Subgroups — C₁₃₁×#29881 #45715765

Order 33536.

688 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2628×C₁₃₁	C₂	1	2	yes	maximal
4	#2088×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2077×C₁₃₁	C₂	1	4	yes	maximal
6	#1993×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1835×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
10	#498×C₁₃₁	—	2	4	no	—
11	#433×C₁₃₁	—	2	4	no	—
12	C₂×#106×C₁₃₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	#400×C₁₃₁	—	2	2	no	—
18	#399×C₁₃₁	—	2	2	no	—
19	#395×C₁₃₁	—	2	2	no	—
20	#393×C₁₃₁	—	2	2	no	—
21	#389×C₁₃₁	C₂²	1	4	yes	—
22	#382×C₁₃₁	C₂²	1	2	yes	—
23	#380×C₁₃₁	C₂²	1	4	yes	—
24	#340×C₁₃₁	C₂²	1	8	yes	—
25	#340×C₁₃₁	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃₁	—	2	2	no	—
27	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
32	#117×C₁₃₁	—	2	2	no	—
33	#117×C₁₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	#106×C₁₃₁	—	2	8	no	—
44	C₄⋊C₈×C₁₃₁	—	2	4	no	—
45	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
46	#97×C₁₃₁	—	2	2	no	—
47	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
48	#95×C₁₃₁	—	2	4	no	—
49	#95×C₁₃₁	—	2	2	no	—
50	#95×C₁₃₁	—	2	4	no	—
51	#95×C₁₃₁	—	2	2	no	—
52	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₃₁	—	4	4	no	—
54	C₂×Q₈×C₁₃₁	—	8	4	no	—
55	C₂²×C₄×C₁₃₁	—	2	1	no	—
56	C₂²×C₄×C₁₃₁	—	2	1	no	—
57	C₂²×C₄×C₁₃₁	—	2	4	no	—
58	C₂²×C₄×C₁₃₁	—	2	4	no	—
59	C₂²×C₄×C₁₃₁	—	2	4	no	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	—	2	4	no	—
65	C₂×C₈×C₁₃₁	—	2	2	no	—
66	C₂×C₈×C₁₃₁	—	2	4	no	—
67	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
68	C₄⋊C₄×C₁₃₁	—	2	4	no	—
69	C₄⋊C₄×C₁₃₁	—	2	4	no	—
70	C₄⋊C₄×C₁₃₁	—	2	4	no	—
71	C₄⋊C₄×C₁₃₁	—	2	8	no	—
72	C₄⋊C₄×C₁₃₁	—	4	4	no	—
73	C₄⋊C₄×C₁₃₁	—	8	4	no	—
74	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
75	C₂²⋊C₄×C₁₃₁	—	8	4	no	—
76	C₄²×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	—	2	1	no	—
78	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃₁	#41	1	2	yes	—
80	C₂³×C₁₃₁	—	2	1	no	—
81	C₂³×C₁₃₁	—	2	2	no	—
82	C₂³×C₁₃₁	—	2	4	no	—
83	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₁₃₁	—	8	8	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	2	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	—	2	4	no	—
90	C₂×C₄×C₁₃₁	—	4	2	no	—
91	C₂×C₄×C₁₃₁	—	4	2	no	—
92	C₂×C₄×C₁₃₁	—	4	4	no	—
93	C₂×C₄×C₁₃₁	—	4	4	no	—
94	C₂×C₄×C₁₃₁	—	4	4	no	—
95	C₂×C₄×C₁₃₁	—	4	4	no	—
96	C₂×C₄×C₁₃₁	—	4	4	no	—
97	C₂×C₄×C₁₃₁	—	4	4	no	—
98	C₂×C₄×C₁₃₁	#121	1	4	yes	—
99	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
102	C₈×C₁₃₁	—	2	4	no	—
103	C₈×C₁₃₁	—	4	2	no	—
104	C₂²×C₁₃₁	—	4	1	no	—
105	C₂²×C₁₃₁	—	4	4	no	—
106	C₂²×C₁₃₁	—	4	2	no	—
107	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
108	C₂²×C₁₃₁	#536	1	2	yes	—
109	C₂²×C₁₃₁	#577	1	4	yes	—
110	C₄×C₁₃₁	—	2	4	no	—
111	C₄×C₁₃₁	—	4	1	no	—
112	C₄×C₁₃₁	—	4	2	no	—
113	C₄×C₁₃₁	—	8	4	no	—
114	C₄×C₁₃₁	—	8	4	no	—
115	C₄×C₁₃₁	#536	1	2	yes	—
116	C₄×C₁₃₁	#541	1	2	yes	—
117	C₂×C₁₃₁	—	8	1	no	—
118	C₂×C₁₃₁	#2621	1	1	yes	characteristic
119	C₂×C₁₃₁	#3057	1	2	yes	—
120	C₂×C₁₃₁	#3244	1	4	yes	—
