Subgroups — C₁₃₁×#33314 #45719198

Order 33536.

910 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×#498×C₁₃₁	C₂	1	2	yes	maximal
3	#2868×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2862×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1847×C₁₃₁	C₂	1	8	yes	maximal
6	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₁₃₁	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
11	#500×C₁₃₁	—	2	8	no	—
12	#498×C₁₃₁	—	2	8	no	—
13	C₂×#107×C₁₃₁	C₂²	1	8	yes	—
14	C₂×#106×C₁₃₁	C₂²	1	8	yes	—
15	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
17	#391×C₁₃₁	—	2	2	no	—
18	#391×C₁₃₁	—	2	2	no	—
19	#384×C₁₃₁	—	2	2	no	—
20	#384×C₁₃₁	C₂²	1	4	yes	—
21	#382×C₁₃₁	C₂²	1	4	yes	—
22	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₃₁	—	2	1	no	—
24	C₂²×Q₈×C₁₃₁	—	2	1	no	—
25	C₂²×Q₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
27	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
28	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
29	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
30	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₁	—	4	8	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
43	#107×C₁₃₁	—	2	16	no	—
44	#106×C₁₃₁	—	2	16	no	—
45	#97×C₁₃₁	Q₈	1	16	yes	complemented
46	C₄×C₈×C₁₃₁	D₄	1	8	yes	—
47	#95×C₁₃₁	—	2	8	no	—
48	C₂×Q₈×C₁₃₁	—	4	4	no	—
49	C₂×Q₈×C₁₃₁	—	4	2	no	—
50	C₂×Q₈×C₁₃₁	—	4	2	no	—
51	C₂×Q₈×C₁₃₁	—	4	8	no	—
52	C₂×Q₈×C₁₃₁	—	4	4	no	—
53	C₂×Q₈×C₁₃₁	—	4	4	no	—
54	C₂²×C₄×C₁₃₁	—	2	2	no	—
55	C₂²×C₄×C₁₃₁	—	2	2	no	—
56	C₂²×C₄×C₁₃₁	—	2	4	no	—
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	C₂×D₄	1	8	yes	—
62	C₂×C₈×C₁₃₁	C₂×Q₈	1	16	yes	—
63	C₄⋊C₄×C₁₃₁	—	2	4	no	—
64	C₄⋊C₄×C₁₃₁	—	2	4	no	—
65	C₄⋊C₄×C₁₃₁	—	2	4	no	—
66	C₄⋊C₄×C₁₃₁	—	2	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	16	no	—
68	C₄⋊C₄×C₁₃₁	—	4	2	no	—
69	C₄⋊C₄×C₁₃₁	—	4	2	no	—
70	C₄⋊C₄×C₁₃₁	—	4	4	no	—
71	C₄²×C₁₃₁	—	2	4	no	—
72	C₄²×C₁₃₁	—	2	4	no	—
73	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
74	C₄²×C₁₃₁	C₂×Q₈	1	8	yes	—
75	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₁₃₁	—	8	8	no	—
77	Q₈×C₁₃₁	—	8	4	no	—
78	Q₈×C₁₃₁	—	8	4	no	—
79	C₂×C₄×C₁₃₁	—	2	4	no	—
80	C₂×C₄×C₁₃₁	—	2	4	no	—
81	C₂×C₄×C₁₃₁	—	4	4	no	—
82	C₂×C₄×C₁₃₁	—	4	2	no	—
83	C₂×C₄×C₁₃₁	—	4	2	no	—
84	C₂×C₄×C₁₃₁	—	4	8	no	—
85	C₂×C₄×C₁₃₁	—	4	4	no	—
86	C₂×C₄×C₁₃₁	—	4	4	no	—
87	C₂×C₄×C₁₃₁	#127	1	2	yes	—
88	C₂×C₄×C₁₃₁	#127	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	8	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×D₄	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂²×Q₈	1	4	yes	—
94	C₈×C₁₃₁	C₄⋊Q₈	1	16	yes	complemented
95	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
96	C₂²×C₁₃₁	C₂×C₄⋊Q₈	1	2	yes	—
97	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
98	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
99	C₄×C₁₃₁	—	2	8	no	—
100	C₄×C₁₃₁	—	8	4	no	—
101	C₄×C₁₃₁	—	8	2	no	—
102	C₄×C₁₃₁	—	8	2	no	—
103	C₄×C₁₃₁	#496	1	4	yes	—
104	C₄×C₁₃₁	#497	1	4	yes	—
105	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	4	yes	—
106	C₂×C₁₃₁	#2868	1	1	yes	characteristic
107	C₂×C₁₃₁	C₂×#496	1	1	yes	characteristic
108	C₂×C₁₃₁	C₂×#497	1	1	yes	characteristic
109	C₂×C₁₃₁	#3165	1	4	yes	—
110	#33314	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₁	#33314	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×#500	C₂×C₁₃₁	1	2	yes	—
113	C₂×#498	C₂×C₁₃₁	1	2	yes	—
