Subgroups — C₁₃₁×#23198 #45709082

Order 33536.

946 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×#492×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2949×C₁₃₁	C₂	1	4	yes	maximal, complemented
3	#1977×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#1887×C₁₃₁	C₂	1	2	yes	maximal
5	#1886×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1772×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₁₃₁	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
10	#519×C₁₃₁	C₂²	1	2	yes	—
11	#518×C₁₃₁	C₂²	1	2	yes	—
12	#492×C₁₃₁	C₄	1	8	yes	complemented
13	#440×C₁₃₁	—	2	8	no	—
14	C₄×SD₁₆×C₁₃₁	—	2	8	no	—
15	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
16	#427×C₁₃₁	C₂²	1	2	yes	—
17	#419×C₁₃₁	C₂²	1	4	yes	—
18	#418×C₁₃₁	C₂²	1	4	yes	—
19	C₂×#101×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#100×C₁₃₁	C₂²	1	2	yes	—
21	#405×C₁₃₁	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	#339×C₁₃₁	C₂²	1	4	yes	—
24	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
25	C₂²×D₄×C₁₃₁	—	4	1	no	—
26	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
27	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
28	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
29	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
30	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	16	yes	complemented
31	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
33	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
34	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
35	#127×C₁₃₁	—	4	1	no	—
36	#127×C₁₃₁	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₃₁	—	2	8	no	—
38	C₄×D₄×C₁₃₁	—	2	8	no	—
39	#117×C₁₃₁	—	4	2	no	—
40	#117×C₁₃₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₁₃₁	—	4	2	no	—
43	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	#108×C₁₃₁	—	2	4	no	—
45	#107×C₁₃₁	D₄	1	4	yes	complemented
46	#106×C₁₃₁	D₄	1	4	yes	complemented
47	#103×C₁₃₁	—	2	8	no	—
48	#102×C₁₃₁	—	2	8	no	—
49	#101×C₁₃₁	—	4	2	no	—
50	#100×C₁₃₁	—	4	2	no	—
51	#97×C₁₃₁	—	2	4	no	—
52	C₄×C₈×C₁₃₁	—	2	4	no	—
53	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
54	C₂⁴×C₁₃₁	—	2	2	no	—
55	C₂×Q₈×C₁₃₁	—	2	2	no	—
56	C₂×Q₈×C₁₃₁	—	2	4	no	—
57	C₂×Q₈×C₁₃₁	—	2	4	no	—
58	C₂×Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
59	C₂×D₄×C₁₃₁	—	2	4	no	—
60	C₂×D₄×C₁₃₁	—	2	4	no	—
61	C₂×D₄×C₁₃₁	—	2	2	no	—
62	C₂×D₄×C₁₃₁	—	4	4	no	—
63	C₂×D₄×C₁₃₁	—	8	4	no	—
64	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₁	—	2	1	no	—
66	C₂²×C₄×C₁₃₁	—	2	2	no	—
67	C₂²×C₄×C₁₃₁	—	4	4	no	—
68	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
69	SD₁₆×C₁₃₁	—	2	16	no	—
70	SD₁₆×C₁₃₁	—	2	16	no	—
71	M₁₆×C₁₃₁	—	2	4	no	—
72	M₁₆×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	2	4	no	—
74	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃₁	#41	1	4	yes	—
78	C₄⋊C₄×C₁₃₁	—	4	4	no	—
79	C₄⋊C₄×C₁₃₁	—	4	4	no	—
80	C₄⋊C₄×C₁₃₁	C₂×D₄	1	8	yes	—
81	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
82	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
83	C₄²×C₁₃₁	—	2	4	no	—
84	C₄²×C₁₃₁	—	2	1	no	—
85	C₄²×C₁₃₁	—	4	4	no	—
86	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
87	C₂³×C₁₃₁	—	2	4	no	—
88	C₂³×C₁₃₁	—	4	2	no	—
89	C₂³×C₁₃₁	—	4	2	no	—
90	C₂³×C₁₃₁	—	8	4	no	—
91	C₂³×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	2	8	no	—
93	Q₈×C₁₃₁	—	4	4	no	—
94	D₄×C₁₃₁	—	2	8	no	—
95	D₄×C₁₃₁	—	4	4	no	—
96	D₄×C₁₃₁	—	8	8	no	—
97	C₂×C₄×C₁₃₁	—	2	4	no	—
98	C₂×C₄×C₁₃₁	—	2	2	no	—
99	C₂×C₄×C₁₃₁	—	2	4	no	—
100	C₂×C₄×C₁₃₁	—	2	8	no	—
101	C₂×C₄×C₁₃₁	—	4	2	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	16	4	no	—
104	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
107	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
108	C₈×C₁₃₁	—	4	4	no	—
109	C₈×C₁₃₁	C₄×D₄	1	4	yes	—
110	C₈×C₁₃₁	C₄×D₄	1	4	yes	—
111	C₂²×C₁₃₁	—	2	2	no	—
112	C₂²×C₁₃₁	—	4	4	no	—
113	C₂²×C₁₃₁	—	8	4	no	—
114	C₂²×C₁₃₁	—	8	4	no	—
115	C₂²×C₁₃₁	—	16	2	no	—
116	C₂²×C₁₃₁	#518	1	1	yes	characteristic
117	C₂²×C₁₃₁	D₄²	1	1	yes	characteristic
118	C₂²×C₁₃₁	#559	1	1	yes	characteristic
119	C₄×C₁₃₁	—	2	4	no	—
120	C₄×C₁₃₁	—	4	4	no	—
121	C₄×C₁₃₁	—	4	8	no	—
122	C₄×C₁₃₁	C₂×C₄×D₄	1	2	yes	—
123	C₄×C₁₃₁	#550	1	2	yes	—
124	C₂×C₁₃₁	—	2	1	no	—
125	C₂×C₁₃₁	—	2	1	no	—
126	C₂×C₁₃₁	—	8	4	no	—
127	C₂×C₁₃₁	#2366	1	1	yes	characteristic
