Subgroups — C₃₁×#13205 #21469964

Order 7936.

522 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×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#358	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#357	C₂	1	2	yes	maximal
4	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×SD₁₆×C₃₁	—	2	1	no	—
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#124	—	2	1	no	—
10	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#495	—	2	2	no	—
12	C₃₁×#495	C₄	1	4	yes	complemented
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#384	—	2	1	no	—
17	C₃₁×#358	—	2	4	no	—
18	C₃₁×#357	—	2	4	no	—
19	C₃₁×#346	—	2	2	no	—
20	C₂²×Q₈×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	2	1	no	—
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	1	no	—
24	C₂×Q₁₆×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₃₁	—	2	4	no	—
27	C₂×SD₁₆×C₃₁	—	4	2	no	—
28	C₂×SD₁₆×C₃₁	—	4	2	no	—
29	C₂×D₈×C₃₁	—	2	2	no	—
30	C₂×D₈×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	—	2	2	no	—
32	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#124	—	2	4	no	—
36	C₃₁×#124	—	4	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	—	4	2	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₃₁	D₄	1	2	yes	—
49	C₂⁴×C₃₁	—	4	1	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	4	1	no	—
52	C₂×Q₈×C₃₁	—	4	1	no	—
53	C₂×Q₈×C₃₁	—	4	1	no	—
54	C₂×Q₈×C₃₁	—	4	1	no	—
55	C₂×Q₈×C₃₁	—	4	1	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	D₈	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₃₁	—	2	4	no	—
67	Q₁₆×C₃₁	—	4	2	no	—
68	SD₁₆×C₃₁	—	4	8	no	—
69	D₈×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₃₁	D₈	1	2	yes	—
73	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	#41	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	4	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	8	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	8	no	—
82	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₃₁	SD₁₆	1	2	yes	—
84	C₁₆×C₃₁	—	2	4	no	—
85	#13205	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	—	8	1	no	—
88	C₂³×C₃₁	—	8	1	no	—
89	C₂³×C₃₁	—	8	2	no	—
90	C₂³×C₃₁	—	8	2	no	—
91	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₃₁	—	4	2	no	—
93	Q₈×C₃₁	—	4	2	no	—
94	Q₈×C₃₁	—	8	1	no	—
95	Q₈×C₃₁	—	8	1	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	8	1	no	—
99	D₄×C₃₁	—	8	1	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	8	1	no	—
104	C₂×C₄×C₃₁	—	8	1	no	—
105	C₂×C₄×C₃₁	—	8	2	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	#102	1	4	yes	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
114	C₈×C₃₁	#102	1	4	yes	—
115	C₈×C₃₁	C₄×D₄	1	2	yes	—
116	C₈×C₃₁	#121	1	2	yes	—
117	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
118	C₂×#358	C₂×C₃₁	1	2	yes	complemented
119	C₂×#357	C₂×C₃₁	1	2	yes	—
120	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂²×C₃₁	—	8	4	no	—
123	C₂²×C₃₁	—	8	2	no	—
124	C₂²×C₃₁	—	16	2	no	—
125	C₂²×C₃₁	—	16	1	no	—
126	C₂²×C₃₁	—	16	1	no	—
127	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
128	C₂²×C₃₁	#486	1	1	yes	characteristic
129	C₂²×C₃₁	#493	1	1	yes	characteristic
130	C₂²×C₃₁	#509	1	2	yes	—
131	C₂²×C₃₁	#510	1	2	yes	—
132	C₄×C₃₁	—	2	4	no	—
133	C₄×C₃₁	—	8	2	no	—
134	C₄×C₃₁	—	16	2	no	—
135	C₄×C₃₁	#390	1	1	yes	characteristic
136	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
137	C₄×C₃₁	#486	1	1	yes	characteristic
138	C₄×C₃₁	#492	1	1	yes	characteristic
139	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
140	C₂²×SD₁₆	—	2	1	no	—
141	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×#124	—	2	1	no	—
143	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
144	#495	—	2	2	no	—
145	#495	C₄×C₃₁	1	4	yes	complemented
146	C₂×#107	—	2	1	no	—
147	C₂×#107	C₂²×C₃₁	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	#384	—	2	1	no	—
150	#358	—	2	4	no	—
151	#357	—	2	4	no	—
152	#346	—	2	2	no	—
153	C₂×C₃₁	—	16	2	no	—
154	C₂×C₃₁	#1957	1	1	yes	characteristic
155	C₂×C₃₁	#2139	1	2	yes	—
156	C₂×C₃₁	#2243	1	2	yes	—
157	C₂×C₃₁	#2250	1	1	yes	characteristic
158	C₂×C₃₁	#2251	1	1	yes	characteristic
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	—	2	1	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
165	C₂×SD₁₆	—	2	4	no	—
166	C₂×SD₁₆	—	4	2	no	—
167	C₂×SD₁₆	—	4	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
172	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
174	#124	—	2	4	no	—
175	#124	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	2	no	—
178	C₂×C₂²⋊C₄	—	4	2	no	—
179	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
183	#107	—	2	2	no	—
184	#107	—	2	2	no	—
185	#107	—	4	2	no	—
186	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
187	C₄×C₈	D₄×C₃₁	1	2	yes	—
188	C₃₁	#13205	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	4	1	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	D₈×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	Q₁₆	—	2	4	no	—
207	Q₁₆	—	4	2	no	—
208	SD₁₆	—	4	8	no	—
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂×C₈	D₈×C₃₁	1	2	yes	—
213	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	#41×C₃₁	1	2	yes	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	8	4	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₄²	SD₁₆×C₃₁	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	1	no	—
227	C₂³	—	8	1	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	8	1	no	—
234	Q₈	—	8	1	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₃₁×#102	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
253	C₈	C₃₁×#102	1	4	yes	—
254	C₈	C₄×D₄×C₃₁	1	2	yes	—
255	C₈	C₃₁×#121	1	2	yes	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	1	no	—
260	C₂²	—	16	1	no	—
261	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
262	C₂²	C₃₁×#486	1	1	yes	characteristic
263	C₂²	C₃₁×#493	1	1	yes	characteristic
264	C₂²	C₃₁×#509	1	2	yes	—
265	C₂²	C₃₁×#510	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	16	2	no	—
269	C₄	C₃₁×#390	1	1	yes	characteristic
270	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
271	C₄	C₃₁×#486	1	1	yes	characteristic
272	C₄	C₃₁×#492	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	C₃₁×#1957	1	1	yes	characteristic
275	C₂	C₃₁×#2139	1	2	yes	—
276	C₂	C₃₁×#2243	1	2	yes	—
277	C₂	C₃₁×#2250	1	1	yes	characteristic
278	C₂	C₃₁×#2251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#358	C₂	semidirect
4. C₂×C₃₁×#346	C₂	semidirect
5. C₃₁×#1845	C₂	semidirect
12. C₃₁×#495	C₄	semidirect
85. #13205	C₃₁	direct (N × H)
118. C₂×#358	C₂×C₃₁	semidirect
120. C₂×#346	C₂×C₃₁	semidirect
121. #1845	C₂×C₃₁	semidirect
145. #495	C₄×C₃₁	semidirect
188. C₃₁	#13205	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#495	2
C₂×C₃₁×#358	2
C₂×C₃₁×#357	2
C₂×C₃₁×#346	2
C₃₁×#1845	2
#13205	31

Subgroups — C31×#13205 #21469964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13205 #21469964