Subgroups — C₁₃₁×#33294 #45719178

Order 33536.

750 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2868×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2859×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1992×C₁₃₁	C₂	1	4	yes	maximal
4	#1990×C₁₃₁	C₂	1	4	yes	maximal
5	#1836×C₁₃₁	C₂	1	4	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	2	yes	—
9	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
10	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
13	#389×C₁₃₁	—	2	2	no	—
14	#389×C₁₃₁	—	2	2	no	—
15	#384×C₁₃₁	—	2	2	no	—
16	#384×C₁₃₁	C₂²	1	2	yes	—
17	#384×C₁₃₁	C₂²	1	2	yes	—
18	#384×C₁₃₁	C₂²	1	2	yes	—
19	#382×C₁₃₁	C₂²	1	2	yes	—
20	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
21	#340×C₁₃₁	C₂²	1	16	yes	—
22	C₂²×Q₈×C₁₃₁	—	2	2	no	—
23	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
24	C₄⋊Q₈×C₁₃₁	—	2	8	no	—
25	C₄⋊Q₈×C₁₃₁	—	4	8	no	—
26	#125×C₁₃₁	—	2	8	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	4	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	#97×C₁₃₁	—	2	4	no	—
41	#97×C₁₃₁	Q₈	1	8	yes	complemented
42	C₄×C₈×C₁₃₁	—	2	4	no	—
43	#95×C₁₃₁	—	2	4	no	—
44	C₂×Q₈×C₁₃₁	—	4	4	no	—
45	C₂×Q₈×C₁₃₁	—	4	8	no	—
46	C₂²×C₄×C₁₃₁	—	2	4	no	—
47	C₂²×C₄×C₁₃₁	—	2	4	no	—
48	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₁₃₁	—	2	4	no	—
54	C₂×C₈×C₁₃₁	—	2	4	no	—
55	C₂×C₈×C₁₃₁	C₂×Q₈	1	8	yes	—
56	C₄⋊C₄×C₁₃₁	—	2	8	no	—
57	C₄⋊C₄×C₁₃₁	—	2	8	no	—
58	C₄⋊C₄×C₁₃₁	—	4	8	no	—
59	C₄⋊C₄×C₁₃₁	—	4	8	no	—
60	C₄⋊C₄×C₁₃₁	—	4	8	no	—
61	C₄⋊C₄×C₁₃₁	—	4	4	no	—
62	C₄⋊C₄×C₁₃₁	—	4	4	no	—
63	C₄⋊C₄×C₁₃₁	—	4	4	no	—
64	C₄²×C₁₃₁	—	2	4	no	—
65	C₄²×C₁₃₁	—	2	4	no	—
66	C₄²×C₁₃₁	C₂×Q₈	1	4	yes	—
67	C₄²×C₁₃₁	#41	1	4	yes	—
68	C₄²×C₁₃₁	#41	1	4	yes	—
69	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₁₃₁	—	8	8	no	—
71	C₂×C₄×C₁₃₁	—	2	2	no	—
72	C₂×C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	4	4	no	—
76	C₂×C₄×C₁₃₁	—	4	4	no	—
77	C₂×C₄×C₁₃₁	—	4	8	no	—
78	C₂×C₄×C₁₃₁	—	4	4	no	—
79	C₂×C₄×C₁₃₁	—	4	4	no	—
80	C₂×C₄×C₁₃₁	#124	1	4	yes	—
81	C₂×C₄×C₁₃₁	#124	1	2	yes	—
82	C₂×C₄×C₁₃₁	#124	1	2	yes	—
83	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
85	C₂×C₄×C₁₃₁	#137	1	4	yes	—
86	C₂×C₄×C₁₃₁	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
88	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
89	C₈×C₁₃₁	—	2	8	no	—
90	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
91	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
92	C₂²×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
94	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
95	C₄×C₁₃₁	—	2	4	no	—
96	C₄×C₁₃₁	—	2	4	no	—
97	C₄×C₁₃₁	—	8	4	no	—
98	C₄×C₁₃₁	—	8	4	no	—
99	C₄×C₁₃₁	#490	1	4	yes	—
100	C₄×C₁₃₁	#491	1	4	yes	—
101	C₄×C₁₃₁	#533	1	4	yes	—
102	C₂×C₁₃₁	#2862	1	1	yes	characteristic
103	C₂×C₁₃₁	C₂×#490	1	1	yes	characteristic
104	C₂×C₁₃₁	C₂×#491	1	1	yes	characteristic
105	C₂×C₁₃₁	#3141	1	2	yes	—
106	C₂×C₁₃₁	#3165	1	2	yes	—
107	#33294	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₃₁	#33294	1	1	yes	characteristic, direct product factor, Sylow
