Subgroups — C₁₃₁×#10274 #45696158

Order 33536.

626 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#474×C₁₃₁	C₂	1	2	yes	maximal
3	#1954×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1277×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₃₁	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
9	#494×C₁₃₁	—	2	4	no	—
10	#474×C₁₃₁	—	2	4	no	—
11	#474×C₁₃₁	—	2	4	no	—
12	C₂×#107×C₁₃₁	—	2	2	no	—
13	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#102×C₁₃₁	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₁₃₁	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	#332×C₁₃₁	—	2	4	no	—
19	C₂²×Q₈×C₁₃₁	—	2	2	no	—
20	C₂²×D₄×C₁₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₁	—	2	1	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	8	no	—
23	C₂×Q₁₆×C₁₃₁	—	4	4	no	—
24	C₂×Q₁₆×C₁₃₁	—	4	4	no	—
25	C₂²×C₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₈×C₁₃₁	—	2	1	no	—
27	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
28	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
29	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
30	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
31	C₄⋊Q₈×C₁₃₁	D₄	1	4	yes	—
32	C₄×D₄×C₁₃₁	—	2	2	no	—
33	C₄×D₄×C₁₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
38	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	#107×C₁₃₁	—	2	8	no	—
40	C₄⋊C₈×C₁₃₁	—	2	2	no	—
41	C₄⋊C₈×C₁₃₁	—	2	2	no	—
42	C₄⋊C₈×C₁₃₁	D₄	1	4	yes	—
43	#102×C₁₃₁	—	2	8	no	—
44	C₂²⋊C₈×C₁₃₁	—	4	4	no	—
45	C₄×C₈×C₁₃₁	—	2	2	no	—
46	C₂⁴×C₁₃₁	—	2	1	no	—
47	C₂×Q₈×C₁₃₁	—	2	4	no	—
48	C₂×Q₈×C₁₃₁	—	4	2	no	—
49	C₂×Q₈×C₁₃₁	—	4	4	no	—
50	C₂×Q₈×C₁₃₁	—	4	4	no	—
51	C₂×D₄×C₁₃₁	—	2	2	no	—
52	C₂×D₄×C₁₃₁	—	2	1	no	—
53	C₂×D₄×C₁₃₁	—	2	2	no	—
54	C₂×D₄×C₁₃₁	Q₁₆	1	2	yes	complemented
55	C₂²×C₄×C₁₃₁	—	2	1	no	—
56	C₂²×C₄×C₁₃₁	—	4	2	no	—
57	C₂²×C₄×C₁₃₁	—	4	1	no	—
58	C₂²×C₄×C₁₃₁	—	4	1	no	—
59	C₂²×C₄×C₁₃₁	—	4	2	no	—
60	C₂²×C₄×C₁₃₁	—	4	2	no	—
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
64	Q₁₆×C₁₃₁	—	4	16	no	—
65	C₂×C₈×C₁₃₁	—	2	2	no	—
66	C₂×C₈×C₁₃₁	—	2	2	no	—
67	C₂×C₈×C₁₃₁	—	2	4	no	—
68	C₂×C₈×C₁₃₁	—	4	2	no	—
69	C₂×C₈×C₁₃₁	—	4	2	no	—
70	C₂×C₈×C₁₃₁	—	4	1	no	—
71	C₂×C₈×C₁₃₁	—	4	2	no	—
72	C₂×C₈×C₁₃₁	D₈	1	4	yes	—
73	C₄⋊C₄×C₁₃₁	—	2	4	no	—
74	C₄⋊C₄×C₁₃₁	—	2	2	no	—
75	C₄⋊C₄×C₁₃₁	—	2	4	no	—
76	C₄⋊C₄×C₁₃₁	—	4	2	no	—
77	C₄⋊C₄×C₁₃₁	—	4	2	no	—
78	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
79	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
80	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
81	C₂³×C₁₃₁	—	2	2	no	—
82	C₂³×C₁₃₁	—	4	1	no	—
83	C₂³×C₁₃₁	—	4	2	no	—
84	C₂³×C₁₃₁	—	4	1	no	—
85	C₂³×C₁₃₁	—	4	2	no	—
86	C₂³×C₁₃₁	#121	1	1	yes	characteristic
87	Q₈×C₁₃₁	—	4	8	no	—
88	Q₈×C₁₃₁	—	8	2	no	—
89	Q₈×C₁₃₁	—	8	2	no	—
90	D₄×C₁₃₁	—	2	4	no	—
91	D₄×C₁₃₁	—	4	2	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	4	1	no	—
94	C₂×C₄×C₁₃₁	—	4	4	no	—
95	C₂×C₄×C₁₃₁	—	4	2	no	—
96	C₂×C₄×C₁₃₁	—	8	4	no	—
97	C₂×C₄×C₁₃₁	—	8	2	no	—
98	C₂×C₄×C₁₃₁	—	8	1	no	—
99	C₂×C₄×C₁₃₁	—	8	1	no	—
100	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	#120	1	2	yes	—
102	C₂×C₄×C₁₃₁	#121	1	2	yes	—
103	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₄×C₁₃₁	#137	1	2	yes	—
109	C₈×C₁₃₁	—	2	4	no	—
110	C₈×C₁₃₁	—	4	4	no	—
111	C₈×C₁₃₁	—	8	1	no	—
112	C₂²×C₁₃₁	—	4	4	no	—
113	C₂²×C₁₃₁	—	4	2	no	—
114	C₂²×C₁₃₁	—	8	2	no	—
115	C₂²×C₁₃₁	—	8	1	no	—
116	C₂²×C₁₃₁	—	8	1	no	—
117	C₂²×C₁₃₁	#392	1	1	yes	characteristic
118	C₂²×C₁₃₁	#459	1	2	yes	—
119	C₂²×C₁₃₁	#467	1	1	yes	characteristic
120	C₂²×C₁₃₁	#470	1	1	yes	characteristic
121	C₂²×C₁₃₁	#473	1	2	yes	—
122	C₄×C₁₃₁	—	2	2	no	—
123	C₄×C₁₃₁	—	8	1	no	—
124	C₄×C₁₃₁	—	8	4	no	—
125	C₄×C₁₃₁	#447	1	2	yes	—
