Subgroups — C₁₃₁×#33319 #45719203

Order 33536.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#501×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2660×C₁₃₁	C₂	1	2	yes	maximal
5	#1847×C₁₃₁	C₂	1	4	yes	maximal
6	#1846×C₁₃₁	C₂	1	2	yes	maximal
7	#1845×C₁₃₁	C₂	1	2	yes	maximal
8	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
11	#501×C₁₃₁	—	2	8	no	—
12	#500×C₁₃₁	—	2	4	no	—
13	#499×C₁₃₁	—	2	4	no	—
14	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#107×C₁₃₁	C₂²	1	4	yes	—
17	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₁	C₂²	1	4	yes	—
20	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
25	#398×C₁₃₁	—	2	4	no	—
26	#395×C₁₃₁	—	2	4	no	—
27	#384×C₁₃₁	C₂²	1	4	yes	—
28	#382×C₁₃₁	C₂²	1	4	yes	—
29	#376×C₁₃₁	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₁₃₁	—	2	2	no	—
31	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
36	#125×C₁₃₁	—	2	4	no	—
37	#125×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
42	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
47	#107×C₁₃₁	—	2	2	no	—
48	#107×C₁₃₁	—	2	2	no	—
49	#107×C₁₃₁	—	2	4	no	—
50	#107×C₁₃₁	—	2	4	no	—
51	#107×C₁₃₁	—	2	8	no	—
52	#106×C₁₃₁	—	2	4	no	—
53	#106×C₁₃₁	—	2	8	no	—
54	#97×C₁₃₁	D₄	1	4	yes	—
55	#97×C₁₃₁	Q₈	1	8	yes	complemented
56	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
57	C₄×C₈×C₁₃₁	Q₈	1	8	yes	complemented
58	#95×C₁₃₁	—	2	8	no	—
59	#95×C₁₃₁	D₄	1	2	yes	—
60	#95×C₁₃₁	D₄	1	2	yes	—
61	C₂×Q₈×C₁₃₁	—	4	4	no	—
62	C₂×Q₈×C₁₃₁	—	8	4	no	—
63	C₂²×C₄×C₁₃₁	—	2	2	no	—
64	C₂²×C₄×C₁₃₁	—	2	4	no	—
65	C₂²×C₄×C₁₃₁	—	2	2	no	—
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
77	C₄⋊C₄×C₁₃₁	—	2	2	no	—
78	C₄⋊C₄×C₁₃₁	—	2	2	no	—
79	C₄⋊C₄×C₁₃₁	—	2	2	no	—
80	C₄⋊C₄×C₁₃₁	—	2	4	no	—
81	C₄⋊C₄×C₁₃₁	—	2	2	no	—
82	C₄⋊C₄×C₁₃₁	—	2	4	no	—
83	C₄⋊C₄×C₁₃₁	—	4	4	no	—
84	C₄⋊C₄×C₁₃₁	—	4	4	no	—
85	C₄⋊C₄×C₁₃₁	—	4	4	no	—
86	C₄⋊C₄×C₁₃₁	—	8	8	no	—
87	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₁	C₂×Q₈	1	4	yes	—
90	C₄²×C₁₃₁	C₂×Q₈	1	4	yes	—
91	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	8	8	no	—
93	C₂×C₄×C₁₃₁	—	2	4	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	4	no	—
97	C₂×C₄×C₁₃₁	—	4	2	no	—
98	C₂×C₄×C₁₃₁	—	4	2	no	—
99	C₂×C₄×C₁₃₁	—	4	4	no	—
100	C₂×C₄×C₁₃₁	—	4	4	no	—
101	C₂×C₄×C₁₃₁	—	4	8	no	—
102	C₂×C₄×C₁₃₁	—	4	4	no	—
103	C₂×C₄×C₁₃₁	#127	1	2	yes	—
104	C₂×C₄×C₁₃₁	#127	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	8	yes	—
106	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃₁	C₂²×Q₈	1	2	yes	—
110	C₈×C₁₃₁	C₄⋊Q₈	1	4	yes	—
111	C₈×C₁₃₁	C₄⋊Q₈	1	4	yes	—
112	C₈×C₁₃₁	C₄⋊Q₈	1	8	yes	—
113	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
114	C₂²×C₁₃₁	C₂×C₄⋊Q₈	1	2	yes	—
115	C₂²×C₁₃₁	#576	1	2	yes	—
116	C₂²×C₁₃₁	#578	1	2	yes	—
117	C₄×C₁₃₁	—	2	2	no	—
118	C₄×C₁₃₁	—	2	2	no	—
119	C₄×C₁₃₁	—	2	8	no	—
120	C₄×C₁₃₁	—	8	2	no	—
121	C₄×C₁₃₁	—	8	4	no	—
122	C₄×C₁₃₁	—	8	2	no	—
123	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	2	yes	—
126	C₂×C₁₃₁	#2868	1	1	yes	characteristic
127	C₂×C₁₃₁	#3155	1	1	yes	characteristic
128	C₂×C₁₃₁	#3156	1	1	yes	characteristic
129	C₂×C₁₃₁	#3164	1	4	yes	—
130	#33319	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₁	#33319	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#501	C₂×C₁₃₁	1	2	yes	—
133	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
134	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic
135	#2660	C₂×C₁₃₁	1	2	yes	—
136	#1847	C₂×C₁₃₁	1	4	yes	—
137	#1846	C₂×C₁₃₁	1	2	yes	—
