Subgroups — C₃₁×#2506 #16842439

Order 3968.

436 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#125	C₂	1	2	yes	maximal
2	C₃₁×#400	C₂	1	2	yes	maximal
3	C₃₁×#398	C₂	1	2	yes	maximal
4	C₃₁×#389	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#389	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#389	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#389	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#387	C₂	1	2	yes	maximal
9	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#125	—	2	8	no	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#95	C₂²	1	2	yes	—
30	C₃₁×#95	C₂²	1	2	yes	—
31	C₃₁×#95	C₂²	1	2	yes	—
32	C₃₁×#95	C₂²	1	1	yes	characteristic
33	C₃₁×#95	C₂²	1	2	yes	—
34	C₃₁×#95	C₂²	1	2	yes	—
35	C₃₁×#95	C₂²	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₃₁	—	2	4	no	—
48	C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₄⋊C₄×C₃₁	—	2	4	no	—
52	C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₄⋊C₄×C₃₁	—	2	4	no	—
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	D₄	1	4	yes	—
58	C₄⋊C₄×C₃₁	Q₈	1	4	yes	—
59	C₄²×C₃₁	—	2	4	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	—	2	2	no	—
62	C₄²×C₃₁	—	2	2	no	—
63	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	—	2	1	no	—
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	—	2	1	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
93	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₃₁	#41	1	2	yes	—
95	C₂×C₄×C₃₁	#41	1	2	yes	—
96	C₂×C₄×C₃₁	#41	1	2	yes	—
97	#2506	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
103	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
105	C₄×C₃₁	—	2	2	no	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	4	no	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	4	2	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	1	no	—
113	C₄×C₃₁	—	4	2	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	#125	1	4	yes	complemented
116	C₂×#125	C₂×C₃₁	1	2	yes	—
117	#400	C₂×C₃₁	1	2	yes	—
118	#398	C₂×C₃₁	1	2	yes	—
119	#389	C₂×C₃₁	1	1	yes	characteristic
120	#389	C₂×C₃₁	1	1	yes	characteristic
121	#389	C₂×C₃₁	1	1	yes	characteristic
122	#389	C₂×C₃₁	1	1	yes	characteristic
123	#387	C₂×C₃₁	1	2	yes	—
124	#384	C₂×C₃₁	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
126	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
127	C₂×C₃₁	C₂×#125	1	1	yes	characteristic
128	C₂×C₃₁	#532	1	1	yes	characteristic
129	C₂×C₃₁	#541	1	1	yes	characteristic
130	C₂×C₃₁	#547	1	1	yes	characteristic
131	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
132	C₂×C₃₁	#552	1	1	yes	characteristic
133	C₂×C₃₁	#557	1	1	yes	characteristic
134	#125	—	2	8	no	—
135	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
148	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
151	#95	C₂²×C₃₁	1	1	yes	characteristic
152	#95	C₂²×C₃₁	1	2	yes	—
153	#95	C₂²×C₃₁	1	1	yes	characteristic
154	#95	C₂²×C₃₁	1	2	yes	—
155	#95	C₂²×C₃₁	1	2	yes	—
156	#95	C₂²×C₃₁	1	2	yes	—
157	#95	C₂²×C₃₁	1	2	yes	—
158	C₃₁	#2506	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
164	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	D₄×C₃₁	1	4	yes	—
181	C₄⋊C₄	Q₈×C₃₁	1	4	yes	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂×C₄	C₂×Q₈×C₃₁	1	4	yes	—
216	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
217	C₂×C₄	#41×C₃₁	1	2	yes	—
218	C₂×C₄	#41×C₃₁	1	2	yes	—
219	C₂×C₄	#41×C₃₁	1	2	yes	—
220	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
221	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
222	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
223	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
224	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
225	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
226	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	C₃₁×#125	1	4	yes	complemented
238	C₂	C₂×C₃₁×#125	1	1	yes	characteristic
239	C₂	C₃₁×#532	1	1	yes	characteristic
240	C₂	C₃₁×#541	1	1	yes	characteristic
241	C₂	C₃₁×#547	1	1	yes	characteristic
242	C₂	D₄×Q₈×C₃₁	1	1	yes	characteristic
243	C₂	C₃₁×#552	1	1	yes	characteristic
244	C₂	C₃₁×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
97. #2506	C₃₁	direct (N × H)
115. C₄×C₃₁	#125	semidirect
158. C₃₁	#2506	direct (N × H)
237. C₄	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#125	2
C₃₁×#400	2
C₃₁×#398	2
C₃₁×#389	2
C₃₁×#387	2
C₃₁×#384	2
C₄×C₄⋊C₄×C₃₁	2
#2506	31

Subgroups — C31×#2506 #16842439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2506 #16842439