Subgroups — C₁₃₁×#33720 #45719604

Order 33536.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#509×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×#508×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#3155×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2250×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2249×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2246×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	2	no	—
10	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₁₃₁	C₂²	1	2	yes	—
12	#535×C₁₃₁	—	2	2	no	—
13	#509×C₁₃₁	—	2	8	no	—
14	#508×C₁₃₁	—	2	4	no	—
15	C₂×Q₃₂×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₂×SD₃₂×C₁₃₁	C₂²	1	4	yes	complemented
17	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	complemented
18	C₂×D₁₆×C₁₃₁	—	2	4	no	—
19	C₂×D₁₆×C₁₃₁	C₂²	1	2	yes	—
20	C₂×D₁₆×C₁₃₁	C₂²	1	2	yes	—
21	C₂×D₁₆×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₁₆×C₁₃₁	C₂²	1	4	yes	complemented
23	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
25	#496×C₁₃₁	—	2	4	no	—
26	#495×C₁₃₁	C₂²	1	4	yes	complemented
27	#493×C₁₃₁	C₂²	1	4	yes	complemented
28	#405×C₁₃₁	C₂²	1	1	yes	characteristic
29	#346×C₁₃₁	C₂²	1	2	yes	complemented
30	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
31	C₂×#41×C₁₃₁	—	2	2	no	—
32	C₂²×D₄×C₁₃₁	—	2	2	no	—
33	AGL(1,ℤ/8)×C₁₃₁	—	4	8	no	—
34	#135×C₁₃₁	—	2	4	no	—
35	#135×C₁₃₁	—	2	4	no	—
36	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
39	C₂×D₈×C₁₃₁	—	2	4	no	—
40	C₂×D₈×C₁₃₁	—	2	4	no	—
41	C₂×D₈×C₁₃₁	—	2	4	no	—
42	C₂×D₈×C₁₃₁	—	2	2	no	—
43	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
44	C₂×D₈×C₁₃₁	C₂³	1	4	yes	—
45	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	#127×C₁₃₁	—	2	4	no	—
48	#124×C₁₃₁	—	2	4	no	—
49	#121×C₁₃₁	—	4	4	no	—
50	#120×C₁₃₁	—	4	4	no	—
51	#117×C₁₃₁	D₄	1	1	yes	characteristic
52	Q₃₂×C₁₃₁	—	2	2	no	—
53	SD₃₂×C₁₃₁	—	2	4	no	—
54	SD₃₂×C₁₃₁	—	2	8	no	—
55	D₁₆×C₁₃₁	—	2	4	no	—
56	D₁₆×C₁₃₁	—	2	4	no	—
57	D₁₆×C₁₃₁	—	2	2	no	—
58	D₁₆×C₁₃₁	—	2	8	no	—
59	M₃₂×C₁₃₁	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₁₃₁	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
64	#97×C₁₃₁	D₄	1	2	yes	—
65	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
66	C₂⁴×C₁₃₁	—	4	2	no	—
67	#41×C₁₃₁	—	4	4	no	—
68	#41×C₁₃₁	—	4	4	no	—
69	C₂×Q₈×C₁₃₁	—	2	2	no	—
70	C₂×D₄×C₁₃₁	—	2	4	no	—
71	C₂×D₄×C₁₃₁	—	2	2	no	—
72	C₂×D₄×C₁₃₁	—	4	2	no	—
73	C₂×D₄×C₁₃₁	—	4	4	no	—
74	C₂×D₄×C₁₃₁	—	4	4	no	—
75	C₂×D₄×C₁₃₁	—	4	2	no	—
76	C₂×D₄×C₁₃₁	—	4	8	no	—
77	C₂²×C₄×C₁₃₁	—	4	2	no	—
78	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₁₃₁	—	2	2	no	—
80	Q₁₆×C₁₃₁	—	2	2	no	—
81	SD₁₆×C₁₃₁	—	4	4	no	—
82	SD₁₆×C₁₃₁	—	4	8	no	—
83	D₈×C₁₃₁	—	2	4	no	—
84	D₈×C₁₃₁	—	2	2	no	—
85	D₈×C₁₃₁	—	2	4	no	—
86	D₈×C₁₃₁	—	2	2	no	—
87	D₈×C₁₃₁	—	4	4	no	—
88	D₈×C₁₃₁	—	4	8	no	—
89	M₁₆×C₁₃₁	D₈	1	4	yes	—
90	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₁₃₁	D₈	1	2	yes	—
96	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
97	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
98	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
99	C₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
101	C₁₆×C₁₃₁	C₂×D₄	1	4	yes	—
102	C₂³×C₁₃₁	—	4	4	no	—
103	C₂³×C₁₃₁	—	4	2	no	—
104	C₂³×C₁₃₁	—	8	4	no	—
105	C₂³×C₁₃₁	—	16	4	no	—
