Subgroups — C₃₁×#19996 #21476755

Order 7936.

650 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3361	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3295	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1619	C₂	1	1	yes	maximal, characteristic
9	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₃₁×#559	C₂²	1	1	yes	characteristic, complemented
11	C₃₁×#554	C₂²	1	2	yes	—
12	D₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	C₂²	1	1	yes	characteristic
15	C₃₁×#488	—	2	2	no	—
16	C₃₁×#486	C₂²	1	2	yes	—
17	C₃₁×#486	C₂²	1	2	yes	complemented
18	C₃₁×#481	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	complemented
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#465	—	2	2	no	—
23	C₃₁×#463	—	2	2	no	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₃₁×#450	—	2	2	no	—
27	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
28	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×D₈×C₃₁	—	2	2	no	—
30	C₈×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	—	2	2	no	—
32	C₃₁×#334	C₂²	1	2	yes	complemented
33	C₃₁×#333	C₂²	1	2	yes	complemented
34	C₃₁×#331	C₂²	1	1	yes	characteristic
35	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	—	4	1	no	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
43	C₂×D₈×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	4	1	no	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×#127	C₂³	1	1	yes	characteristic
48	C₃₁×#127	C₂³	1	1	yes	characteristic
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×#123	—	4	1	no	—
52	C₃₁×C₂²⋊Q₈	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	4	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#120	—	4	1	no	—
56	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₃₁×#107	—	2	2	no	—
62	C₃₁×#106	—	2	2	no	—
63	C₃₁×#106	D₄	1	2	yes	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	D₄	1	2	yes	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	D₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	2	yes	—
81	C₂⁴×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	4	no	—
90	C₂×D₄×C₃₁	—	8	2	no	—
91	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	4	1	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	Q₁₆×C₃₁	—	8	2	no	—
96	SD₁₆×C₃₁	—	2	4	no	—
97	SD₁₆×C₃₁	—	8	2	no	—
98	D₈×C₃₁	—	4	4	no	—
99	D₈×C₃₁	—	8	2	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	8	2	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	1	no	—
116	C₂²⋊C₄×C₃₁	—	4	1	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19996	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	2	2	no	—
122	C₂³×C₃₁	—	4	1	no	—
123	C₂³×C₃₁	—	4	1	no	—
124	C₂³×C₃₁	—	8	4	no	—
125	C₂³×C₃₁	—	16	1	no	—
126	Q₈×C₃₁	—	2	2	no	—
127	Q₈×C₃₁	—	2	2	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	4	2	no	—
134	D₄×C₃₁	—	8	1	no	—
135	D₄×C₃₁	—	8	1	no	—
136	D₄×C₃₁	—	8	4	no	—
137	D₄×C₃₁	#135	1	4	yes	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	16	1	no	—
145	C₂×C₄×C₃₁	—	16	2	no	—
146	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	2	no	—
151	C₈×C₃₁	—	4	2	no	—
152	C₈×C₃₁	—	4	2	no	—
153	#3361	C₂×C₃₁	1	2	yes	complemented
154	#3295	C₂×C₃₁	1	2	yes	complemented
155	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#1678	C₂×C₃₁	1	2	yes	complemented
157	#1677	C₂×C₃₁	1	2	yes	complemented
158	#1632	C₂×C₃₁	1	2	yes	complemented
159	#1620	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#1619	C₂×C₃₁	1	1	yes	characteristic
161	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
162	C₂²×C₃₁	—	2	4	no	—
163	C₂²×C₃₁	—	4	2	no	—
164	C₂²×C₃₁	—	4	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	1	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	—	16	4	no	—
170	C₂²×C₃₁	D₄²	1	1	yes	characteristic
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	4	2	no	—
173	C₄×C₃₁	—	4	2	no	—
174	C₄×C₃₁	—	4	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	C₂×#135	1	2	yes	—
177	C₄×C₃₁	#577	1	2	yes	—
178	#559	C₂²×C₃₁	1	1	yes	characteristic, complemented
179	#554	C₂²×C₃₁	1	2	yes	—
180	D₄²	C₂²×C₃₁	1	1	yes	characteristic
181	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
182	#493	C₂²×C₃₁	1	1	yes	characteristic
183	#488	—	2	2	no	—
184	#486	C₂²×C₃₁	1	2	yes	—
185	#486	C₂²×C₃₁	1	2	yes	complemented
186	#481	—	2	2	no	—
187	#478	—	2	2	no	—
188	#477	C₂²×C₃₁	1	2	yes	complemented
