Subgroups — C₃₁×#33869 #21490628

Order 7936.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 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, complemented
2	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	—
13	C₃₁×#556	—	2	1	no	—
14	C₃₁×#547	—	2	1	no	—
15	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
16	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#500	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#485	—	2	2	no	—
21	C₃₁×#475	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#467	C₂²	1	2	yes	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#452	—	2	2	no	—
26	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#427	C₂²	1	2	yes	—
30	C₃₁×#367	C₂²	1	1	yes	characteristic
31	C₃₁×#366	C₂²	1	2	yes	—
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₃₁×#135	—	2	4	no	—
43	C₃₁×#135	—	4	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	1	no	—
45	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	—	2	1	no	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×#126	—	4	2	no	—
53	C₃₁×#124	—	4	1	no	—
54	C₃₁×#123	—	4	2	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×#121	—	4	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	4	1	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×#117	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
64	C₃₁×Q₃₂	—	4	2	no	—
65	C₃₁×SD₃₂	—	2	2	no	—
66	C₂×C₁₆×C₃₁	—	4	2	no	—
67	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	C₂³	1	1	yes	characteristic
72	C₃₁×#107	C₂³	1	2	yes	—
73	C₃₁×#107	C₂³	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₃₁×#103	—	2	1	no	—
77	C₃₁×#103	—	2	2	no	—
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#102	—	2	1	no	—
80	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	8	2	no	—
84	C₂×Q₈×C₃₁	—	2	1	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	—	4	1	no	—
91	C₂²×C₄×C₃₁	—	4	2	no	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	Q₁₆×C₃₁	—	2	1	no	—
94	Q₁₆×C₃₁	—	2	2	no	—
95	Q₁₆×C₃₁	—	2	1	no	—
96	Q₁₆×C₃₁	—	2	2	no	—
97	Q₁₆×C₃₁	—	8	1	no	—
98	SD₁₆×C₃₁	—	4	4	no	—
99	D₈×C₃₁	—	2	1	no	—
100	D₈×C₃₁	#41	1	2	yes	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	—	4	1	no	—
114	C₄⋊C₄×C₃₁	—	4	1	no	—
115	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
121	C₄²×C₃₁	—	4	2	no	—
122	C₄²×C₃₁	—	4	1	no	—
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₃₁	—	2	1	no	—
125	C₁₆×C₃₁	—	2	1	no	—
126	C₁₆×C₃₁	—	2	2	no	—
127	#33869	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	4	1	no	—
129	C₂³×C₃₁	C₂×D₈	1	2	yes	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	4	1	no	—
132	Q₈×C₃₁	—	4	2	no	—
133	Q₈×C₃₁	—	4	1	no	—
134	Q₈×C₃₁	—	8	2	no	—
135	D₄×C₃₁	—	2	2	no	—
136	D₄×C₃₁	—	2	2	no	—
137	D₄×C₃₁	—	4	1	no	—
138	D₄×C₃₁	—	8	4	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	8	4	no	—
147	C₂×C₄×C₃₁	—	8	2	no	—
148	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
155	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
156	#3342	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#3283	C₂×C₃₁	1	1	yes	characteristic
158	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2237	C₂×C₃₁	1	2	yes	complemented
160	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2216	C₂×C₃₁	1	2	yes	complemented
162	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#2189	C₂×C₃₁	1	2	yes	complemented
164	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
165	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2138	C₂×C₃₁	1	2	yes	complemented
167	C₂²×C₃₁	—	2	2	no	—
168	C₂²×C₃₁	—	2	2	no	—
169	C₂²×C₃₁	—	2	2	no	—
170	C₂²×C₃₁	—	4	2	no	—
171	C₂²×C₃₁	—	8	1	no	—
172	C₂²×C₃₁	—	8	1	no	—
173	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	8	1	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	—	8	1	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	1	no	—
181	C₄×C₃₁	#508	1	2	yes	—
182	C₄×C₃₁	#546	1	1	yes	characteristic
183	C₄×C₃₁	#577	1	1	yes	characteristic
184	C₂×#135	C₂²×C₃₁	1	2	yes	—
185	#556	—	2	1	no	—
186	#547	—	2	1	no	—
187	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