109	#2868	C₂×C₁₃₁	1	1	yes	characteristic
110	#2859	C₂×C₁₃₁	1	1	yes	characteristic
111	#1992	C₂×C₁₃₁	1	4	yes	—
112	#1990	C₂×C₁₃₁	1	4	yes	—
113	#1836	C₂×C₁₃₁	1	4	yes	—
114	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	—	2	2	no	—
116	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
117	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
118	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
119	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
121	#389	—	2	2	no	—
122	#389	—	2	2	no	—
123	#384	—	2	2	no	—
124	#384	C₂²×C₁₃₁	1	2	yes	—
125	#384	C₂²×C₁₃₁	1	2	yes	—
126	#384	C₂²×C₁₃₁	1	2	yes	—
127	#382	C₂²×C₁₃₁	1	2	yes	—
128	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
129	#340	C₂²×C₁₃₁	1	16	yes	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	—	4	8	no	—
134	#125	—	2	8	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
142	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
143	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
144	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
145	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
148	#97	—	2	4	no	—
149	#97	Q₈×C₁₃₁	1	8	yes	complemented
150	C₄×C₈	—	2	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂×Q₈×C₁₃₁	1	8	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	8	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	C₂×Q₈×C₁₃₁	1	4	yes	—
175	C₄²	#41×C₁₃₁	1	4	yes	—
176	C₄²	#41×C₁₃₁	1	4	yes	—
177	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
178	Q₈	—	8	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	#124×C₁₃₁	1	4	yes	—
189	C₂×C₄	#124×C₁₃₁	1	2	yes	—
190	C₂×C₄	#124×C₁₃₁	1	2	yes	—
191	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	4	yes	—
192	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
193	C₂×C₄	#137×C₁₃₁	1	4	yes	—
194	C₂×C₄	C₂²×Q₈×C₁₃₁	1	2	yes	—
195	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
196	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
197	C₈	—	2	8	no	—
198	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
199	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
200	C₂²	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
201	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
202	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	8	4	no	—
206	C₄	—	8	4	no	—
207	C₄	#490×C₁₃₁	1	4	yes	—
208	C₄	#491×C₁₃₁	1	4	yes	—
209	C₄	#533×C₁₃₁	1	4	yes	—
210	C₂	#2862×C₁₃₁	1	1	yes	characteristic
211	C₂	C₂×#490×C₁₃₁	1	1	yes	characteristic
212	C₂	C₂×#491×C₁₃₁	1	1	yes	characteristic
213	C₂	#3141×C₁₃₁	1	2	yes	—
214	C₂	#3165×C₁₃₁	1	2	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
41. #97×C₁₃₁	Q₈	semidirect
107. #33294	C₁₃₁	direct (N × H)
108. C₁₃₁	#33294	direct (N × H)
149. #97	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2868×C₁₃₁	2
#2859×C₁₃₁	2
#1992×C₁₃₁	2
#1990×C₁₃₁	2
#1836×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33294	131

Subgroups — C131×#33294 #45719178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33294 #45719178