Subgroups — C₁₃₁×#33381 #45719265

Order 33536.

1176 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#504×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2913×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2140×C₁₃₁	C₂	1	3	yes	maximal, complemented
5	#2139×C₁₃₁	C₂	1	3	yes	maximal, complemented
6	#2138×C₁₃₁	C₂	1	3	yes	maximal, complemented
7	C₂×#357×C₁₃₁	C₂	1	3	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₁₃₁	—	2	1	no	—
9	#576×C₁₃₁	—	2	4	no	—
10	#576×C₁₃₁	C₄	1	8	yes	complemented
11	C₂×AGL(1,ℤ/8)×C₁₃₁	—	2	3	no	—
12	C₂×#135×C₁₃₁	C₂²	1	3	yes	—
13	C₂²×D₈×C₁₃₁	C₂²	1	3	yes	—
14	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
15	#520×C₁₃₁	—	2	1	no	—
16	#504×C₁₃₁	—	2	4	no	—
17	C₂×M₃₂×C₁₃₁	C₂²	1	3	yes	complemented
18	C₂²×C₁₆×C₁₃₁	C₂²	1	3	yes	complemented
19	#428×C₁₃₁	—	2	3	no	—
20	#427×C₁₃₁	C₂²	1	3	yes	complemented
21	C₂×#107×C₁₃₁	C₂²	1	3	yes	complemented
22	#358×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
23	#358×C₁₃₁	C₂²	1	3	yes	complemented
24	#357×C₁₃₁	C₂²	1	3	yes	complemented
25	#357×C₁₃₁	C₂²	1	6	yes	complemented
26	#357×C₁₃₁	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₁₃₁	—	2	8	no	—
28	H(5,𝔽₂)×C₁₃₁	—	4	4	no	—
29	C₂×#41×C₁₃₁	—	2	3	no	—
30	C₂×#41×C₁₃₁	—	4	1	no	—
31	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂²×D₄×C₁₃₁	—	2	3	no	—
33	C₂²×D₄×C₁₃₁	—	4	3	no	—
34	AGL(1,ℤ/8)×C₁₃₁	—	2	24	no	—
35	#135×C₁₃₁	—	2	6	no	—
36	#135×C₁₃₁	C₂×C₄	1	12	yes	complemented
37	C₂×Q₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂×SD₁₆×C₁₃₁	—	2	3	no	—
39	C₂×D₈×C₁₃₁	—	2	3	no	—
40	C₂×D₈×C₁₃₁	—	2	6	no	—
41	C₂×D₈×C₁₃₁	C₂×C₄	1	12	yes	complemented
42	C₂×D₈×C₁₃₁	C₂³	1	3	yes	—
43	#131×C₁₃₁	C₂×C₄	1	4	yes	—
44	#131×C₁₃₁	D₄	1	4	yes	—
45	C₂×M₁₆×C₁₃₁	D₄	1	3	yes	—
46	C₂²×C₈×C₁₃₁	C₂³	1	3	yes	—
47	C₄×Q₈×C₁₃₁	—	4	1	no	—
48	C₄×D₄×C₁₃₁	—	4	3	no	—
49	#117×C₁₃₁	—	2	3	no	—
50	C₂×C₄⋊C₄×C₁₃₁	—	2	3	no	—
51	M₃₂×C₁₃₁	—	2	6	no	—
52	C₂×C₁₆×C₁₃₁	—	2	6	no	—
53	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₁₃₁	C₂³	1	3	yes	—
55	#107×C₁₃₁	—	2	3	no	—
56	#107×C₁₃₁	C₂³	1	3	yes	—
57	#107×C₁₃₁	C₂³	1	1	yes	characteristic
58	#106×C₁₃₁	—	2	3	no	—
59	C₂⁴×C₁₃₁	—	4	3	no	—
60	#41×C₁₃₁	—	2	12	no	—
61	#41×C₁₃₁	—	4	8	no	—
62	#41×C₁₃₁	—	4	6	no	—
63	#41×C₁₃₁	C₂²⋊C₄	1	4	yes	—
64	#41×C₁₃₁	SD₁₆	1	4	yes	—
65	C₂×Q₈×C₁₃₁	—	2	1	no	—
66	C₂×Q₈×C₁₃₁	D₈	1	1	yes	characteristic
67	C₂×D₄×C₁₃₁	—	2	12	no	—
68	C₂×D₄×C₁₃₁	—	2	3	no	—
69	C₂×D₄×C₁₃₁	—	4	3	no	—
70	C₂×D₄×C₁₃₁	—	4	24	no	—
71	C₂×D₄×C₁₃₁	—	4	3	no	—
72	C₂×D₄×C₁₃₁	—	4	6	no	—
73	C₂×D₄×C₁₃₁	D₈	1	3	yes	—
74	C₂²×C₄×C₁₃₁	—	4	3	no	—
75	C₂²×C₄×C₁₃₁	—	4	3	no	—
76	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