188	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
189	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
190	#500	C₂²×C₃₁	1	1	yes	characteristic
191	#494	C₂²×C₃₁	1	1	yes	characteristic
192	#485	—	2	2	no	—
193	#475	—	2	1	no	—
194	#467	C₂²×C₃₁	1	2	yes	—
195	#467	C₂²×C₃₁	1	2	yes	—
196	#461	—	2	1	no	—
197	#452	—	2	2	no	—
198	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
199	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
201	#427	C₂²×C₃₁	1	2	yes	—
202	#367	C₂²×C₃₁	1	1	yes	characteristic
203	#366	C₂²×C₃₁	1	2	yes	—
204	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
205	#358	C₂²×C₃₁	1	2	yes	—
206	#358	C₂²×C₃₁	1	2	yes	—
207	#358	C₂²×C₃₁	1	1	yes	characteristic
208	#358	C₂²×C₃₁	1	2	yes	—
209	#357	C₂²×C₃₁	1	1	yes	characteristic
210	#357	C₂²×C₃₁	1	2	yes	—
211	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
212	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₃₁	—	4	2	no	—
214	C₂×C₃₁	—	8	2	no	—
215	C₂×C₃₁	#3294	1	1	yes	characteristic
216	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
217	C₂×C₃₁	#3417	1	1	yes	characteristic
218	C₂×#41	—	2	2	no	—
219	#135	—	2	4	no	—
220	#135	—	4	2	no	—
221	C₂×Q₁₆	—	2	1	no	—
222	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
223	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	#126	—	4	2	no	—
230	#124	—	4	1	no	—
231	#123	—	4	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	#121	—	4	1	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×Q₈	—	4	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	—	4	1	no	—
241	Q₃₂	—	4	2	no	—
242	SD₃₂	—	2	2	no	—
243	C₂×C₁₆	—	4	2	no	—
244	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
245	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
246	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
247	#107	—	2	1	no	—
248	#107	C₂³×C₃₁	1	2	yes	—
249	#107	C₂³×C₃₁	1	1	yes	characteristic
250	#107	C₂³×C₃₁	1	1	yes	characteristic
251	#106	—	2	2	no	—
252	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
253	#103	—	2	1	no	—
254	#103	—	2	2	no	—
255	#103	—	2	2	no	—
256	#102	—	2	1	no	—
257	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
258	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₃₁	#33869	1	1	yes	characteristic, direct product factor, Sylow
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	8	1	no	—
276	SD₁₆	—	4	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#41×C₃₁	1	2	yes	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
283	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₂×D₈×C₃₁	1	2	yes	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
326	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
332	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	C₃₁×#508	1	2	yes	—
348	C₄	C₃₁×#546	1	1	yes	characteristic
349	C₄	C₃₁×#577	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	8	2	no	—
352	C₂	C₃₁×#3294	1	1	yes	characteristic
353	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
354	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	C₃₁×#3283	C₂
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
1. C₃₁×#3342	C₂	semidirect
3. C₃₁×#2241	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2224	C₂	semidirect
6. C₃₁×#2216	C₂	semidirect
7. C₃₁×#2209	C₂	semidirect
8. C₃₁×#2189	C₂	semidirect
9. C₄×C₃₁×SD₃₂	C₂	semidirect
10. D₄×C₁₆×C₃₁	C₂	semidirect
11. C₃₁×#2138	C₂	semidirect
127. #33869	C₃₁	direct (N × H)
156. #3342	C₂×C₃₁	semidirect
158. #2241	C₂×C₃₁	semidirect
159. #2237	C₂×C₃₁	semidirect
160. #2224	C₂×C₃₁	semidirect
161. #2216	C₂×C₃₁	semidirect
162. #2209	C₂×C₃₁	semidirect
163. #2189	C₂×C₃₁	semidirect
164. C₄×SD₃₂	C₂×C₃₁	semidirect
165. D₄×C₁₆	C₂×C₃₁	semidirect
166. #2138	C₂×C₃₁	semidirect
259. C₃₁	#33869	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×#3283	2
C₃₁×#2241	2
C₃₁×#2237	2
C₃₁×#2224	2
C₃₁×#2216	2
C₃₁×#2209	2
C₃₁×#2189	2
C₄×C₃₁×SD₃₂	2
D₄×C₁₆×C₃₁	2
C₃₁×#2138	2
#33869	31

Subgroups — C31×#33869 #21490628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33869 #21490628