114	#2868	C₂×C₁₃₁	1	1	yes	characteristic
115	#2862	C₂×C₁₃₁	1	1	yes	characteristic
116	#1847	C₂×C₁₃₁	1	8	yes	—
117	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	2	2	no	—
119	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
122	#500	—	2	8	no	—
123	#498	—	2	8	no	—
124	C₂×#107	C₂²×C₁₃₁	1	8	yes	—
125	C₂×#106	C₂²×C₁₃₁	1	8	yes	—
126	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
128	#391	—	2	2	no	—
129	#391	—	2	2	no	—
130	#384	—	2	2	no	—
131	#384	C₂²×C₁₃₁	1	4	yes	—
132	#382	C₂²×C₁₃₁	1	4	yes	—
133	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
134	C₂²×Q₈	—	2	2	no	—
135	C₂²×Q₈	—	2	1	no	—
136	C₂²×Q₈	—	2	1	no	—
137	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
138	C₄⋊Q₈	—	2	4	no	—
139	C₄⋊Q₈	—	2	4	no	—
140	C₄⋊Q₈	—	2	4	no	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	4	8	no	—
143	C₂×C₄⋊C₄	—	2	8	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
148	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
149	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
150	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
151	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
152	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
154	#107	—	2	16	no	—
155	#106	—	2	16	no	—
156	#97	Q₈×C₁₃₁	1	16	yes	complemented
157	C₄×C₈	D₄×C₁₃₁	1	8	yes	—
158	#95	—	2	8	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	8	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
169	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₁₃₁	1	8	yes	—
173	C₂×C₈	C₂×Q₈×C₁₃₁	1	16	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	16	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	4	no	—
184	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
185	C₄²	C₂×Q₈×C₁₃₁	1	8	yes	—
186	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
187	Q₈	—	8	8	no	—
188	Q₈	—	8	4	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	#127×C₁₃₁	1	2	yes	—
199	C₂×C₄	#127×C₁₃₁	1	2	yes	—
200	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	8	yes	—
201	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
202	C₂×C₄	#137×C₁₃₁	1	4	yes	—
203	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
204	C₂×C₄	C₂²×Q₈×C₁₃₁	1	4	yes	—
205	C₈	C₄⋊Q₈×C₁₃₁	1	16	yes	complemented
206	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
207	C₂²	C₂×C₄⋊Q₈×C₁₃₁	1	2	yes	—
208	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
209	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
210	C₄	—	2	8	no	—
211	C₄	—	8	4	no	—
212	C₄	—	8	2	no	—
213	C₄	—	8	2	no	—
214	C₄	#496×C₁₃₁	1	4	yes	—
215	C₄	#497×C₁₃₁	1	4	yes	—
216	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	4	yes	—
217	C₂	#2868×C₁₃₁	1	1	yes	characteristic
218	C₂	C₂×#496×C₁₃₁	1	1	yes	characteristic
219	C₂	C₂×#497×C₁₃₁	1	1	yes	characteristic
220	C₂	#3165×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 ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
45. #97×C₁₃₁	Q₈	semidirect
94. C₈×C₁₃₁	C₄⋊Q₈	semidirect
110. #33314	C₁₃₁	direct (N × H)
111. C₁₃₁	#33314	direct (N × H)
156. #97	Q₈×C₁₃₁	semidirect
205. C₈	C₄⋊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×#500×C₁₃₁	2
C₂×#498×C₁₃₁	2
#2868×C₁₃₁	2
#2862×C₁₃₁	2
#1847×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33314	131

Subgroups — C131×#33314 #45719198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33314 #45719198