Subgroups — C₃₁×#33939 #21490698

Order 7936.

476 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3380	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2245	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2208	C₂	1	2	yes	maximal
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal
11	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#557	—	2	1	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#499	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	2	yes	—
20	C₃₁×#478	—	2	2	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#462	—	2	1	no	—
23	C₃₁×#462	—	2	2	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×D₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₃₁	C₂²	1	2	yes	—
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×#358	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	1	yes	characteristic
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×Q₁₆×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
45	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#125	—	4	1	no	—
51	C₃₁×#125	—	4	2	no	—
52	C₃₁×C₂²⋊Q₈	—	4	1	no	—
53	C₃₁×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	—	4	1	no	—
57	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
61	C₃₁×SD₃₂	—	2	2	no	—
62	C₃₁×SD₃₂	—	2	4	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	2	yes	—
69	C₃₁×#107	C₂³	1	1	yes	characteristic
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	1	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	1	no	—
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	1	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	C₂²×C₄×C₃₁	—	4	2	no	—
86	Q₁₆×C₃₁	—	2	1	no	—
87	Q₁₆×C₃₁	—	2	2	no	—
88	Q₁₆×C₃₁	—	2	1	no	—
89	Q₁₆×C₃₁	—	2	2	no	—
90	Q₁₆×C₃₁	—	8	2	no	—
91	Q₁₆×C₃₁	—	8	1	no	—
92	D₈×C₃₁	—	2	1	no	—
93	D₈×C₃₁	#41	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	4	2	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	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
111	C₂²⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	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	1	no	—
118	C₁₆×C₃₁	—	2	2	no	—
119	C₁₆×C₃₁	—	2	1	no	—
120	#33939	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	—	4	1	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	4	1	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	16	1	no	—
128	D₄×C₃₁	—	2	2	no	—
129	D₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	4	2	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	—	8	2	no	—
136	C₂×C₄×C₃₁	—	16	2	no	—
137	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
146	#3395	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#3380	C₂×C₃₁	1	1	yes	characteristic
148	#2245	C₂×C₃₁	1	2	yes	complemented
149	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2229	C₂×C₃₁	1	2	yes	complemented
151	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2208	C₂×C₃₁	1	2	yes	—
153	#2208	C₂×C₃₁	1	1	yes	characteristic
154	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2175	C₂×C₃₁	1	2	yes	—
156	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
157	C₂²×C₃₁	—	4	2	no	—
158	C₂²×C₃₁	—	8	1	no	—
159	C₂²×C₃₁	—	8	1	no	—
160	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	1	no	—
166	C₄×C₃₁	—	8	1	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	#510	1	2	yes	—
170	C₄×C₃₁	#548	1	1	yes	characteristic
171	C₄×C₃₁	#577	1	1	yes	characteristic
172	#557	—	2	1	no	—
173	#554	—	2	1	no	—
174	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
176	#500	C₂²×C₃₁	1	1	yes	characteristic
177	#499	C₂²×C₃₁	1	2	yes	—
178	#494	C₂²×C₃₁	1	1	yes	characteristic
179	#494	C₂²×C₃₁	1	2	yes	—
180	#478	—	2	2	no	—
181	#476	—	2	1	no	—
182	#462	—	2	1	no	—
183	#462	—	2	2	no	—
184	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
187	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
188	C₄×D₈	C₂²×C₃₁	1	2	yes	—
189	#367	C₂²×C₃₁	1	1	yes	characteristic
190	#366	C₂²×C₃₁	1	2	yes	—
191	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
192	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
193	#358	C₂²×C₃₁	1	1	yes	characteristic
194	#358	C₂²×C₃₁	1	2	yes	—
195	#358	C₂²×C₃₁	1	2	yes	—
196	#358	C₂²×C₃₁	1	2	yes	—
197	#358	C₂²×C₃₁	1	2	yes	—
198	#357	C₂²×C₃₁	1	1	yes	characteristic
199	#346	C₂²×C₃₁	1	2	yes	—
200	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₃₁	—	8	2	no	—
202	C₂×C₃₁	#3358	1	1	yes	characteristic
203	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
204	C₂×C₃₁	#3416	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
209	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#125	—	2	2	no	—
214	#125	—	4	1	no	—
215	#125	—	4	2	no	—
216	C₂²⋊Q₈	—	4	1	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	4	1	no	—
225	SD₃₂	—	2	2	no	—
226	SD₃₂	—	2	4	no	—
227	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	C₂³×C₃₁	1	1	yes	characteristic
232	#107	C₂³×C₃₁	1	2	yes	—
233	#107	C₂³×C₃₁	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
236	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	1	no	—
239	#102	—	2	2	no	—
240	#102	—	2	1	no	—
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	2	yes	—
243	C₃₁	#33939	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	2	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	8	2	no	—
256	Q₁₆	—	8	1	no	—
257	D₈	—	2	1	no	—
258	D₈	#41×C₃₁	1	2	yes	—
259	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
275	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₃₁	1	2	yes	—
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₂³	—	4	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	16	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	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	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
304	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
309	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	C₃₁×#510	1	2	yes	—
323	C₄	C₃₁×#548	1	1	yes	characteristic
324	C₄	C₃₁×#577	1	1	yes	characteristic
325	C₂	—	8	2	no	—
326	C₂	C₃₁×#3358	1	1	yes	characteristic
327	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
328	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×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
1. C₃₁×#3395	C₂	semidirect
3. C₃₁×#2245	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
6. C₃₁×#2227	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
120. #33939	C₃₁	direct (N × H)
146. #3395	C₂×C₃₁	semidirect
148. #2245	C₂×C₃₁	semidirect
149. #2241	C₂×C₃₁	semidirect
150. #2229	C₂×C₃₁	semidirect
151. #2227	C₂×C₃₁	semidirect
154. #2183	C₂×C₃₁	semidirect
243. C₃₁	#33939	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×#3395	2
C₃₁×#3380	2
C₃₁×#2245	2
C₃₁×#2241	2
C₃₁×#2229	2
C₃₁×#2227	2
C₃₁×#2208	2
C₃₁×#2183	2
C₃₁×#2175	2
C₄×C₃₁×SD₃₂	2
#33939	31

Subgroups — C31×#33939 #21490698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33939 #21490698