Subgroups — C₃₁×#2735 #16842668

Order 3968.

552 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	C₃₁×#399	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#397	C₂	1	2	yes	maximal, complemented
3	C₃₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#396	C₂	1	2	yes	maximal, complemented
5	C₃₁×#388	C₂	1	2	yes	maximal, complemented
6	C₃₁×#388	C₂	1	2	yes	maximal, complemented
7	C₃₁×#388	C₂	1	2	yes	maximal, complemented
8	C₃₁×#379	C₂	1	2	yes	maximal
9	C₃₁×#379	C₂	1	1	yes	maximal, characteristic
10	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂²	1	2	yes	complemented
26	C₃₁×#95	C₂²	1	2	yes	complemented
27	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#95	C₂²	1	2	yes	complemented
29	C₃₁×#95	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#95	C₂²	1	2	yes	complemented
31	C₂⁴×C₃₁	—	2	2	no	—
32	C₂⁴×C₃₁	—	2	4	no	—
33	C₂⁴×C₃₁	—	4	2	no	—
34	C₂⁴×C₃₁	—	4	1	no	—
35	C₂⁴×C₃₁	—	4	1	no	—
36	C₂⁴×C₃₁	C₂³	1	2	yes	—
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₃₁	—	4	1	no	—
46	C₄⋊C₄×C₃₁	—	4	2	no	—
47	C₄⋊C₄×C₃₁	—	4	2	no	—
48	C₄⋊C₄×C₃₁	—	4	1	no	—
49	C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂²⋊C₄×C₃₁	—	2	4	no	—
51	C₂²⋊C₄×C₃₁	—	2	4	no	—
52	C₂²⋊C₄×C₃₁	—	4	2	no	—
53	C₂²⋊C₄×C₃₁	—	4	2	no	—
54	C₂²⋊C₄×C₃₁	—	4	2	no	—
55	C₂²⋊C₄×C₃₁	—	4	2	no	—
56	C₂²⋊C₄×C₃₁	—	4	2	no	—
57	C₂²⋊C₄×C₃₁	—	4	1	no	—
58	C₂²⋊C₄×C₃₁	—	4	1	no	—
59	C₄²×C₃₁	—	4	2	no	—
60	C₂³×C₃₁	—	2	2	no	—
61	C₂³×C₃₁	—	2	2	no	—
62	C₂³×C₃₁	—	2	2	no	—
63	C₂³×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	—	2	1	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	1	no	—
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	2	2	no	—
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	4	8	no	—
71	C₂³×C₃₁	—	4	4	no	—
72	C₂³×C₃₁	—	4	8	no	—
73	C₂³×C₃₁	—	4	4	no	—
74	C₂³×C₃₁	—	4	4	no	—
75	C₂³×C₃₁	#41	1	4	yes	—
76	C₂³×C₃₁	#41	1	2	yes	—
77	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₃₁	—	2	2	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	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	#2735	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₃₁	—	2	4	no	—
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	—	2	4	no	—
101	C₂²×C₃₁	—	2	4	no	—
102	C₂²×C₃₁	—	2	4	no	—
103	C₂²×C₃₁	—	4	16	no	—
104	C₂²×C₃₁	—	4	2	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	2	no	—
110	C₂²×C₃₁	—	4	1	no	—
111	C₂²×C₃₁	C₂×#41	1	2	yes	—
112	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
113	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	—	4	1	no	—
123	#399	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#397	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#397	C₂×C₃₁	1	2	yes	complemented
126	#396	C₂×C₃₁	1	2	yes	complemented
127	#388	C₂×C₃₁	1	2	yes	complemented
128	#388	C₂×C₃₁	1	2	yes	complemented
129	#388	C₂×C₃₁	1	2	yes	complemented
130	#379	C₂×C₃₁	1	1	yes	characteristic
131	#379	C₂×C₃₁	1	2	yes	—
132	C₂×C₃₁	—	4	4	no	—
133	C₂×C₃₁	—	4	2	no	—
134	C₂×C₃₁	#538	1	2	yes	—
135	C₂×C₃₁	#538	1	1	yes	characteristic
136	C₂×C₃₁	#551	1	2	yes	—
137	C₂×C₃₁	#551	1	1	yes	characteristic
138	C₂×C₃₁	#560	1	1	yes	characteristic
139	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
141	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
142	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
143	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
144	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₃₁	1	2	yes	complemented
155	#95	C₂²×C₃₁	1	2	yes	complemented
156	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
157	#95	C₂²×C₃₁	1	2	yes	complemented
158	#95	C₂²×C₃₁	1	2	yes	complemented
159	#95	C₂²×C₃₁	1	1	yes	characteristic, complemented
160	C₃₁	#2735	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	2	no	—
162	C₂⁴	—	2	4	no	—
163	C₂⁴	—	4	2	no	—
164	C₂⁴	—	4	1	no	—
165	C₂⁴	—	4	1	no	—
166	C₂⁴	C₂³×C₃₁	1	2	yes	—
167	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
172	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	4	1	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₄²	—	4	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	4	no	—
205	C₂³	#41×C₃₁	1	4	yes	—
206	C₂³	#41×C₃₁	1	2	yes	—
207	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×#41×C₃₁	1	2	yes	—
241	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
242	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
244	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	4	no	—
253	C₂	—	4	2	no	—
254	C₂	C₃₁×#538	1	2	yes	—
255	C₂	C₃₁×#538	1	1	yes	characteristic
256	C₂	C₃₁×#551	1	2	yes	—
257	C₂	C₃₁×#551	1	1	yes	characteristic
258	C₂	C₃₁×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂²
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
1. C₃₁×#399	C₂	semidirect
2. C₃₁×#397	C₂	semidirect
3. C₃₁×#397	C₂	semidirect
4. C₃₁×#396	C₂	semidirect
5. C₃₁×#388	C₂	semidirect
6. C₃₁×#388	C₂	semidirect
7. C₃₁×#388	C₂	semidirect
11. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
12. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
13. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
14. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
25. C₂×C₄²×C₃₁	C₂²	semidirect
26. C₃₁×#95	C₂²	semidirect
27. C₃₁×#95	C₂²	semidirect
28. C₃₁×#95	C₂²	semidirect
29. C₃₁×#95	C₂²	semidirect
30. C₃₁×#95	C₂²	semidirect
97. #2735	C₃₁	direct (N × H)
123. #399	C₂×C₃₁	semidirect
124. #397	C₂×C₃₁	semidirect
125. #397	C₂×C₃₁	semidirect
126. #396	C₂×C₃₁	semidirect
127. #388	C₂×C₃₁	semidirect
128. #388	C₂×C₃₁	semidirect
129. #388	C₂×C₃₁	semidirect
140. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
141. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
142. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
143. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
154. C₂×C₄²	C₂²×C₃₁	semidirect
155. #95	C₂²×C₃₁	semidirect
156. #95	C₂²×C₃₁	semidirect
157. #95	C₂²×C₃₁	semidirect
158. #95	C₂²×C₃₁	semidirect
159. #95	C₂²×C₃₁	semidirect
160. C₃₁	#2735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#399	2
C₃₁×#397	2
C₃₁×#396	2
C₃₁×#388	2
C₃₁×#379	2
#2735	31

Subgroups — C31×#2735 #16842668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2735 #16842668