Subgroups — C₁₃₁×#33286 #45719170

Order 33536.

834 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2859×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#2858×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1990×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1957×C₁₃₁	C₂	1	2	yes	maximal
5	#1936×C₁₃₁	C₂	1	8	yes	maximal, complemented
6	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₁	—	2	1	no	—
10	C₂×#124×C₁₃₁	—	2	2	no	—
11	C₂×#102×C₁₃₁	C₂²	1	8	yes	—
12	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
14	#390×C₁₃₁	—	2	1	no	—
15	#390×C₁₃₁	—	2	1	no	—
16	#390×C₁₃₁	C₂²	1	4	yes	—
17	#389×C₁₃₁	—	2	2	no	—
18	#389×C₁₃₁	C₂²	1	4	yes	—
19	#384×C₁₃₁	C₂²	1	2	yes	—
20	#382×C₁₃₁	—	2	1	no	—
21	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
22	#340×C₁₃₁	C₂²	1	8	yes	—
23	C₂²×Q₈×C₁₃₁	—	2	1	no	—
24	C₂²×D₄×C₁₃₁	—	2	1	no	—
25	C₂²×D₄×C₁₃₁	—	2	1	no	—
26	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
29	#127×C₁₃₁	—	2	4	no	—
30	#125×C₁₃₁	—	4	4	no	—
31	#124×C₁₃₁	—	4	8	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
38	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
45	#102×C₁₃₁	—	2	16	no	—
46	#97×C₁₃₁	—	2	8	no	—
47	C₄×C₈×C₁₃₁	—	2	4	no	—
48	#95×C₁₃₁	—	2	4	no	—
49	C₂⁴×C₁₃₁	—	2	2	no	—
50	C₂×Q₈×C₁₃₁	—	4	2	no	—
51	C₂×Q₈×C₁₃₁	—	4	4	no	—
52	C₂×D₄×C₁₃₁	—	2	8	no	—
53	C₂×D₄×C₁₃₁	—	2	4	no	—
54	C₂×D₄×C₁₃₁	—	4	2	no	—
55	C₂×D₄×C₁₃₁	—	4	2	no	—
56	C₂×D₄×C₁₃₁	—	4	4	no	—
57	C₂×D₄×C₁₃₁	—	4	2	no	—
58	C₂×D₄×C₁₃₁	—	4	2	no	—
59	C₂²×C₄×C₁₃₁	—	2	4	no	—
60	C₂²×C₄×C₁₃₁	—	2	2	no	—
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	—	2	4	no	—
67	C₂×C₈×C₁₃₁	—	2	4	no	—
68	C₂×C₈×C₁₃₁	#41	1	8	yes	—
69	C₄⋊C₄×C₁₃₁	—	2	8	no	—
70	C₄⋊C₄×C₁₃₁	—	2	4	no	—
71	C₄⋊C₄×C₁₃₁	—	4	8	no	—
72	C₄⋊C₄×C₁₃₁	—	4	2	no	—
73	C₄⋊C₄×C₁₃₁	—	4	4	no	—
74	C₄⋊C₄×C₁₃₁	—	4	4	no	—
75	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
76	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
77	C₄²×C₁₃₁	—	2	4	no	—
78	C₄²×C₁₃₁	—	2	2	no	—
79	C₄²×C₁₃₁	—	2	2	no	—
80	C₄²×C₁₃₁	—	2	4	no	—
81	C₄²×C₁₃₁	#41	1	4	yes	—
82	C₂³×C₁₃₁	—	4	2	no	—
83	C₂³×C₁₃₁	—	4	4	no	—
84	C₂³×C₁₃₁	—	4	4	no	—
85	C₂³×C₁₃₁	—	4	2	no	—
86	C₂³×C₁₃₁	—	4	2	no	—
87	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₁₃₁	—	8	4	no	—
89	D₄×C₁₃₁	—	4	8	no	—
90	D₄×C₁₃₁	—	8	4	no	—
91	D₄×C₁₃₁	—	8	4	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	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	4	no	—
102	C₂×C₄×C₁₃₁	—	4	4	no	—
103	C₂×C₄×C₁₃₁	#124	1	2	yes	—
104	C₂×C₄×C₁₃₁	#124	1	2	yes	—
105	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₁₃₁	#137	1	2	yes	—
108	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
109	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
110	C₈×C₁₃₁	—	2	8	no	—
111	C₂²×C₁₃₁	—	8	2	no	—
112	C₂²×C₁₃₁	—	8	2	no	—
113	C₂²×C₁₃₁	—	8	2	no	—
114	C₂²×C₁₃₁	—	8	4	no	—
115	C₂²×C₁₃₁	—	8	4	no	—
116	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
117	C₂²×C₁₃₁	#532	1	2	yes	—
118	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₁₃₁	C₂×#137	1	1	yes	characteristic
121	C₄×C₁₃₁	—	4	2	no	—
122	C₄×C₁₃₁	—	4	2	no	—
123	C₄×C₁₃₁	—	8	2	no	—
124	C₄×C₁₃₁	—	8	4	no	—
125	C₄×C₁₃₁	#489	1	4	yes	—
126	C₄×C₁₃₁	#490	1	4	yes	—
127	C₄×C₁₃₁	#529	1	4	yes	—
128	C₂×C₁₃₁	—	16	2	no	—
129	C₂×C₁₃₁	#2861	1	1	yes	characteristic
130	C₂×C₁₃₁	C₂×#489	1	1	yes	characteristic
