Subgroups — C₃₁×#33686 #21490445

Order 7936.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3176	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2207	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2185	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#540	—	2	1	no	—
12	C₃₁×#519	—	2	1	no	—
13	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₃₁×#499	C₂²	1	1	yes	characteristic
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#490	—	2	1	no	—
18	C₃₁×#469	—	2	1	no	—
19	C₃₁×#466	C₂²	1	2	yes	—
20	C₃₁×#442	—	2	1	no	—
21	C₃₁×#441	—	2	1	no	—
22	C₃₁×#441	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#419	—	2	1	no	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₃₁×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×Q₁₆×C₃₁	—	2	1	no	—
42	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#125	—	2	1	no	—
49	C₃₁×#123	—	4	2	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	4	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₃₁×#117	—	2	1	no	—
57	C₃₁×#117	—	4	2	no	—
58	C₃₁×#117	D₄	1	1	yes	characteristic
59	C₃₁×SD₃₂	—	2	4	no	—
60	C₃₁×SD₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#106	—	2	1	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#97	D₄	1	1	yes	characteristic
72	C₃₁×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	2	no	—
78	C₂×Q₈×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	—	4	1	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₃₁	—	2	4	no	—
86	Q₁₆×C₃₁	—	2	2	no	—
87	Q₁₆×C₃₁	#41	1	4	yes	—
88	D₈×C₃₁	—	2	2	no	—
89	D₈×C₃₁	—	2	2	no	—
90	D₈×C₃₁	—	4	2	no	—
91	M₁₆×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
107	C₄²×C₃₁	—	4	1	no	—
108	C₄²×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	—	4	2	no	—
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#33686	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	2	no	—
116	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₃₁	—	2	4	no	—
118	Q₈×C₃₁	—	4	4	no	—
119	Q₈×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	16	2	no	—
123	D₄×C₃₁	—	16	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	16	2	no	—
131	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	C₂×#41	1	2	yes	—
139	#3176	C₂×C₃₁	1	1	yes	characteristic
140	#2950	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#2234	C₂×C₃₁	1	2	yes	complemented
142	#2229	C₂×C₃₁	1	2	yes	complemented
143	#2207	C₂×C₃₁	1	2	yes	complemented
144	#2185	C₂×C₃₁	1	2	yes	complemented
145	#2175	C₂×C₃₁	1	2	yes	complemented
146	C₄×SD₃₂	C₂×C₃₁	1	2	yes	complemented
147	#2164	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	#525	1	1	yes	characteristic
161	C₄×C₃₁	#575	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	#540	—	2	1	no	—
164	#519	—	2	1	no	—
165	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
166	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
167	#499	C₂²×C₃₁	1	1	yes	characteristic
168	#493	C₂²×C₃₁	1	1	yes	characteristic
169	#490	—	2	1	no	—
170	#469	—	2	1	no	—
171	#466	C₂²×C₃₁	1	2	yes	—
172	#442	—	2	1	no	—
173	#441	—	2	1	no	—
174	#441	—	2	1	no	—
175	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
176	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
177	C₄×D₈	C₂²×C₃₁	1	2	yes	—
178	#427	C₂²×C₃₁	1	1	yes	characteristic
179	#419	—	2	1	no	—
180	#405	C₂²×C₃₁	1	1	yes	characteristic
181	#367	C₂²×C₃₁	1	2	yes	—
182	#366	C₂²×C₃₁	1	2	yes	—
183	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
184	#358	C₂²×C₃₁	1	2	yes	—
185	#358	C₂²×C₃₁	1	2	yes	—
186	#357	C₂²×C₃₁	1	2	yes	—
187	#357	C₂²×C₃₁	1	2	yes	—
188	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
189	#346	C₂²×C₃₁	1	1	yes	characteristic
190	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₃₁	—	4	1	no	—
192	C₂×C₃₁	—	16	2	no	—
193	C₂×C₃₁	#3100	1	1	yes	characteristic
194	C₂×C₃₁	#3416	1	2	yes	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	C₂³×C₃₁	1	2	yes	—
201	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	#127	—	2	1	no	—
204	#125	—	2	1	no	—
205	#123	—	4	2	no	—
206	#121	—	2	2	no	—
207	#121	—	4	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	2	no	—
211	C₄×D₄	—	2	2	no	—
212	#117	—	2	1	no	—
213	#117	—	4	2	no	—
214	#117	D₄×C₃₁	1	1	yes	characteristic
215	SD₃₂	—	2	4	no	—
216	SD₃₂	—	2	4	no	—
217	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
218	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
219	#107	C₂³×C₃₁	1	2	yes	—
220	#107	C₂³×C₃₁	1	2	yes	—
221	#106	—	2	1	no	—
222	#106	—	2	1	no	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	#97	D₄×C₃₁	1	1	yes	characteristic
228	#97	D₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₃₁	#33686	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	#41×C₃₁	1	4	yes	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	16	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
288	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
289	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₂×#41×C₃₁	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	C₃₁×#525	1	1	yes	characteristic
308	C₄	C₃₁×#575	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	16	2	no	—
311	C₂	C₃₁×#3100	1	1	yes	characteristic
312	C₂	C₃₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2950	C₂	semidirect
3. C₃₁×#2234	C₂	semidirect
4. C₃₁×#2229	C₂	semidirect
5. C₃₁×#2207	C₂	semidirect
6. C₃₁×#2185	C₂	semidirect
7. C₃₁×#2175	C₂	semidirect
8. C₄×C₃₁×SD₃₂	C₂	semidirect
9. C₃₁×#2164	C₂	semidirect
114. #33686	C₃₁	direct (N × H)
140. #2950	C₂×C₃₁	semidirect
141. #2234	C₂×C₃₁	semidirect
142. #2229	C₂×C₃₁	semidirect
143. #2207	C₂×C₃₁	semidirect
144. #2185	C₂×C₃₁	semidirect
145. #2175	C₂×C₃₁	semidirect
146. C₄×SD₃₂	C₂×C₃₁	semidirect
147. #2164	C₂×C₃₁	semidirect
231. C₃₁	#33686	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×#3176	2
C₃₁×#2950	2
C₃₁×#2234	2
C₃₁×#2229	2
C₃₁×#2207	2
C₃₁×#2185	2
C₃₁×#2175	2
C₄×C₃₁×SD₃₂	2
C₃₁×#2164	2
#33686	31

Subgroups — C31×#33686 #21490445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33686 #21490445