128	C₂×C₁₃₁	#3291	1	2	yes	—
129	#23198	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₁	#23198	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#492	C₂×C₁₃₁	1	1	yes	characteristic
132	#2949	C₂×C₁₃₁	1	4	yes	complemented
133	#1977	C₂×C₁₃₁	1	4	yes	complemented
134	#1887	C₂×C₁₃₁	1	2	yes	—
135	#1886	C₂×C₁₃₁	1	2	yes	complemented
136	#1772	C₂×C₁₃₁	1	2	yes	complemented
137	C₂²×SD₁₆	C₂²×C₁₃₁	1	4	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
140	#519	C₂²×C₁₃₁	1	2	yes	—
141	#518	C₂²×C₁₃₁	1	2	yes	—
142	#492	C₄×C₁₃₁	1	8	yes	complemented
143	#440	—	2	8	no	—
144	C₄×SD₁₆	—	2	8	no	—
145	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
146	#427	C₂²×C₁₃₁	1	2	yes	—
147	#419	C₂²×C₁₃₁	1	4	yes	—
148	#418	C₂²×C₁₃₁	1	4	yes	—
149	C₂×#101	C₂²×C₁₃₁	1	2	yes	—
150	C₂×#100	C₂²×C₁₃₁	1	2	yes	—
151	#405	C₂²×C₁₃₁	1	4	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
153	#339	C₂²×C₁₃₁	1	4	yes	—
154	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
155	C₂²×D₄	—	4	1	no	—
156	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
157	C₂×SD₁₆	—	2	4	no	—
158	C₂×SD₁₆	—	2	4	no	—
159	C₂×SD₁₆	—	2	8	no	—
160	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	16	yes	complemented
161	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
162	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
163	C₄⋊Q₈	—	4	1	no	—
164	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
165	#127	—	4	1	no	—
166	#127	C₂×C₄×C₁₃₁	1	4	yes	—
167	C₄×Q₈	—	2	8	no	—
168	C₄×D₄	—	2	8	no	—
169	#117	—	4	2	no	—
170	#117	C₂³×C₁₃₁	1	4	yes	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
174	#108	—	2	4	no	—
175	#107	D₄×C₁₃₁	1	4	yes	complemented
176	#106	D₄×C₁₃₁	1	4	yes	complemented
177	#103	—	2	8	no	—
178	#102	—	2	8	no	—
179	#101	—	4	2	no	—
180	#100	—	4	2	no	—
181	#97	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
184	C₂⁴	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	C₂²×C₄×C₁₃₁	1	4	yes	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	8	4	no	—
194	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
199	SD₁₆	—	2	16	no	—
200	SD₁₆	—	2	16	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
205	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
207	C₂×C₈	#41×C₁₃₁	1	4	yes	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	C₂×D₄×C₁₃₁	1	8	yes	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
217	C₂³	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	8	4	no	—
221	C₂³	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
222	Q₈	—	2	8	no	—
223	Q₈	—	4	4	no	—
224	D₄	—	2	8	no	—
225	D₄	—	4	4	no	—
226	D₄	—	8	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
236	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
237	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
238	C₈	—	4	4	no	—
239	C₈	C₄×D₄×C₁₃₁	1	4	yes	—
240	C₈	C₄×D₄×C₁₃₁	1	4	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	16	2	no	—
246	C₂²	#518×C₁₃₁	1	1	yes	characteristic
247	C₂²	D₄²×C₁₃₁	1	1	yes	characteristic
248	C₂²	#559×C₁₃₁	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	4	no	—
252	C₄	C₂×C₄×D₄×C₁₃₁	1	2	yes	—
253	C₄	#550×C₁₃₁	1	2	yes	—
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	8	4	no	—
257	C₂	#2366×C₁₃₁	1	1	yes	characteristic
258	C₂	#3291×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2949×C₁₃₁	C₂	semidirect
3. #1977×C₁₃₁	C₂	semidirect
5. #1886×C₁₃₁	C₂	semidirect
6. #1772×C₁₃₁	C₂	semidirect
12. #492×C₁₃₁	C₄	semidirect
30. C₂×SD₁₆×C₁₃₁	C₂×C₄	semidirect
34. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
45. #107×C₁₃₁	D₄	semidirect
46. #106×C₁₃₁	D₄	semidirect
53. C₄×C₈×C₁₃₁	C₂×C₄	semidirect
129. #23198	C₁₃₁	direct (N × H)
130. C₁₃₁	#23198	direct (N × H)
132. #2949	C₂×C₁₃₁	semidirect
133. #1977	C₂×C₁₃₁	semidirect
135. #1886	C₂×C₁₃₁	semidirect
136. #1772	C₂×C₁₃₁	semidirect
142. #492	C₄×C₁₃₁	semidirect
160. C₂×SD₁₆	C₂×C₄×C₁₃₁	semidirect
164. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
175. #107	D₄×C₁₃₁	semidirect
176. #106	D₄×C₁₃₁	semidirect
183. C₄×C₈	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×#492×C₁₃₁	2
#2949×C₁₃₁	2
#1977×C₁₃₁	2
#1887×C₁₃₁	2
#1886×C₁₃₁	2
#1772×C₁₃₁	2
#23198	131

Subgroups — C131×#23198 #45709082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23198 #45709082