131	C₂×C₁₃₁	C₂×#490	1	1	yes	characteristic
132	C₂×C₁₃₁	#3182	1	2	yes	—
133	C₂×C₁₃₁	#3184	1	2	yes	—
134	#33286	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₃₁	#33286	1	1	yes	characteristic, direct product factor, Sylow
136	#2859	C₂×C₁₃₁	1	1	yes	characteristic, complemented
137	#2858	C₂×C₁₃₁	1	1	yes	characteristic
138	#1990	C₂×C₁₃₁	1	2	yes	complemented
139	#1957	C₂×C₁₃₁	1	2	yes	—
140	#1936	C₂×C₁₃₁	1	8	yes	complemented
141	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic, complemented
142	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
143	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
144	C₂×#125	—	2	1	no	—
145	C₂×#124	—	2	2	no	—
146	C₂×#102	C₂²×C₁₃₁	1	8	yes	—
147	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
149	#390	—	2	1	no	—
150	#390	—	2	1	no	—
151	#390	C₂²×C₁₃₁	1	4	yes	—
152	#389	—	2	2	no	—
153	#389	C₂²×C₁₃₁	1	4	yes	—
154	#384	C₂²×C₁₃₁	1	2	yes	—
155	#382	—	2	1	no	—
156	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
157	#340	C₂²×C₁₃₁	1	8	yes	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
162	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
163	C₄⋊Q₈	—	2	4	no	—
164	#127	—	2	4	no	—
165	#125	—	4	4	no	—
166	#124	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
176	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
179	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
180	#102	—	2	16	no	—
181	#97	—	2	8	no	—
182	C₄×C₈	—	2	4	no	—
183	#95	—	2	4	no	—
184	C₂⁴	—	2	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	#41×C₁₃₁	1	8	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	8	no	—
211	C₂²⋊C₄	—	4	8	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	#41×C₁₃₁	1	4	yes	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
223	Q₈	—	8	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	#124×C₁₃₁	1	2	yes	—
239	C₂×C₄	#124×C₁₃₁	1	2	yes	—
240	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
242	C₂×C₄	#137×C₁₃₁	1	2	yes	—
243	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
244	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
245	C₈	—	2	8	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
252	C₂²	#532×C₁₃₁	1	2	yes	—
253	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
254	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
255	C₂²	C₂×#137×C₁₃₁	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	4	no	—
260	C₄	#489×C₁₃₁	1	4	yes	—
261	C₄	#490×C₁₃₁	1	4	yes	—
262	C₄	#529×C₁₃₁	1	4	yes	—
263	C₂	—	16	2	no	—
264	C₂	#2861×C₁₃₁	1	1	yes	characteristic
265	C₂	C₂×#489×C₁₃₁	1	1	yes	characteristic
266	C₂	C₂×#490×C₁₃₁	1	1	yes	characteristic
267	C₂	#3182×C₁₃₁	1	2	yes	—
268	C₂	#3184×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
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
1. #2859×C₁₃₁	C₂	semidirect
3. #1990×C₁₃₁	C₂	semidirect
5. #1936×C₁₃₁	C₂	semidirect
6. C₄×#97×C₁₃₁	C₂	semidirect
134. #33286	C₁₃₁	direct (N × H)
135. C₁₃₁	#33286	direct (N × H)
136. #2859	C₂×C₁₃₁	semidirect
138. #1990	C₂×C₁₃₁	semidirect
140. #1936	C₂×C₁₃₁	semidirect
141. C₄×#97	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2859×C₁₃₁	2
#2858×C₁₃₁	2
#1990×C₁₃₁	2
#1957×C₁₃₁	2
#1936×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33286	131

Subgroups — C131×#33286 #45719170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33286 #45719170