Subgroups — C₃₁×#33942 #21490701

Order 7936.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3396	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2224	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2207	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	—	2	1	no	—
13	Q₈²×C₃₁	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#477	—	2	1	no	—
21	C₃₁×#477	—	2	2	no	—
22	C₃₁×#463	—	2	2	no	—
23	C₃₁×#460	—	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	1	yes	characteristic
28	C₄×D₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#366	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	1	yes	characteristic
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#357	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	4	1	no	—
49	C₃₁×C₄⋊Q₈	—	4	2	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×#124	—	4	1	no	—
53	C₃₁×#123	—	4	2	no	—
54	C₄×Q₈×C₃₁	—	2	1	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	1	no	—
57	C₄×Q₈×C₃₁	—	4	1	no	—
58	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₃₁×Q₃₂	—	2	2	no	—
62	C₃₁×SD₃₂	—	2	4	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#107	—	2	1	no	—
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#103	—	2	2	no	—
75	C₃₁×#103	—	2	1	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×Q₈×C₃₁	—	2	1	no	—
81	C₂×Q₈×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	—	4	1	no	—
83	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	Q₁₆×C₃₁	—	2	1	no	—
89	Q₁₆×C₃₁	—	2	1	no	—
90	Q₁₆×C₃₁	—	2	1	no	—
91	Q₁₆×C₃₁	#41	1	2	yes	—
92	SD₁₆×C₃₁	—	8	2	no	—
93	D₈×C₃₁	—	2	2	no	—
94	D₈×C₃₁	—	2	2	no	—
95	D₈×C₃₁	—	8	1	no	—
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	2	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	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₄²×C₃₁	—	4	1	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂×D₄	1	2	yes	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₃₁	—	2	2	no	—
118	C₁₆×C₃₁	—	2	1	no	—
119	C₁₆×C₃₁	—	2	1	no	—
120	#33942	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	2	2	no	—
123	Q₈×C₃₁	—	2	2	no	—
124	Q₈×C₃₁	—	4	1	no	—
125	Q₈×C₃₁	—	4	1	no	—
126	Q₈×C₃₁	—	4	1	no	—
127	Q₈×C₃₁	—	16	2	no	—
128	Q₈×C₃₁	—	16	1	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	16	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	16	2	no	—
139	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
147	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
148	#3396	C₂×C₃₁	1	1	yes	characteristic
149	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2243	C₂×C₃₁	1	2	yes	complemented
151	#2240	C₂×C₃₁	1	1	yes	characteristic
152	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2224	C₂×C₃₁	1	2	yes	complemented
154	#2211	C₂×C₃₁	1	2	yes	complemented
155	#2207	C₂×C₃₁	1	1	yes	characteristic
156	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2175	C₂×C₃₁	1	2	yes	complemented
158	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	2	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	1	no	—
169	C₄×C₃₁	—	8	1	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	#510	1	2	yes	—
172	C₄×C₃₁	#548	1	1	yes	characteristic
173	C₄×C₃₁	#577	1	1	yes	characteristic
174	#559	—	2	1	no	—
175	Q₈²	—	2	1	no	—
176	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
178	#500	C₂²×C₃₁	1	2	yes	—
179	#500	C₂²×C₃₁	1	1	yes	characteristic
180	#495	C₂²×C₃₁	1	2	yes	—
181	#493	C₂²×C₃₁	1	1	yes	characteristic
182	#477	—	2	1	no	—
183	#477	—	2	2	no	—
184	#463	—	2	2	no	—
185	#460	—	2	1	no	—
186	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
187	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
188	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
189	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₄×D₈	C₂²×C₃₁	1	2	yes	—
191	#366	C₂²×C₃₁	1	1	yes	characteristic
192	#366	C₂²×C₃₁	1	2	yes	—
193	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
195	#358	C₂²×C₃₁	1	1	yes	characteristic
196	#358	C₂²×C₃₁	1	1	yes	characteristic
197	#358	C₂²×C₃₁	1	2	yes	—
198	#357	C₂²×C₃₁	1	2	yes	—
199	#357	C₂²×C₃₁	1	2	yes	—
200	#357	C₂²×C₃₁	1	2	yes	—
201	#346	C₂²×C₃₁	1	2	yes	—
202	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×C₃₁	—	16	2	no	—
204	C₂×C₃₁	#3358	1	1	yes	characteristic
205	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
206	C₂×C₃₁	#3415	1	1	yes	characteristic
207	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂³×C₃₁	1	2	yes	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	4	1	no	—
215	C₄⋊Q₈	—	4	2	no	—
216	#127	—	2	1	no	—
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	4	1	no	—
227	Q₃₂	—	2	2	no	—
228	SD₃₂	—	2	4	no	—
229	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	C₂³×C₃₁	1	2	yes	—
235	#107	C₂³×C₃₁	1	1	yes	characteristic
236	#107	C₂³×C₃₁	1	1	yes	characteristic
237	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
238	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
239	#103	—	2	1	no	—
240	#103	—	2	2	no	—
241	#103	—	2	1	no	—
242	#102	—	2	2	no	—
243	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₃₁	1	2	yes	—
245	C₃₁	#33942	1	1	yes	characteristic, direct product factor, Sylow
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	Q₁₆	#41×C₃₁	1	2	yes	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	8	1	no	—
263	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
275	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×D₄×C₃₁	1	2	yes	—
283	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₂³	—	4	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	16	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	16	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
313	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	C₃₁×#510	1	2	yes	—
327	C₄	C₃₁×#548	1	1	yes	characteristic
328	C₄	C₃₁×#577	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	C₃₁×#3358	1	1	yes	characteristic
331	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
332	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#493	C₂²
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. Q₈×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2243	C₂	semidirect
5. C₃₁×#2227	C₂	semidirect
6. C₃₁×#2224	C₂	semidirect
7. C₃₁×#2211	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
10. C₃₁×#2175	C₂	semidirect
11. C₄×C₃₁×Q₃₂	C₂	semidirect
120. #33942	C₃₁	direct (N × H)
149. Q₈×Q₁₆	C₂×C₃₁	semidirect
150. #2243	C₂×C₃₁	semidirect
152. #2227	C₂×C₃₁	semidirect
153. #2224	C₂×C₃₁	semidirect
154. #2211	C₂×C₃₁	semidirect
156. #2183	C₂×C₃₁	semidirect
157. #2175	C₂×C₃₁	semidirect
158. C₄×Q₃₂	C₂×C₃₁	semidirect
245. C₃₁	#33942	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×#3396	2
Q₈×Q₁₆×C₃₁	2
C₃₁×#2243	2
C₃₁×#2240	2
C₃₁×#2227	2
C₃₁×#2224	2
C₃₁×#2211	2
C₃₁×#2207	2
C₃₁×#2183	2
C₃₁×#2175	2
C₄×C₃₁×Q₃₂	2
#33942	31

Subgroups — C31×#33942 #21490701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33942 #21490701