189	#476	C₂²×C₃₁	1	2	yes	—
190	#465	—	2	2	no	—
191	#463	—	2	2	no	—
192	#460	C₂²×C₃₁	1	2	yes	—
193	#459	C₂²×C₃₁	1	2	yes	—
194	#450	—	2	2	no	—
195	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
196	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
197	C₄×D₈	—	2	2	no	—
198	C₈×D₄	C₂²×C₃₁	1	2	yes	—
199	C₂×#102	—	2	2	no	—
200	#334	C₂²×C₃₁	1	2	yes	complemented
201	#333	C₂²×C₃₁	1	2	yes	complemented
202	#331	C₂²×C₃₁	1	1	yes	characteristic
203	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
204	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
205	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
206	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
207	C₂×C₃₁	—	4	4	no	—
208	C₂×C₃₁	—	16	1	no	—
209	C₂×C₃₁	—	16	1	no	—
210	C₂×C₃₁	#3282	1	2	yes	—
211	C₂×C₃₁	#3292	1	1	yes	characteristic
212	C₂²×D₄	—	2	2	no	—
213	C₂×Q₁₆	—	4	1	no	—
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	—	4	1	no	—
218	C₂²×C₈	—	2	2	no	—
219	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
220	#127	C₂³×C₃₁	1	1	yes	characteristic
221	#127	C₂³×C₃₁	1	1	yes	characteristic
222	#125	—	2	2	no	—
223	#124	—	2	2	no	—
224	#123	—	4	1	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	C₂²⋊Q₈	—	4	2	no	—
227	#121	—	2	2	no	—
228	#120	—	4	1	no	—
229	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	C₄×D₄	C₂³×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	#107	—	2	2	no	—
235	#106	—	2	2	no	—
236	#106	D₄×C₃₁	1	2	yes	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
239	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
240	#103	—	2	2	no	—
241	#103	D₄×C₃₁	1	2	yes	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	D₄×C₃₁	1	2	yes	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₃₁	1	2	yes	—
254	C₃₁	#19996	1	1	yes	characteristic, direct product factor, Sylow
255	C₂⁴	—	4	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	4	no	—
264	C₂×D₄	—	8	2	no	—
265	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	Q₁₆	—	8	2	no	—
270	SD₁₆	—	2	4	no	—
271	SD₁₆	—	8	2	no	—
272	D₈	—	4	4	no	—
273	D₈	—	8	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	8	2	no	—
278	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
286	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	4	1	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	16	1	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	4	no	—
310	D₄	C₃₁×#135	1	4	yes	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	16	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
320	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	16	4	no	—
334	C₂²	D₄²×C₃₁	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₂×C₃₁×#135	1	2	yes	—
341	C₄	C₃₁×#577	1	2	yes	—
342	C₂	—	4	4	no	—
343	C₂	—	16	1	no	—
344	C₂	—	16	1	no	—
345	C₂	C₃₁×#3282	1	2	yes	—
346	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#1619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3361	C₂	semidirect
2. C₃₁×#3295	C₂	semidirect
3. C₃₁×#1709	C₂	semidirect
4. C₃₁×#1678	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1632	C₂	semidirect
7. C₃₁×#1620	C₂	semidirect
9. C₈×SD₁₆×C₃₁	C₂	semidirect
10. C₃₁×#559	C₂²	semidirect
13. C₃₁×#494	C₂²	semidirect
17. C₃₁×#486	C₂²	semidirect
20. C₃₁×#477	C₂²	semidirect
27. C₈×Q₈×C₃₁	C₂²	semidirect
32. C₃₁×#334	C₂²	semidirect
33. C₃₁×#333	C₂²	semidirect
35. C₃₁×#331	C₂²	semidirect
36. C₃₁×Q₈⋊C₈	C₂²	semidirect
38. C₈²×C₃₁	C₂²	semidirect
120. #19996	C₃₁	direct (N × H)
153. #3361	C₂×C₃₁	semidirect
154. #3295	C₂×C₃₁	semidirect
155. #1709	C₂×C₃₁	semidirect
156. #1678	C₂×C₃₁	semidirect
157. #1677	C₂×C₃₁	semidirect
158. #1632	C₂×C₃₁	semidirect
159. #1620	C₂×C₃₁	semidirect
161. C₈×SD₁₆	C₂×C₃₁	semidirect
178. #559	C₂²×C₃₁	semidirect
181. #494	C₂²×C₃₁	semidirect
185. #486	C₂²×C₃₁	semidirect
188. #477	C₂²×C₃₁	semidirect
195. C₈×Q₈	C₂²×C₃₁	semidirect
200. #334	C₂²×C₃₁	semidirect
201. #333	C₂²×C₃₁	semidirect
203. #331	C₂²×C₃₁	semidirect
204. Q₈⋊C₈	C₂²×C₃₁	semidirect
206. C₈²	C₂²×C₃₁	semidirect
254. C₃₁	#19996	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×#3361	2
C₃₁×#3295	2
C₃₁×#1709	2
C₃₁×#1678	2
C₃₁×#1677	2
C₃₁×#1632	2
C₃₁×#1620	2
C₃₁×#1619	2
C₈×SD₁₆×C₃₁	2
#19996	31

Subgroups — C31×#19996 #21476755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19996 #21476755