Subgroups — C₃₁×#33923 #21490682

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3342	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3338	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2235	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2219	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#2159	C₂	1	2	yes	maximal, complemented
12	C₃₁×#556	—	2	1	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#485	—	2	2	no	—
19	C₃₁×#484	—	2	2	no	—
20	C₃₁×#477	—	2	1	no	—
21	C₃₁×#475	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#467	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#427	C₂²	1	2	yes	—
28	C₂×C₃₁×#107	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×#367	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	1	yes	characteristic
33	C₃₁×#366	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	1	yes	characteristic
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#358	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	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	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×#126	—	4	2	no	—
47	C₃₁×#125	—	4	1	no	—
48	C₃₁×#124	—	4	1	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	2	1	no	—
54	C₄×Q₈×C₃₁	—	4	1	no	—
55	C₄×D₄×C₃₁	—	4	1	no	—
56	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₃₁×Q₃₂	—	2	2	no	—
60	C₃₁×M₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	2	no	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#103	—	2	1	no	—
75	C₃₁×#103	—	2	2	no	—
76	C₃₁×#103	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	D₄	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	1	no	—
82	C₂×Q₈×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	—	4	2	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
86	Q₁₆×C₃₁	—	2	1	no	—
87	Q₁₆×C₃₁	—	2	1	no	—
88	Q₁₆×C₃₁	—	2	1	no	—
89	Q₁₆×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	1	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	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
111	C₄²×C₃₁	—	4	2	no	—
112	C₄²×C₃₁	—	4	1	no	—
113	C₄²×C₃₁	—	4	1	no	—
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₃₁	—	2	2	no	—
116	C₁₆×C₃₁	—	2	1	no	—
117	C₁₆×C₃₁	#41	1	2	yes	—
118	#33923	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	C₂×D₈	1	2	yes	—
120	Q₈×C₃₁	—	4	1	no	—
121	Q₈×C₃₁	—	4	1	no	—
122	Q₈×C₃₁	—	4	1	no	—
123	Q₈×C₃₁	—	4	1	no	—
124	Q₈×C₃₁	—	8	2	no	—
125	Q₈×C₃₁	—	16	1	no	—
126	D₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	1	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	16	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	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
143	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	#3342	C₂×C₃₁	1	1	yes	characteristic
145	#3338	C₂×C₃₁	1	1	yes	characteristic
146	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2237	C₂×C₃₁	1	2	yes	complemented
148	#2235	C₂×C₃₁	1	2	yes	complemented
149	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2225	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2219	C₂×C₃₁	1	2	yes	complemented
152	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2169	C₂×C₃₁	1	1	yes	characteristic
154	#2159	C₂×C₃₁	1	2	yes	complemented
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	8	1	no	—
162	C₄×C₃₁	—	8	1	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	1	no	—
165	C₄×C₃₁	—	8	1	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	#510	1	2	yes	—
168	C₄×C₃₁	#547	1	1	yes	characteristic
169	C₄×C₃₁	#578	1	1	yes	characteristic
170	#556	—	2	1	no	—
171	#554	—	2	1	no	—
172	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#500	C₂²×C₃₁	1	1	yes	characteristic
176	#485	—	2	2	no	—
177	#484	—	2	2	no	—
178	#477	—	2	1	no	—
179	#475	—	2	1	no	—
180	#467	C₂²×C₃₁	1	2	yes	—
181	#467	C₂²×C₃₁	1	2	yes	—
182	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
183	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
184	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
185	#427	C₂²×C₃₁	1	2	yes	—
186	C₂×#107	C₂²×C₃₁	1	2	yes	—
187	#367	C₂²×C₃₁	1	2	yes	—
188	#367	C₂²×C₃₁	1	2	yes	—
189	#367	C₂²×C₃₁	1	2	yes	—
190	#366	C₂²×C₃₁	1	1	yes	characteristic
191	#366	C₂²×C₃₁	1	2	yes	—
192	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	#358	C₂²×C₃₁	1	1	yes	characteristic
194	#358	C₂²×C₃₁	1	1	yes	characteristic
195	#358	C₂²×C₃₁	1	2	yes	—
196	#358	C₂²×C₃₁	1	2	yes	—
197	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
198	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
199	C₂×C₃₁	—	4	2	no	—
200	C₂×C₃₁	#3334	1	1	yes	characteristic
201	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
202	C₂×C₃₁	#3416	1	1	yes	characteristic
203	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	#126	—	4	2	no	—
209	#125	—	4	1	no	—
210	#124	—	4	1	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	4	2	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×D₄	—	4	1	no	—
218	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
219	#117	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	Q₃₂	—	2	2	no	—
222	M₃₂	—	2	4	no	—
223	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
225	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	C₂³×C₃₁	1	1	yes	characteristic
230	#107	C₂³×C₃₁	1	1	yes	characteristic
231	#107	C₂³×C₃₁	1	2	yes	—
232	#107	C₂³×C₃₁	1	2	yes	—
233	#106	—	2	2	no	—
234	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
235	#103	—	2	1	no	—
236	#103	—	2	1	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₃₁	#33923	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
257	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₃₁	1	2	yes	—
281	C₂³	C₂×D₈×C₃₁	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	1	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	C₃₁×#510	1	2	yes	—
319	C₄	C₃₁×#547	1	1	yes	characteristic
320	C₄	C₃₁×#578	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₃₁×#3334	1	1	yes	characteristic
323	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
324	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2235	C₂	semidirect
6. C₃₁×#2227	C₂	semidirect
7. C₃₁×#2225	C₂	semidirect
8. C₃₁×#2219	C₂	semidirect
9. C₄×C₃₁×Q₃₂	C₂	semidirect
11. C₃₁×#2159	C₂	semidirect
118. #33923	C₃₁	direct (N × H)
146. #2254	C₂×C₃₁	semidirect
147. #2237	C₂×C₃₁	semidirect
148. #2235	C₂×C₃₁	semidirect
149. #2227	C₂×C₃₁	semidirect
150. #2225	C₂×C₃₁	semidirect
151. #2219	C₂×C₃₁	semidirect
152. C₄×Q₃₂	C₂×C₃₁	semidirect
154. #2159	C₂×C₃₁	semidirect
241. C₃₁	#33923	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×#3342	2
C₃₁×#3338	2
C₃₁×#2254	2
C₃₁×#2237	2
C₃₁×#2235	2
C₃₁×#2227	2
C₃₁×#2225	2
C₃₁×#2219	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2169	2
C₃₁×#2159	2
#33923	31

Subgroups — C31×#33923 #21490682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33923 #21490682