106	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₁₃₁	—	4	2	no	—
108	Q₈×C₁₃₁	—	4	2	no	—
109	D₄×C₁₃₁	—	4	4	no	—
110	D₄×C₁₃₁	—	4	2	no	—
111	D₄×C₁₃₁	—	4	2	no	—
112	D₄×C₁₃₁	—	4	4	no	—
113	D₄×C₁₃₁	—	8	4	no	—
114	D₄×C₁₃₁	—	8	4	no	—
115	D₄×C₁₃₁	—	16	8	no	—
116	C₂×C₄×C₁₃₁	—	4	2	no	—
117	C₂×C₄×C₁₃₁	—	8	4	no	—
118	C₂×C₄×C₁₃₁	#127	1	2	yes	—
119	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
120	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃₁	#127	1	2	yes	—
124	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
125	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₃₁	—	2	2	no	—
128	C₂²×C₁₃₁	—	8	2	no	—
129	C₂²×C₁₃₁	—	8	4	no	—
130	C₂²×C₁₃₁	—	8	4	no	—
131	C₂²×C₁₃₁	—	8	8	no	—
132	C₂²×C₁₃₁	—	16	8	no	—
133	C₂²×C₁₃₁	#493	1	2	yes	—
134	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₁₃₁	—	2	4	no	—
136	C₄×C₁₃₁	—	8	2	no	—
137	C₄×C₁₃₁	#493	1	2	yes	—
138	C₄×C₁₃₁	C₂×#127	1	1	yes	characteristic
139	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
140	C₂×C₁₃₁	—	2	2	no	—
141	C₂×C₁₃₁	—	16	2	no	—
142	C₂×C₁₃₁	—	16	4	no	—
143	C₂×C₁₃₁	C₂×#493	1	1	yes	characteristic
144	C₂×C₁₃₁	#3415	1	2	yes	—
145	#33720	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₁	#33720	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#509	C₂×C₁₃₁	1	2	yes	complemented
148	C₂×#508	C₂×C₁₃₁	1	1	yes	characteristic, complemented
149	C₂²×D₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
150	#3155	C₂×C₁₃₁	1	2	yes	complemented
151	#2250	C₂×C₁₃₁	1	4	yes	complemented
152	#2249	C₂×C₁₃₁	1	2	yes	complemented
153	#2246	C₂×C₁₃₁	1	2	yes	complemented
154	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
155	C₂×AGL(1,ℤ/8)	—	2	2	no	—
156	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
157	C₂²×D₈	C₂²×C₁₃₁	1	2	yes	—
158	#535	—	2	2	no	—
159	#509	—	2	8	no	—
160	#508	—	2	4	no	—
161	C₂×Q₃₂	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
162	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	complemented
163	C₂×SD₃₂	C₂²×C₁₃₁	1	4	yes	complemented
164	C₂×D₁₆	—	2	4	no	—
165	C₂×D₁₆	C₂²×C₁₃₁	1	2	yes	—
166	C₂×D₁₆	C₂²×C₁₃₁	1	2	yes	—
167	C₂×D₁₆	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
168	C₂×D₁₆	C₂²×C₁₃₁	1	4	yes	complemented
169	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
171	#496	—	2	4	no	—
172	#495	C₂²×C₁₃₁	1	4	yes	complemented
173	#493	C₂²×C₁₃₁	1	4	yes	complemented
174	#405	C₂²×C₁₃₁	1	1	yes	characteristic
175	#346	C₂²×C₁₃₁	1	2	yes	complemented
176	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
177	C₂×#41	—	2	2	no	—
178	C₂²×D₄	—	2	2	no	—
179	AGL(1,ℤ/8)	—	4	8	no	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×SD₁₆	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	4	no	—
189	C₂×D₈	C₂³×C₁₃₁	1	4	yes	—
190	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
191	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
193	#127	—	2	4	no	—
194	#124	—	2	4	no	—
195	#121	—	4	4	no	—
196	#120	—	4	4	no	—
197	#117	D₄×C₁₃₁	1	1	yes	characteristic
198	Q₃₂	—	2	2	no	—
199	SD₃₂	—	2	4	no	—
200	SD₃₂	—	2	8	no	—
201	D₁₆	—	2	4	no	—
202	D₁₆	—	2	4	no	—
203	D₁₆	—	2	2	no	—
204	D₁₆	—	2	8	no	—
205	M₃₂	D₄×C₁₃₁	1	4	yes	complemented
206	C₂×C₁₆	D₄×C₁₃₁	1	4	yes	complemented
207	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