138	#1845	C₂×C₁₃₁	1	2	yes	—
139	#1726	C₂×C₁₃₁	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
141	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
142	#501	—	2	8	no	—
143	#500	—	2	4	no	—
144	#499	—	2	4	no	—
145	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
146	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
147	C₂×#107	C₂²×C₁₃₁	1	4	yes	—
148	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
149	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
150	C₂×#106	C₂²×C₁₃₁	1	4	yes	—
151	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
152	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
153	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
156	#398	—	2	4	no	—
157	#395	—	2	4	no	—
158	#384	C₂²×C₁₃₁	1	4	yes	—
159	#382	C₂²×C₁₃₁	1	4	yes	—
160	#376	C₂²×C₁₃₁	1	1	yes	characteristic
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
165	C₄⋊Q₈	—	2	4	no	—
166	C₄⋊Q₈	—	2	4	no	—
167	#125	—	2	4	no	—
168	#125	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	8	no	—
173	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
176	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	—	2	4	no	—
181	#107	—	2	4	no	—
182	#107	—	2	8	no	—
183	#106	—	2	4	no	—
184	#106	—	2	8	no	—
185	#97	D₄×C₁₃₁	1	4	yes	—
186	#97	Q₈×C₁₃₁	1	8	yes	complemented
187	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
188	C₄×C₈	Q₈×C₁₃₁	1	8	yes	complemented
189	#95	—	2	8	no	—
190	#95	D₄×C₁₃₁	1	2	yes	—
191	#95	D₄×C₁₃₁	1	2	yes	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	8	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
204	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
205	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
206	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
207	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	8	8	no	—
218	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
219	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
220	C₄²	C₂×Q₈×C₁₃₁	1	4	yes	—
221	C₄²	C₂×Q₈×C₁₃₁	1	4	yes	—
222	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#127×C₁₃₁	1	2	yes	—
235	C₂×C₄	#127×C₁₃₁	1	2	yes	—
236	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	8	yes	—
237	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×Q₈×C₁₃₁	1	2	yes	—
240	C₂×C₄	C₂²×Q₈×C₁₃₁	1	2	yes	—
241	C₈	C₄⋊Q₈×C₁₃₁	1	4	yes	—
242	C₈	C₄⋊Q₈×C₁₃₁	1	4	yes	—
243	C₈	C₄⋊Q₈×C₁₃₁	1	8	yes	—
244	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
245	C₂²	C₂×C₄⋊Q₈×C₁₃₁	1	2	yes	—
246	C₂²	#576×C₁₃₁	1	2	yes	—
247	C₂²	#578×C₁₃₁	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
255	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
256	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	2	yes	—
257	C₂	#2868×C₁₃₁	1	1	yes	characteristic
258	C₂	#3155×C₁₃₁	1	1	yes	characteristic
259	C₂	#3156×C₁₃₁	1	1	yes	characteristic
260	C₂	#3164×C₁₃₁	1	4	yes	—

	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 ◁ #376×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
55. #97×C₁₃₁	Q₈	semidirect
57. C₄×C₈×C₁₃₁	Q₈	semidirect
130. #33319	C₁₃₁	direct (N × H)
131. C₁₃₁	#33319	direct (N × H)
186. #97	Q₈×C₁₃₁	semidirect
188. C₄×C₈	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#501×C₁₃₁	2
C₂×#500×C₁₃₁	2
C₂×#499×C₁₃₁	2
#2660×C₁₃₁	2
#1847×C₁₃₁	2
#1846×C₁₃₁	2
#1845×C₁₃₁	2
#1726×C₁₃₁	2
#33319	131

Subgroups — C131×#33319 #45719203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33319 #45719203