Subgroups — C₃₁×#33931 #21490690

Order 7936.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2242	C₂	1	2	yes	maximal
4	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2211	C₂	1	2	yes	maximal
8	C₃₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#557	—	2	1	no	—
13	C₃₁×#556	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#499	C₂²	1	2	yes	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#479	—	2	2	no	—
21	C₃₁×#475	—	2	1	no	—
22	C₃₁×#464	—	2	2	no	—
23	C₃₁×#461	—	2	1	no	—
24	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
28	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	1	yes	characteristic
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₃₁	—	2	1	no	—
42	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	4	1	no	—
49	C₃₁×#126	—	4	2	no	—
50	C₃₁×#125	—	2	2	no	—
51	C₃₁×#125	—	4	2	no	—
52	C₃₁×#124	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	4	1	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	4	1	no	—
58	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₃₁×#117	—	4	1	no	—
61	C₃₁×Q₃₂	—	2	4	no	—
62	C₃₁×SD₃₂	—	2	2	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#107	C₂³	1	2	yes	—
70	C₃₁×#106	—	2	2	no	—
71	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×#103	—	2	2	no	—
74	C₃₁×#103	—	2	2	no	—
75	C₃₁×#103	—	2	1	no	—
76	C₃₁×#102	—	2	1	no	—
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	Q₁₆×C₃₁	—	2	2	no	—
86	Q₁₆×C₃₁	—	2	1	no	—
87	Q₁₆×C₃₁	—	2	2	no	—
88	Q₁₆×C₃₁	—	8	1	no	—
89	Q₁₆×C₃₁	#41	1	2	yes	—
90	SD₁₆×C₃₁	—	8	2	no	—
91	D₈×C₃₁	—	2	1	no	—
92	D₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	2	yes	—
117	C₁₆×C₃₁	—	2	1	no	—
118	C₁₆×C₃₁	—	2	2	no	—
119	C₁₆×C₃₁	—	2	1	no	—
120	#33931	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	1	no	—
122	Q₈×C₃₁	—	2	2	no	—
123	Q₈×C₃₁	—	2	2	no	—
124	Q₈×C₃₁	—	4	1	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	8	2	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	4	1	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	8	2	no	—
136	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
145	#3393	C₂×C₃₁	1	1	yes	characteristic
146	#3383	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2242	C₂×C₃₁	1	2	yes	—
148	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2229	C₂×C₃₁	1	2	yes	complemented
150	#2224	C₂×C₃₁	1	1	yes	characteristic
151	#2211	C₂×C₃₁	1	2	yes	—
152	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
153	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₄×Q₃₂	C₂×C₃₁	1	2	yes	complemented
155	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
156	C₂²×C₃₁	—	8	1	no	—
157	C₂²×C₃₁	—	8	1	no	—
158	C₂²×C₃₁	—	8	1	no	—
159	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	1	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	#508	1	2	yes	—
169	C₄×C₃₁	#548	1	1	yes	characteristic
170	C₄×C₃₁	#577	1	1	yes	characteristic
171	#557	—	2	1	no	—
172	#556	—	2	1	no	—
173	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
174	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
175	#500	C₂²×C₃₁	1	1	yes	characteristic
176	#499	C₂²×C₃₁	1	2	yes	—
177	#495	C₂²×C₃₁	1	2	yes	—
178	#494	C₂²×C₃₁	1	1	yes	characteristic
179	#479	—	2	2	no	—
180	#475	—	2	1	no	—
181	#464	—	2	2	no	—
182	#461	—	2	1	no	—
183	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
186	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
187	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
188	#367	C₂²×C₃₁	1	1	yes	characteristic
189	#366	C₂²×C₃₁	1	2	yes	—
190	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
192	#358	C₂²×C₃₁	1	1	yes	characteristic
193	#358	C₂²×C₃₁	1	2	yes	—
194	#358	C₂²×C₃₁	1	2	yes	—
195	#358	C₂²×C₃₁	1	2	yes	—
196	#357	C₂²×C₃₁	1	1	yes	characteristic
197	#357	C₂²×C₃₁	1	2	yes	—
198	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
199	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
200	C₂×C₃₁	—	16	1	no	—
201	C₂×C₃₁	#3358	1	1	yes	characteristic
202	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
203	C₂×C₃₁	#3417	1	1	yes	characteristic
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
206	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	4	1	no	—
212	#126	—	4	2	no	—
213	#125	—	2	2	no	—
214	#125	—	4	2	no	—
215	#124	—	2	2	no	—
216	C₂²⋊Q₈	—	4	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×D₄	—	2	1	no	—
223	#117	—	4	1	no	—
224	Q₃₂	—	2	4	no	—
225	SD₃₂	—	2	2	no	—
226	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
227	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
228	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
229	#107	—	2	1	no	—
230	#107	C₂³×C₃₁	1	2	yes	—
231	#107	C₂³×C₃₁	1	1	yes	characteristic
232	#107	C₂³×C₃₁	1	1	yes	characteristic
233	#106	—	2	2	no	—
234	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
235	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	1	no	—
239	#102	—	2	1	no	—
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₃₁	1	2	yes	—
242	C₃₁	#33931	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	8	1	no	—
253	Q₁₆	#41×C₃₁	1	2	yes	—
254	SD₁₆	—	8	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
259	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
272	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₃₁	1	2	yes	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₂³	—	4	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
307	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	C₃₁×#508	1	2	yes	—
321	C₄	C₃₁×#548	1	1	yes	characteristic
322	C₄	C₃₁×#577	1	1	yes	characteristic
323	C₂	—	16	1	no	—
324	C₂	C₃₁×#3358	1	1	yes	characteristic
325	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
326	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3383	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
8. C₃₁×#2209	C₂	semidirect
9. Q₈×C₁₆×C₃₁	C₂	semidirect
10. C₄×C₃₁×Q₃₂	C₂	semidirect
120. #33931	C₃₁	direct (N × H)
146. #3383	C₂×C₃₁	semidirect
148. #2241	C₂×C₃₁	semidirect
149. #2229	C₂×C₃₁	semidirect
152. #2209	C₂×C₃₁	semidirect
153. Q₈×C₁₆	C₂×C₃₁	semidirect
154. C₄×Q₃₂	C₂×C₃₁	semidirect
242. C₃₁	#33931	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×#3393	2
C₃₁×#3383	2
C₃₁×#2242	2
C₃₁×#2241	2
C₃₁×#2229	2
C₃₁×#2224	2
C₃₁×#2211	2
C₃₁×#2209	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×Q₃₂	2
C₄×C₃₁×SD₃₂	2
#33931	31

Subgroups — C31×#33931 #21490690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33931 #21490690