209	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
210	#97	D₄×C₁₃₁	1	2	yes	—
211	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
212	C₂⁴	—	4	2	no	—
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	SD₁₆	—	4	8	no	—
229	D₈	—	2	4	no	—
230	D₈	—	2	2	no	—
231	D₈	—	2	4	no	—
232	D₈	—	2	2	no	—
233	D₈	—	4	4	no	—
234	D₈	—	4	8	no	—
235	M₁₆	D₈×C₁₃₁	1	4	yes	—
236	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
240	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
241	C₄⋊C₄	D₈×C₁₃₁	1	2	yes	—
242	C₂²⋊C₄	—	4	8	no	—
243	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
244	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
245	C₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
246	C₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
247	C₁₆	C₂×D₄×C₁₃₁	1	4	yes	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	4	no	—
251	C₂³	—	16	4	no	—
252	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	4	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	#127×C₁₃₁	1	2	yes	—
265	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
266	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
269	C₈	#127×C₁₃₁	1	2	yes	—
270	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
271	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
272	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	8	no	—
278	C₂²	—	16	8	no	—
279	C₂²	#493×C₁₃₁	1	2	yes	—
280	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	8	2	no	—
283	C₄	#493×C₁₃₁	1	2	yes	—
284	C₄	C₂×#127×C₁₃₁	1	1	yes	characteristic
285	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	2	no	—
289	C₂	C₂×#493×C₁₃₁	1	1	yes	characteristic
290	C₂	#3415×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ #117×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#509×C₁₃₁	C₂	semidirect
2. C₂×#508×C₁₃₁	C₂	semidirect
3. C₂²×D₁₆×C₁₃₁	C₂	semidirect
4. #3155×C₁₃₁	C₂	semidirect
5. #2250×C₁₃₁	C₂	semidirect
6. #2249×C₁₃₁	C₂	semidirect
7. #2246×C₁₃₁	C₂	semidirect
8. #2108×C₁₃₁	C₂	semidirect
15. C₂×Q₃₂×C₁₃₁	C₂²	semidirect
16. C₂×SD₃₂×C₁₃₁	C₂²	semidirect
17. C₂×SD₃₂×C₁₃₁	C₂²	semidirect
21. C₂×D₁₆×C₁₃₁	C₂²	semidirect
22. C₂×D₁₆×C₁₃₁	C₂²	semidirect
26. #495×C₁₃₁	C₂²	semidirect
27. #493×C₁₃₁	C₂²	semidirect
29. #346×C₁₃₁	C₂²	semidirect
30. C₄×C₁₆×C₁₃₁	C₂²	semidirect
59. M₃₂×C₁₃₁	D₄	semidirect
60. C₂×C₁₆×C₁₃₁	D₄	semidirect
145. #33720	C₁₃₁	direct (N × H)
146. C₁₃₁	#33720	direct (N × H)
147. C₂×#509	C₂×C₁₃₁	semidirect
148. C₂×#508	C₂×C₁₃₁	semidirect
149. C₂²×D₁₆	C₂×C₁₃₁	semidirect
150. #3155	C₂×C₁₃₁	semidirect
151. #2250	C₂×C₁₃₁	semidirect
152. #2249	C₂×C₁₃₁	semidirect
153. #2246	C₂×C₁₃₁	semidirect
154. #2108	C₂×C₁₃₁	semidirect
161. C₂×Q₃₂	C₂²×C₁₃₁	semidirect
162. C₂×SD₃₂	C₂²×C₁₃₁	semidirect
163. C₂×SD₃₂	C₂²×C₁₃₁	semidirect
167. C₂×D₁₆	C₂²×C₁₃₁	semidirect
168. C₂×D₁₆	C₂²×C₁₃₁	semidirect
172. #495	C₂²×C₁₃₁	semidirect
173. #493	C₂²×C₁₃₁	semidirect
175. #346	C₂²×C₁₃₁	semidirect
176. C₄×C₁₆	C₂²×C₁₃₁	semidirect
205. M₃₂	D₄×C₁₃₁	semidirect
206. C₂×C₁₆	D₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509×C₁₃₁	2
C₂×#508×C₁₃₁	2
C₂²×D₁₆×C₁₃₁	2
#3155×C₁₃₁	2
#2250×C₁₃₁	2
#2249×C₁₃₁	2
#2246×C₁₃₁	2
#2108×C₁₃₁	2
#33720	131

Subgroups — C131×#33720 #45719604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33720 #45719604