Subgroups — C₁₃₁×#9832 #45695716

Order 33536.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1957×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1957×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1834×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₁	—	2	2	no	—
7	C₂×#127×C₁₃₁	C₂²	1	2	yes	—
8	#493×C₁₃₁	—	2	2	no	—
9	#493×C₁₃₁	C₄	1	4	yes	complemented
10	C₂×#102×C₁₃₁	—	2	4	no	—
11	C₂×#102×C₁₃₁	—	2	4	no	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#384×C₁₃₁	C₂²	1	2	yes	—
16	#340×C₁₃₁	—	2	2	no	—
17	#322×C₁₃₁	—	2	2	no	—
18	C₂²×D₄×C₁₃₁	—	2	2	no	—
19	C₂²×D₄×C₁₃₁	—	2	2	no	—
20	C₂²×D₄×C₁₃₁	—	2	2	no	—
21	C₂×D₈×C₁₃₁	—	2	8	no	—
22	C₂×D₈×C₁₃₁	—	4	8	no	—
23	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
25	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
26	#127×C₁₃₁	—	2	2	no	—
27	#127×C₁₃₁	—	2	2	no	—
28	#127×C₁₃₁	—	2	2	no	—
29	#127×C₁₃₁	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
34	#102×C₁₃₁	—	4	8	no	—
35	#102×C₁₃₁	—	4	8	no	—
36	C₄×C₈×C₁₃₁	—	2	1	no	—
37	C₄×C₈×C₁₃₁	—	2	1	no	—
38	C₄×C₈×C₁₃₁	—	2	1	no	—
39	C₄×C₈×C₁₃₁	—	2	1	no	—
40	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
42	C₂⁴×C₁₃₁	—	4	2	no	—
43	C₂×D₄×C₁₃₁	—	2	4	no	—
44	C₂×D₄×C₁₃₁	—	2	4	no	—
45	C₂×D₄×C₁₃₁	—	2	4	no	—
46	C₂×D₄×C₁₃₁	—	4	2	no	—
47	C₂×D₄×C₁₃₁	—	4	2	no	—
48	C₂×D₄×C₁₃₁	—	4	2	no	—
49	C₂×D₄×C₁₃₁	—	4	4	no	—
50	C₂×D₄×C₁₃₁	—	4	4	no	—
51	C₂×D₄×C₁₃₁	—	4	4	no	—
52	C₂×D₄×C₁₃₁	—	4	4	no	—
53	C₂×D₄×C₁₃₁	—	4	4	no	—
54	C₂×D₄×C₁₃₁	—	4	4	no	—
55	C₂²×C₄×C₁₃₁	—	4	2	no	—
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	D₈×C₁₃₁	—	4	16	no	—
60	C₂×C₈×C₁₃₁	—	2	2	no	—
61	C₂×C₈×C₁₃₁	—	2	2	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	—	2	2	no	—
64	C₂×C₈×C₁₃₁	—	2	4	no	—
65	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₃₁	D₈	1	4	yes	complemented
67	C₄⋊C₄×C₁₃₁	—	4	4	no	—
68	C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄⋊C₄×C₁₃₁	—	4	4	no	—
70	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃₁	#41	1	1	yes	characteristic
73	C₄²×C₁₃₁	#41	1	1	yes	characteristic
74	C₂³×C₁₃₁	—	4	4	no	—
75	C₂³×C₁₃₁	—	8	2	no	—
76	C₂³×C₁₃₁	—	8	2	no	—
77	C₂³×C₁₃₁	—	8	2	no	—
78	C₂³×C₁₃₁	—	8	2	no	—
79	C₂³×C₁₃₁	—	8	2	no	—
80	C₂³×C₁₃₁	—	8	2	no	—
81	C₂³×C₁₃₁	#127	1	1	yes	characteristic
82	D₄×C₁₃₁	—	4	8	no	—
83	D₄×C₁₃₁	—	4	8	no	—
84	D₄×C₁₃₁	—	4	8	no	—
85	D₄×C₁₃₁	—	8	4	no	—
86	D₄×C₁₃₁	—	8	4	no	—
87	D₄×C₁₃₁	—	8	4	no	—
88	C₂×C₄×C₁₃₁	—	8	2	no	—
89	C₂×C₄×C₁₃₁	—	8	2	no	—
90	C₂×C₄×C₁₃₁	—	8	2	no	—
91	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₁₃₁	#135	1	2	yes	—
99	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
100	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
101	C₈×C₁₃₁	—	2	4	no	—
102	C₈×C₁₃₁	—	2	4	no	—
103	C₈×C₁₃₁	—	4	2	no	—
104	C₂²×C₁₃₁	—	8	4	no	—
105	C₂²×C₁₃₁	—	8	4	no	—
106	C₂²×C₁₃₁	—	8	4	no	—
107	C₂²×C₁₃₁	—	16	2	no	—
108	C₂²×C₁₃₁	—	16	2	no	—
109	C₂²×C₁₃₁	—	16	2	no	—
110	C₂²×C₁₃₁	—	16	2	no	—
111	C₂²×C₁₃₁	#390	1	1	yes	characteristic
112	C₂²×C₁₃₁	#493	1	1	yes	characteristic
113	C₂²×C₁₃₁	#495	1	1	yes	characteristic
114	C₂²×C₁₃₁	#496	1	2	yes	—
115	C₂²×C₁₃₁	#496	1	2	yes	—
116	C₄×C₁₃₁	—	16	2	no	—
117	C₄×C₁₃₁	#418	1	2	yes	—
118	C₄×C₁₃₁	C₄×D₈	1	2	yes	—
119	C₄×C₁₃₁	#442	1	2	yes	—
120	C₄×C₁₃₁	#466	1	2	yes	—
121	C₄×C₁₃₁	#469	1	2	yes	—
122	C₄×C₁₃₁	#490	1	2	yes	—
123	C₂×C₁₃₁	—	16	4	no	—
124	C₂×C₁₃₁	#1717	1	2	yes	—
125	C₂×C₁₃₁	#1721	1	2	yes	—
126	C₂×C₁₃₁	#1961	1	1	yes	characteristic
127	C₂×C₁₃₁	#1970	1	1	yes	characteristic