77	Q₁₆×C₁₃₁	—	2	1	no	—
78	Q₁₆×C₁₃₁	C₂²×C₄	1	2	yes	—
79	SD₁₆×C₁₃₁	—	2	12	no	—
80	D₈×C₁₃₁	—	2	12	no	—
81	D₈×C₁₃₁	—	2	3	no	—
82	D₈×C₁₃₁	C₂²×C₄	1	6	yes	—
83	M₁₆×C₁₃₁	C₂²⋊C₄	1	12	yes	—
84	C₂×C₈×C₁₃₁	C₂²×C₄	1	6	yes	—
85	C₂×C₈×C₁₃₁	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₁₃₁	C₂×D₄	1	6	yes	—
87	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₁₃₁	—	2	1	no	—
89	C₄⋊C₄×C₁₃₁	—	2	3	no	—
90	C₄⋊C₄×C₁₃₁	—	4	3	no	—
91	C₂²⋊C₄×C₁₃₁	—	4	3	no	—
92	C₄²×C₁₃₁	—	4	3	no	—
93	C₁₆×C₁₃₁	—	2	1	no	—
94	C₁₆×C₁₃₁	—	2	3	no	—
95	C₂³×C₁₃₁	—	4	3	no	—
96	C₂³×C₁₃₁	—	4	12	no	—
97	C₂³×C₁₃₁	—	16	3	no	—
98	C₂³×C₁₃₁	—	16	3	no	—
99	C₂³×C₁₃₁	C₂×D₈	1	3	yes	—
100	Q₈×C₁₃₁	—	2	2	no	—
101	Q₈×C₁₃₁	—	4	1	no	—
102	Q₈×C₁₃₁	#102	1	4	yes	—
103	D₄×C₁₃₁	—	2	6	no	—
104	D₄×C₁₃₁	—	4	12	no	—
105	D₄×C₁₃₁	—	4	3	no	—
106	D₄×C₁₃₁	—	4	12	no	—
107	D₄×C₁₃₁	#102	1	12	yes	—
108	C₂×C₄×C₁₃₁	—	4	1	no	—
109	C₂×C₄×C₁₃₁	—	4	12	no	—
110	C₂×C₄×C₁₃₁	—	4	1	no	—
111	C₂×C₄×C₁₃₁	—	4	3	no	—
112	C₂×C₄×C₁₃₁	—	16	3	no	—
113	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₄×C₁₃₁	C₂×D₈	1	3	yes	—
115	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	6	yes	—
116	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	6	yes	—
118	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₁₃₁	—	2	3	no	—
121	C₂²×C₁₃₁	—	2	3	no	—
122	C₂²×C₁₃₁	—	4	6	no	—
123	C₂²×C₁₃₁	—	8	3	no	—
124	C₂²×C₁₃₁	—	8	3	no	—
125	C₂²×C₁₃₁	—	16	12	no	—
126	C₂²×C₁₃₁	C₂×#102	1	6	yes	—
127	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₁₃₁	—	4	2	no	—
129	C₄×C₁₃₁	—	8	3	no	—
130	C₄×C₁₃₁	—	8	1	no	—
131	C₄×C₁₃₁	C₂×#102	1	6	yes	—
132	C₄×C₁₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₁₃₁	C₂²×SD₁₆	1	1	yes	characteristic
134	C₂×C₁₃₁	—	2	6	no	—
135	C₂×C₁₃₁	—	8	6	no	—
136	C₂×C₁₃₁	C₂²×#102	1	1	yes	characteristic
137	C₂×C₁₃₁	#3415	1	1	yes	characteristic
138	C₂×C₁₃₁	#3416	1	1	yes	characteristic
139	#33381	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	#33381	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#576	C₂×C₁₃₁	1	1	yes	characteristic
142	C₂×#504	C₂×C₁₃₁	1	1	yes	characteristic, complemented
143	#2913	C₂×C₁₃₁	1	1	yes	characteristic, complemented
144	#2140	C₂×C₁₃₁	1	3	yes	complemented
145	#2139	C₂×C₁₃₁	1	3	yes	complemented
146	#2138	C₂×C₁₃₁	1	3	yes	complemented
147	C₂×#357	C₂×C₁₃₁	1	3	yes	complemented
148	C₂×H(5,𝔽₂)	—	2	1	no	—
149	#576	—	2	4	no	—
150	#576	C₄×C₁₃₁	1	8	yes	complemented
151	C₂×AGL(1,ℤ/8)	—	2	3	no	—
152	C₂×#135	C₂²×C₁₃₁	1	3	yes	—
153	C₂²×D₈	C₂²×C₁₃₁	1	3	yes	—
154	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
155	#520	—	2	1	no	—
156	#504	—	2	4	no	—
157	C₂×M₃₂	C₂²×C₁₃₁	1	3	yes	complemented
