 #45719147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33263 #45719147