128	C₂×C₁₃₁	#1972	1	1	yes	characteristic
129	#9832	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₁	#9832	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#493	C₂×C₁₃₁	1	1	yes	characteristic
132	#1957	C₂×C₁₃₁	1	2	yes	complemented
133	#1957	C₂×C₁₃₁	1	2	yes	complemented
134	#1834	C₂×C₁₃₁	1	1	yes	characteristic, complemented
135	C₂×#322	C₂×C₁₃₁	1	1	yes	characteristic, complemented
136	C₂²×D₈	—	2	2	no	—
137	C₂×#127	C₂²×C₁₃₁	1	2	yes	—
138	#493	—	2	2	no	—
139	#493	C₄×C₁₃₁	1	4	yes	complemented
140	C₂×#102	—	2	4	no	—
141	C₂×#102	—	2	4	no	—
142	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
145	#384	C₂²×C₁₃₁	1	2	yes	—
146	#340	—	2	2	no	—
147	#322	—	2	2	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂×D₈	—	2	8	no	—
152	C₂×D₈	—	4	8	no	—
153	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
154	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
155	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	C₂×C₄×C₁₃₁	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
164	#102	—	4	8	no	—
165	#102	—	4	8	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
171	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
172	C₂⁴	—	4	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
189	D₈	—	4	16	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
196	C₂×C₈	D₈×C₁₃₁	1	4	yes	complemented
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
202	C₄²	#41×C₁₃₁	1	1	yes	characteristic
203	C₄²	#41×C₁₃₁	1	1	yes	characteristic
204	C₂³	—	4	4	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	—	8	2	no	—
208	C₂³	—	8	2	no	—
209	C₂³	—	8	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	#127×C₁₃₁	1	1	yes	characteristic
212	D₄	—	4	8	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
224	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
225	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
228	C₂×C₄	#135×C₁₃₁	1	2	yes	—
229	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
230	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	16	2	no	—
238	C₂²	—	16	2	no	—
239	C₂²	—	16	2	no	—
240	C₂²	—	16	2	no	—
241	C₂²	#390×C₁₃₁	1	1	yes	characteristic
242	C₂²	#493×C₁₃₁	1	1	yes	characteristic
243	C₂²	#495×C₁₃₁	1	1	yes	characteristic
244	C₂²	#496×C₁₃₁	1	2	yes	—
245	C₂²	#496×C₁₃₁	1	2	yes	—
246	C₄	—	16	2	no	—
247	C₄	#418×C₁₃₁	1	2	yes	—
248	C₄	C₄×D₈×C₁₃₁	1	2	yes	—
249	C₄	#442×C₁₃₁	1	2	yes	—
250	C₄	#466×C₁₃₁	1	2	yes	—
251	C₄	#469×C₁₃₁	1	2	yes	—
252	C₄	#490×C₁₃₁	1	2	yes	—
253	C₂	—	16	4	no	—
254	C₂	#1717×C₁₃₁	1	2	yes	—
255	C₂	#1721×C₁₃₁	1	2	yes	—
256	C₂	#1961×C₁₃₁	1	1	yes	characteristic
257	C₂	#1970×C₁₃₁	1	1	yes	characteristic
258	C₂	#1972×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	#127
commutator	C₄²	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂×#493×C₁₃₁	C₂
LCS	G ▷ 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
2. #1957×C₁₃₁	C₂	semidirect
3. #1957×C₁₃₁	C₂	semidirect
4. #1834×C₁₃₁	C₂	semidirect
5. C₂×#322×C₁₃₁	C₂	semidirect
9. #493×C₁₃₁	C₄	semidirect
66. C₂×C₈×C₁₃₁	D₈	semidirect
129. #9832	C₁₃₁	direct (N × H)
130. C₁₃₁	#9832	direct (N × H)
132. #1957	C₂×C₁₃₁	semidirect
133. #1957	C₂×C₁₃₁	semidirect
134. #1834	C₂×C₁₃₁	semidirect
135. C₂×#322	C₂×C₁₃₁	semidirect
139. #493	C₄×C₁₃₁	semidirect
196. C₂×C₈	D₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₁	2
#1957×C₁₃₁	2
#1834×C₁₃₁	2
C₂×#322×C₁₃₁	2
#9832	131

Subgroups — C131×#9832 #45695716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#9832 #45695716