158	C₂²×C₁₆	C₂²×C₁₃₁	1	3	yes	complemented
159	#428	—	2	3	no	—
160	#427	C₂²×C₁₃₁	1	3	yes	complemented
161	C₂×#107	C₂²×C₁₃₁	1	3	yes	complemented
162	#358	C₂²×C₁₃₁	1	1	yes	characteristic, complemented
163	#358	C₂²×C₁₃₁	1	3	yes	complemented
164	#357	C₂²×C₁₃₁	1	3	yes	complemented
165	#357	C₂²×C₁₃₁	1	6	yes	complemented
166	#357	C₂²×C₁₃₁	1	3	yes	complemented
167	H(5,𝔽₂)	—	2	8	no	—
168	H(5,𝔽₂)	—	4	4	no	—
169	C₂×#41	—	2	3	no	—
170	C₂×#41	—	4	1	no	—
171	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
172	C₂²×D₄	—	2	3	no	—
173	C₂²×D₄	—	4	3	no	—
174	AGL(1,ℤ/8)	—	2	24	no	—
175	#135	—	2	6	no	—
176	#135	C₂×C₄×C₁₃₁	1	12	yes	complemented
177	C₂×Q₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
178	C₂×SD₁₆	—	2	3	no	—
179	C₂×D₈	—	2	3	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂×D₈	C₂×C₄×C₁₃₁	1	12	yes	complemented
182	C₂×D₈	C₂³×C₁₃₁	1	3	yes	—
183	#131	C₂×C₄×C₁₃₁	1	4	yes	—
184	#131	D₄×C₁₃₁	1	4	yes	—
185	C₂×M₁₆	D₄×C₁₃₁	1	3	yes	—
186	C₂²×C₈	C₂³×C₁₃₁	1	3	yes	—
187	C₄×Q₈	—	4	1	no	—
188	C₄×D₄	—	4	3	no	—
189	#117	—	2	3	no	—
190	C₂×C₄⋊C₄	—	2	3	no	—
191	M₃₂	—	2	6	no	—
192	C₂×C₁₆	—	2	6	no	—
193	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
194	C₂×C₁₆	C₂³×C₁₃₁	1	3	yes	—
195	#107	—	2	3	no	—
196	#107	C₂³×C₁₃₁	1	3	yes	—
197	#107	C₂³×C₁₃₁	1	1	yes	characteristic
198	#106	—	2	3	no	—
199	C₂⁴	—	4	3	no	—
200	#41	—	2	12	no	—
201	#41	—	4	8	no	—
202	#41	—	4	6	no	—
203	#41	C₂²⋊C₄×C₁₃₁	1	4	yes	—
204	#41	SD₁₆×C₁₃₁	1	4	yes	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	D₈×C₁₃₁	1	1	yes	characteristic
207	C₂×D₄	—	2	12	no	—
208	C₂×D₄	—	2	3	no	—
209	C₂×D₄	—	4	3	no	—
210	C₂×D₄	—	4	24	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂×D₄	—	4	6	no	—
213	C₂×D₄	D₈×C₁₃₁	1	3	yes	—
214	C₂²×C₄	—	4	3	no	—
215	C₂²×C₄	—	4	3	no	—
216	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
217	Q₁₆	—	2	1	no	—
218	Q₁₆	C₂²×C₄×C₁₃₁	1	2	yes	—
219	SD₁₆	—	2	12	no	—
220	D₈	—	2	12	no	—
221	D₈	—	2	3	no	—
222	D₈	C₂²×C₄×C₁₃₁	1	6	yes	—
223	M₁₆	C₂²⋊C₄×C₁₃₁	1	12	yes	—
224	C₂×C₈	C₂²×C₄×C₁₃₁	1	6	yes	—
225	C₂×C₈	C₂×D₄×C₁₃₁	1	3	yes	—
226	C₂×C₈	C₂×D₄×C₁₃₁	1	6	yes	—
227	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	3	no	—
230	C₄⋊C₄	—	4	3	no	—
231	C₂²⋊C₄	—	4	3	no	—
232	C₄²	—	4	3	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	—	2	3	no	—
235	C₂³	—	4	3	no	—
236	C₂³	—	4	12	no	—
237	C₂³	—	16	3	no	—
238	C₂³	—	16	3	no	—
239	C₂³	C₂×D₈×C₁₃₁	1	3	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	#102×C₁₃₁	1	4	yes	—
243	D₄	—	2	6	no	—
244	D₄	—	4	12	no	—
245	D₄	—	4	12	no	—
246	D₄	—	4	3	no	—
