Subgroups — C₃₁×#33927 #21490686

Order 7936.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 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, complemented
2	C₃₁×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2242	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×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2210	C₂	1	2	yes	maximal
8	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#556	—	2	1	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	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×#478	—	2	2	no	—
21	C₃₁×#476	—	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	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#357	C₂²	1	1	yes	characteristic
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×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
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×#126	—	4	2	no	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	1	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	D₄	1	1	yes	characteristic
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₃₁×#117	—	4	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
61	C₃₁×SD₃₂	—	2	4	no	—
62	C₃₁×D₁₆	—	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×#102	—	2	1	no	—
75	C₃₁×#102	—	2	2	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	2	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	Q₁₆×C₃₁	—	2	2	no	—
88	Q₁₆×C₃₁	—	2	2	no	—
89	Q₁₆×C₃₁	—	8	1	no	—
90	SD₁₆×C₃₁	—	8	2	no	—
91	D₈×C₃₁	—	2	1	no	—
92	D₈×C₃₁	—	2	1	no	—
93	D₈×C₃₁	—	2	1	no	—
94	D₈×C₃₁	#41	1	2	yes	—
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	—	4	2	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	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	1	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	2	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	#33927	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	1	no	—
122	C₂³×C₃₁	—	4	1	no	—
123	Q₈×C₃₁	—	2	2	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	16	1	no	—
127	D₄×C₃₁	—	2	2	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	4	1	no	—
130	D₄×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	—	8	2	no	—
137	C₂×C₄×C₃₁	—	16	2	no	—
138	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₃₁	—	2	2	no	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
146	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
147	#3393	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#3380	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2242	C₂×C₃₁	1	2	yes	complemented
150	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2229	C₂×C₃₁	1	2	yes	complemented
152	#2226	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2210	C₂×C₃₁	1	2	yes	—
154	#2208	C₂×C₃₁	1	1	yes	characteristic, complemented
155	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₄×SD₃₂	C₂×C₃₁	1	2	yes	complemented
157	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic
158	C₂²×C₃₁	—	4	2	no	—
159	C₂²×C₃₁	—	8	1	no	—
160	C₂²×C₃₁	—	8	1	no	—
161	C₂²×C₃₁	—	8	1	no	—
162	C₂²×C₃₁	—	8	1	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	1	no	—
170	C₄×C₃₁	—	8	1	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	#508	1	2	yes	—
173	C₄×C₃₁	#548	1	1	yes	characteristic
174	C₄×C₃₁	#577	1	1	yes	characteristic
175	#556	—	2	1	no	—
176	#554	—	2	1	no	—
177	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
178	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	#500	C₂²×C₃₁	1	1	yes	characteristic
180	#499	C₂²×C₃₁	1	2	yes	—
181	#495	C₂²×C₃₁	1	2	yes	—
182	#494	C₂²×C₃₁	1	1	yes	characteristic
183	#478	—	2	2	no	—
184	#476	—	2	1	no	—
185	#464	—	2	2	no	—
186	#461	—	2	1	no	—
187	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
188	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
189	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₄×D₈	C₂²×C₃₁	1	2	yes	—
191	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
192	#367	C₂²×C₃₁	1	2	yes	—
193	#366	C₂²×C₃₁	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
196	#358	C₂²×C₃₁	1	2	yes	—
197	#358	C₂²×C₃₁	1	2	yes	—
198	#358	C₂²×C₃₁	1	2	yes	—
199	#357	C₂²×C₃₁	1	1	yes	characteristic
200	#357	C₂²×C₃₁	1	1	yes	characteristic
201	#357	C₂²×C₃₁	1	2	yes	—
202	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
204	C₂×C₃₁	—	8	2	no	—
205	C₂×C₃₁	—	16	1	no	—
206	C₂×C₃₁	#3358	1	1	yes	characteristic
207	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
208	C₂×C₃₁	#3416	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
213	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	#126	—	4	2	no	—
217	#125	—	2	2	no	—
218	#124	—	2	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	2	1	no	—
227	#117	—	4	1	no	—
228	C₂×C₄⋊C₄	—	4	1	no	—
229	SD₃₂	—	2	4	no	—
230	D₁₆	—	2	2	no	—
231	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
233	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
234	#107	—	2	1	no	—
235	#107	C₂³×C₃₁	1	1	yes	characteristic
236	#107	C₂³×C₃₁	1	2	yes	—
237	#107	C₂³×C₃₁	1	1	yes	characteristic
238	#106	—	2	2	no	—
239	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
240	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
241	#103	—	2	2	no	—
242	#102	—	2	1	no	—
243	#102	—	2	2	no	—
244	#102	—	2	1	no	—
245	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₃₁	1	2	yes	—
247	C₃₁	#33927	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	#41×C₃₁	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
277	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₃₁	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	16	2	no	—
306	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
308	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
314	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	C₃₁×#508	1	2	yes	—
330	C₄	C₃₁×#548	1	1	yes	characteristic
331	C₄	C₃₁×#577	1	1	yes	characteristic
332	C₂	—	8	2	no	—
333	C₂	—	16	1	no	—
334	C₂	C₃₁×#3358	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
336	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×C₃₁×D₁₆	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
1. C₃₁×#3393	C₂	semidirect
2. C₃₁×#3380	C₂	semidirect
3. C₃₁×#2242	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
6. C₃₁×#2226	C₂	semidirect
8. C₃₁×#2208	C₂	semidirect
9. Q₈×C₁₆×C₃₁	C₂	semidirect
10. C₄×C₃₁×SD₃₂	C₂	semidirect
120. #33927	C₃₁	direct (N × H)
147. #3393	C₂×C₃₁	semidirect
148. #3380	C₂×C₃₁	semidirect
149. #2242	C₂×C₃₁	semidirect
150. #2241	C₂×C₃₁	semidirect
151. #2229	C₂×C₃₁	semidirect
152. #2226	C₂×C₃₁	semidirect
154. #2208	C₂×C₃₁	semidirect
155. Q₈×C₁₆	C₂×C₃₁	semidirect
156. C₄×SD₃₂	C₂×C₃₁	semidirect
247. C₃₁	#33927	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×#3380	2
C₃₁×#2242	2
C₃₁×#2241	2
C₃₁×#2229	2
C₃₁×#2226	2
C₃₁×#2210	2
C₃₁×#2208	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×SD₃₂	2
C₄×C₃₁×D₁₆	2
#33927	31

Subgroups — C31×#33927 #21490686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33927 #21490686