121	#29881	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₁	#29881	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#498	C₂×C₁₃₁	1	1	yes	characteristic, complemented
124	C₂×#433	C₂×C₁₃₁	1	1	yes	characteristic
125	#2628	C₂×C₁₃₁	1	2	yes	—
126	#2088	C₂×C₁₃₁	1	4	yes	complemented
127	#2077	C₂×C₁₃₁	1	4	yes	—
128	#1993	C₂×C₁₃₁	1	2	yes	complemented
129	#1835	C₂×C₁₃₁	1	1	yes	characteristic, complemented
130	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
131	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
132	#498	—	2	4	no	—
133	#433	—	2	4	no	—
134	C₂×#106	C₂²×C₁₃₁	1	4	yes	—
135	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
136	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
137	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	#400	—	2	2	no	—
140	#399	—	2	2	no	—
141	#395	—	2	2	no	—
142	#393	—	2	2	no	—
143	#389	C₂²×C₁₃₁	1	4	yes	—
144	#382	C₂²×C₁₃₁	1	2	yes	—
145	#380	C₂²×C₁₃₁	1	4	yes	—
146	#340	C₂²×C₁₃₁	1	4	yes	—
147	#340	C₂²×C₁₃₁	1	8	yes	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
151	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	—	2	4	no	—
154	#117	—	2	2	no	—
155	#117	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
161	C₂×C₂²⋊C₄	—	2	4	no	—
162	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
165	#106	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₂²⋊C₈	—	2	4	no	—
168	#97	—	2	2	no	—
169	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
170	#95	—	2	4	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	#95	—	2	4	no	—
174	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	8	4	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
185	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	8	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₂²⋊C₄	—	8	4	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
201	C₄²	#41×C₁₃₁	1	2	yes	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
206	Q₈	—	8	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	#121×C₁₃₁	1	4	yes	—
221	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
223	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
224	C₈	—	2	4	no	—
225	C₈	—	4	2	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
230	C₂²	#536×C₁₃₁	1	2	yes	—
231	C₂²	#577×C₁₃₁	1	4	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	8	4	no	—
237	C₄	#536×C₁₃₁	1	2	yes	—
238	C₄	#541×C₁₃₁	1	2	yes	—
239	C₂	—	8	1	no	—
240	C₂	#2621×C₁₃₁	1	1	yes	characteristic
241	C₂	#3057×C₁₃₁	1	2	yes	—
242	C₂	#3244×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×#117×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#498×C₁₃₁	C₂	semidirect
4. #2088×C₁₃₁	C₂	semidirect
6. #1993×C₁₃₁	C₂	semidirect
7. #1835×C₁₃₁	C₂	semidirect
121. #29881	C₁₃₁	direct (N × H)
122. C₁₃₁	#29881	direct (N × H)
123. C₂×#498	C₂×C₁₃₁	semidirect
126. #2088	C₂×C₁₃₁	semidirect
128. #1993	C₂×C₁₃₁	semidirect
129. #1835	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498×C₁₃₁	2
C₂×#433×C₁₃₁	2
#2628×C₁₃₁	2
#2088×C₁₃₁	2
#2077×C₁₃₁	2
#1993×C₁₃₁	2
#1835×C₁₃₁	2
#29881	131

Subgroups — C131×#29881 #45715765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29881 #45715765