Subgroups — C₁₃₁×#13274 #45699158

Order 33536.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1382×C₁₃₁	C₂	1	4	yes	maximal
5	#533×C₁₃₁	—	2	1	no	—
6	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
7	#500×C₁₃₁	C₄	1	1	yes	characteristic
8	#500×C₁₃₁	C₄	1	1	yes	characteristic
9	#499×C₁₃₁	—	2	1	no	—
10	#499×C₁₃₁	C₄	1	2	yes	—
11	#498×C₁₃₁	—	2	1	no	—
12	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
13	#427×C₁₃₁	—	2	1	no	—
14	#427×C₁₃₁	C₂²	1	2	yes	—
15	C₄×M₁₆×C₁₃₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
18	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
19	#341×C₁₃₁	—	2	1	no	—
20	C₂×M₁₆×C₁₃₁	—	2	2	no	—
21	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
24	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
25	#125×C₁₃₁	—	2	2	no	—
26	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
27	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
28	C₄×Q₈×C₁₃₁	—	4	1	no	—
29	C₄×Q₈×C₁₃₁	—	4	1	no	—
30	#117×C₁₃₁	—	2	2	no	—
31	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₃₁	—	2	2	no	—
33	C₂×C₁₆×C₁₃₁	—	2	4	no	—
34	#108×C₁₃₁	—	2	2	no	—
35	#108×C₁₃₁	Q₈	1	4	yes	complemented
36	#107×C₁₃₁	—	2	1	no	—
37	#107×C₁₃₁	—	2	1	no	—
38	#107×C₁₃₁	C₂×C₄	1	2	yes	—
39	#107×C₁₃₁	C₂×C₄	1	2	yes	—
40	#106×C₁₃₁	—	2	2	no	—
41	#106×C₁₃₁	—	2	2	no	—
42	#97×C₁₃₁	—	4	1	no	—
43	C₄×C₈×C₁₃₁	—	4	1	no	—
44	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₃₁	—	4	1	no	—
49	C₂×Q₈×C₁₃₁	—	4	1	no	—
50	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₁₃₁	—	2	4	no	—
54	M₁₆×C₁₃₁	—	4	2	no	—
55	C₂×C₈×C₁₃₁	—	2	2	no	—
56	C₂×C₈×C₁₃₁	—	4	2	no	—
57	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₁₃₁	#41	1	2	yes	—
63	C₄⋊C₄×C₁₃₁	—	2	2	no	—
64	C₄⋊C₄×C₁₃₁	—	2	2	no	—
65	C₄⋊C₄×C₁₃₁	—	4	2	no	—
66	C₄⋊C₄×C₁₃₁	—	4	2	no	—
67	C₄⋊C₄×C₁₃₁	—	4	1	no	—
68	C₄⋊C₄×C₁₃₁	—	4	1	no	—
69	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
70	C₄²×C₁₃₁	—	4	2	no	—
71	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
74	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
75	C₁₆×C₁₃₁	—	4	2	no	—
76	C₂³×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
77	Q₈×C₁₃₁	—	8	2	no	—
78	Q₈×C₁₃₁	—	8	2	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	4	1	no	—
81	C₂×C₄×C₁₃₁	—	4	2	no	—
82	C₂×C₄×C₁₃₁	—	4	2	no	—
83	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	#137	1	2	yes	—
91	C₈×C₁₃₁	—	2	1	no	—
92	C₈×C₁₃₁	—	2	1	no	—
93	C₈×C₁₃₁	—	4	4	no	—
94	C₈×C₁₃₁	C₄×Q₈	1	2	yes	—
95	C₈×C₁₃₁	#124	1	1	yes	characteristic
96	C₈×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₁₃₁	—	2	1	no	—
98	C₂²×C₁₃₁	—	2	1	no	—
99	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
100	C₂²×C₁₃₁	#475	1	2	yes	—
101	C₄×C₁₃₁	—	2	1	no	—
102	C₄×C₁₃₁	—	2	1	no	—
103	C₄×C₁₃₁	—	2	1	no	—
104	C₄×C₁₃₁	—	8	2	no	—
105	C₄×C₁₃₁	—	8	2	no	—
106	C₄×C₁₃₁	#359	1	2	yes	—
107	C₄×C₁₃₁	#391	1	1	yes	characteristic
108	C₄×C₁₃₁	#419	1	1	yes	characteristic
109	C₄×C₁₃₁	#474	1	2	yes	—
110	C₂×C₁₃₁	—	2	2	no	—
111	C₂×C₁₃₁	#1988	1	1	yes	characteristic
112	C₂×C₁₃₁	C₂×#359	1	1	yes	characteristic
113	C₂×C₁₃₁	#2143	1	1	yes	characteristic
114	#13274	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₁₃₁	#13274	1	1	yes	characteristic, direct product factor, Sylow
116	#3160	C₂×C₁₃₁	1	1	yes	characteristic
117	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
118	C₄×#108	C₂×C₁₃₁	1	1	yes	characteristic
119	#1382	C₂×C₁₃₁	1	4	yes	—
120	#533	—	2	1	no	—
121	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
122	#500	C₄×C₁₃₁	1	1	yes	characteristic
123	#500	C₄×C₁₃₁	1	1	yes	characteristic
124	#499	—	2	1	no	—
125	#499	C₄×C₁₃₁	1	2	yes	—
126	#498	—	2	1	no	—
127	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
128	#427	—	2	1	no	—
129	#427	C₂²×C₁₃₁	1	2	yes	—
130	C₄×M₁₆	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	—	2	1	no	—
134	#341	—	2	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	#125	—	2	2	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₂²⋊Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	C₄×Q₈	—	4	1	no	—
145	#117	—	2	2	no	—
146	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	4	no	—
149	#108	—	2	2	no	—
150	#108	Q₈×C₁₃₁	1	4	yes	complemented
151	#107	—	2	1	no	—
152	#107	—	2	1	no	—
153	#107	C₂×C₄×C₁₃₁	1	2	yes	—
154	#107	C₂×C₄×C₁₃₁	1	2	yes	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	#97	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
162	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
166	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
177	C₂×C₈	#41×C₁₃₁	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
188	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
189	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
200	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
201	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
204	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄	#137×C₁₃₁	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Q₈×C₁₃₁	1	2	yes	—
210	C₈	#124×C₁₃₁	1	1	yes	characteristic
211	C₈	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
215	C₂²	#475×C₁₃₁	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	#359×C₁₃₁	1	2	yes	—
222	C₄	#391×C₁₃₁	1	1	yes	characteristic
223	C₄	#419×C₁₃₁	1	1	yes	characteristic
224	C₄	#474×C₁₃₁	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#1988×C₁₃₁	1	1	yes	characteristic
227	C₂	C₂×#359×C₁₃₁	1	1	yes	characteristic
228	C₂	#2143×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₂×#102
prime radical	C₂³×C₁₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
35. #108×C₁₃₁	Q₈	semidirect
114. #13274	C₁₃₁	direct (N × H)
115. C₁₃₁	#13274	direct (N × H)
150. #108	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₃₁	2
C₂×C₄⋊C₁₆×C₁₃₁	2
C₄×#108×C₁₃₁	2
#1382×C₁₃₁	2
#13274	131

Subgroups — C131×#13274 #45699158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13274 #45699158