247	D₄	#102×C₁₃₁	1	12	yes	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	12	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	16	3	no	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	6	yes	—
254	C₂×C₄	C₂×D₈×C₁₃₁	1	3	yes	—
255	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	6	yes	—
256	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
257	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	6	yes	—
258	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
259	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
260	C₂²	—	2	3	no	—
261	C₂²	—	2	3	no	—
262	C₂²	—	4	6	no	—
263	C₂²	—	8	3	no	—
264	C₂²	—	8	3	no	—
265	C₂²	—	16	12	no	—
266	C₂²	C₂×#102×C₁₃₁	1	6	yes	—
267	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
268	C₄	—	4	2	no	—
269	C₄	—	8	3	no	—
270	C₄	—	8	1	no	—
271	C₄	C₂×#102×C₁₃₁	1	6	yes	—
272	C₄	C₂²×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
273	C₄	C₂²×SD₁₆×C₁₃₁	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	—	8	6	no	—
276	C₂	C₂²×#102×C₁₃₁	1	1	yes	characteristic
277	C₂	#3415×C₁₃₁	1	1	yes	characteristic
278	C₂	#3416×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂×#576×C₁₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×#41×C₁₃₁ ◁ C₂×#131×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#504×C₁₃₁	C₂	semidirect
3. #2913×C₁₃₁	C₂	semidirect
4. #2140×C₁₃₁	C₂	semidirect
5. #2139×C₁₃₁	C₂	semidirect
6. #2138×C₁₃₁	C₂	semidirect
7. C₂×#357×C₁₃₁	C₂	semidirect
10. #576×C₁₃₁	C₄	semidirect
17. C₂×M₃₂×C₁₃₁	C₂²	semidirect
18. C₂²×C₁₆×C₁₃₁	C₂²	semidirect
20. #427×C₁₃₁	C₂²	semidirect
21. C₂×#107×C₁₃₁	C₂²	semidirect
22. #358×C₁₃₁	C₂²	semidirect
23. #358×C₁₃₁	C₂²	semidirect
24. #357×C₁₃₁	C₂²	semidirect
25. #357×C₁₃₁	C₂²	semidirect
26. #357×C₁₃₁	C₂²	semidirect
36. #135×C₁₃₁	C₂×C₄	semidirect
41. C₂×D₈×C₁₃₁	C₂×C₄	semidirect
139. #33381	C₁₃₁	direct (N × H)
140. C₁₃₁	#33381	direct (N × H)
142. C₂×#504	C₂×C₁₃₁	semidirect
143. #2913	C₂×C₁₃₁	semidirect
144. #2140	C₂×C₁₃₁	semidirect
145. #2139	C₂×C₁₃₁	semidirect
146. #2138	C₂×C₁₃₁	semidirect
147. C₂×#357	C₂×C₁₃₁	semidirect
150. #576	C₄×C₁₃₁	semidirect
157. C₂×M₃₂	C₂²×C₁₃₁	semidirect
158. C₂²×C₁₆	C₂²×C₁₃₁	semidirect
160. #427	C₂²×C₁₃₁	semidirect
161. C₂×#107	C₂²×C₁₃₁	semidirect
162. #358	C₂²×C₁₃₁	semidirect
163. #358	C₂²×C₁₃₁	semidirect
164. #357	C₂²×C₁₃₁	semidirect
165. #357	C₂²×C₁₃₁	semidirect
166. #357	C₂²×C₁₃₁	semidirect
176. #135	C₂×C₄×C₁₃₁	semidirect
181. C₂×D₈	C₂×C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₁₃₁	2
C₂×#504×C₁₃₁	2
#2913×C₁₃₁	2
#2140×C₁₃₁	2
#2139×C₁₃₁	2
#2138×C₁₃₁	2
C₂×#357×C₁₃₁	2
#33381	131

Subgroups — C131×#33381 #45719265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33381 #45719265