126	C₄×C₁₃₁	#451	1	2	yes	—
127	C₄×C₁₃₁	#452	1	2	yes	—
128	C₄×C₁₃₁	#493	1	1	yes	characteristic
129	C₄×C₁₃₁	#496	1	1	yes	characteristic
130	C₂×C₁₃₁	—	8	2	no	—
131	C₂×C₁₃₁	#1669	1	2	yes	—
132	C₂×C₁₃₁	#1689	1	2	yes	—
133	C₂×C₁₃₁	#2001	1	1	yes	characteristic
134	C₂×C₁₃₁	#2011	1	1	yes	characteristic
135	C₂×C₁₃₁	#2020	1	1	yes	characteristic
136	#10274	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₁	#10274	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#494	C₂×C₁₃₁	1	1	yes	characteristic, complemented
139	C₂×#474	C₂×C₁₃₁	1	2	yes	—
140	#1954	C₂×C₁₃₁	1	1	yes	characteristic
141	C₂×#332	C₂×C₁₃₁	1	2	yes	complemented
142	#1277	C₂×C₁₃₁	1	1	yes	characteristic, complemented
143	C₂²×Q₁₆	—	2	2	no	—
144	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
145	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
146	#494	—	2	4	no	—
147	#474	—	2	4	no	—
148	#474	—	2	4	no	—
149	C₂×#107	—	2	2	no	—
150	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
152	C₂×#102	—	2	2	no	—
153	C₂×C₂²⋊C₈	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
155	#332	—	2	4	no	—
156	C₂²×Q₈	—	2	2	no	—
157	C₂²×D₄	D₄×C₁₃₁	1	1	yes	characteristic
158	C₂³×C₄	—	2	1	no	—
159	C₂×Q₁₆	—	2	8	no	—
160	C₂×Q₁₆	—	4	4	no	—
161	C₂×Q₁₆	—	4	4	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	D₄×C₁₃₁	1	4	yes	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	1	no	—
175	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
176	#107	—	2	8	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	D₄×C₁₃₁	1	4	yes	—
180	#102	—	2	8	no	—
181	C₂²⋊C₈	—	4	4	no	—
182	C₄×C₈	—	2	2	no	—
183	C₂⁴	—	2	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	Q₁₆×C₁₃₁	1	2	yes	complemented
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
201	Q₁₆	—	4	16	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	D₈×C₁₃₁	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
217	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	4	1	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	2	no	—
223	C₂³	#121×C₁₃₁	1	1	yes	characteristic
224	Q₈	—	4	8	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
238	C₂×C₄	#120×C₁₃₁	1	2	yes	—
239	C₂×C₄	#121×C₁₃₁	1	2	yes	—
240	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
242	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
243	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
245	C₂×C₄	#137×C₁₃₁	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	8	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	2	no	—
254	C₂²	#392×C₁₃₁	1	1	yes	characteristic
255	C₂²	#459×C₁₃₁	1	2	yes	—
256	C₂²	#467×C₁₃₁	1	1	yes	characteristic
257	C₂²	#470×C₁₃₁	1	1	yes	characteristic
258	C₂²	#473×C₁₃₁	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	4	no	—
262	C₄	#447×C₁₃₁	1	2	yes	—
263	C₄	#451×C₁₃₁	1	2	yes	—
264	C₄	#452×C₁₃₁	1	2	yes	—
265	C₄	#493×C₁₃₁	1	1	yes	characteristic
266	C₄	#496×C₁₃₁	1	1	yes	characteristic
267	C₂	—	8	2	no	—
268	C₂	#1669×C₁₃₁	1	2	yes	—
269	C₂	#1689×C₁₃₁	1	2	yes	—
270	C₂	#2001×C₁₃₁	1	1	yes	characteristic
271	C₂	#2011×C₁₃₁	1	1	yes	characteristic
272	C₂	#2020×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#494×C₁₃₁	C₂	semidirect
4. C₂×#332×C₁₃₁	C₂	semidirect
5. #1277×C₁₃₁	C₂	semidirect
54. C₂×D₄×C₁₃₁	Q₁₆	semidirect
136. #10274	C₁₃₁	direct (N × H)
137. C₁₃₁	#10274	direct (N × H)
138. C₂×#494	C₂×C₁₃₁	semidirect
141. C₂×#332	C₂×C₁₃₁	semidirect
142. #1277	C₂×C₁₃₁	semidirect
191. C₂×D₄	Q₁₆×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₃₁	2
C₂×#474×C₁₃₁	2
#1954×C₁₃₁	2
C₂×#332×C₁₃₁	2
#1277×C₁₃₁	2
#10274	131

Subgroups — C131×#10274 #45696158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#10274 #45696158