Subgroups — C₃₁×#20040 #21476799

Order 7936.

628 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×#3385	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3295	C₂	1	2	yes	maximal
3	C₃₁×#1722	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1675	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1629	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1623	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1585	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
11	C₃₁×#554	C₂²	1	2	yes	—
12	D₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#501	—	2	2	no	—
14	C₃₁×#500	C₂²	1	2	yes	complemented
15	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#492	C₂²	1	1	yes	characteristic
17	C₃₁×#486	C₂²	1	2	yes	—
18	C₃₁×#481	—	2	2	no	—
19	C₃₁×#479	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	complemented
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#474	—	2	2	no	—
24	C₃₁×#465	—	2	2	no	—
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#450	—	2	2	no	—
27	C₃₁×#446	C₂²	1	2	yes	complemented
28	C₃₁×#440	—	2	2	no	—
29	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
30	C₈×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	—	2	2	no	—
32	C₃₁×#335	C₂²	1	2	yes	complemented
33	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#333	C₂²	1	2	yes	complemented
35	C₃₁×#331	C₂²	1	1	yes	characteristic
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₃₁×#322	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	—	4	1	no	—
43	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	—	4	1	no	—
47	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×#127	C₂³	1	1	yes	characteristic
50	C₃₁×#125	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	4	2	no	—
53	C₃₁×#121	—	2	2	no	—
54	C₃₁×#120	—	4	1	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₃₁×#107	—	2	2	no	—
61	C₃₁×#107	—	2	2	no	—
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#106	—	2	2	no	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	D₄	1	2	yes	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	D₄	1	2	yes	—
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	C₃₁×#97	—	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	—	2	1	no	—
84	C₂×Q₈×C₃₁	—	4	1	no	—
85	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	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	2	no	—
94	Q₁₆×C₃₁	—	8	2	no	—
95	SD₁₆×C₃₁	—	2	4	no	—
96	SD₁₆×C₃₁	—	8	2	no	—
97	SD₁₆×C₃₁	—	8	2	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	8	2	no	—
102	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	4	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	1	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	4	1	no	—
118	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
119	#20040	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	2	2	no	—
121	C₂³×C₃₁	—	4	1	no	—
122	C₂³×C₃₁	—	8	4	no	—
123	C₂³×C₃₁	—	16	1	no	—
124	Q₈×C₃₁	—	2	2	no	—
125	Q₈×C₃₁	—	2	2	no	—
126	Q₈×C₃₁	—	8	1	no	—
127	Q₈×C₃₁	—	8	2	no	—
128	Q₈×C₃₁	—	16	1	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	8	1	no	—
133	D₄×C₃₁	—	8	4	no	—
134	D₄×C₃₁	#137	1	4	yes	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	16	2	no	—
143	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	2	2	no	—
147	C₈×C₃₁	—	2	2	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	—	4	2	no	—
150	#3385	C₂×C₃₁	1	2	yes	complemented
151	#3295	C₂×C₃₁	1	2	yes	—
152	#1722	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1698	C₂×C₃₁	1	2	yes	complemented
154	#1675	C₂×C₃₁	1	2	yes	complemented
155	#1632	C₂×C₃₁	1	2	yes	complemented
156	#1629	C₂×C₃₁	1	1	yes	characteristic
157	#1623	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#1585	C₂×C₃₁	1	2	yes	complemented
159	C₂²×C₃₁	—	2	4	no	—
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	—	4	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	—	16	4	no	—
165	C₂²×C₃₁	D₄²	1	1	yes	characteristic
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	4	2	no	—
168	C₄×C₃₁	—	4	2	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	1	no	—
172	C₄×C₃₁	C₂×#137	1	2	yes	—
173	C₄×C₃₁	#577	1	2	yes	—
174	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
175	#554	C₂²×C₃₁	1	2	yes	—
176	D₄²	C₂²×C₃₁	1	1	yes	characteristic
177	#501	—	2	2	no	—
178	#500	C₂²×C₃₁	1	2	yes	complemented
179	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
180	#492	C₂²×C₃₁	1	1	yes	characteristic
181	#486	C₂²×C₃₁	1	2	yes	—
182	#481	—	2	2	no	—
183	#479	—	2	2	no	—
184	#477	C₂²×C₃₁	1	2	yes	complemented
185	#476	C₂²×C₃₁	1	2	yes	—
186	#476	C₂²×C₃₁	1	2	yes	—
187	#474	—	2	2	no	—
188	#465	—	2	2	no	—
189	#460	C₂²×C₃₁	1	2	yes	—
190	#450	—	2	2	no	—
191	#446	C₂²×C₃₁	1	2	yes	complemented
192	#440	—	2	2	no	—
193	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
194	C₈×D₄	C₂²×C₃₁	1	2	yes	—
195	C₂×#102	—	2	2	no	—
196	#335	C₂²×C₃₁	1	2	yes	complemented
197	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
198	#333	C₂²×C₃₁	1	2	yes	complemented
199	#331	C₂²×C₃₁	1	1	yes	characteristic
200	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
201	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
202	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
203	C₂×C₃₁	—	4	4	no	—
204	C₂×C₃₁	—	16	1	no	—
205	C₂×C₃₁	#3266	1	2	yes	—
206	C₂×C₃₁	#3292	1	1	yes	characteristic
207	C₂²×D₄	—	2	2	no	—
208	C₂×Q₁₆	—	4	1	no	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	4	1	no	—
211	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
212	C₂²×C₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	4	1	no	—
215	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
217	#127	C₂³×C₃₁	1	1	yes	characteristic
218	#125	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	4	2	no	—
221	#121	—	2	2	no	—
222	#120	—	4	1	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	C₂³×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	2	no	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	#106	D₄×C₃₁	1	2	yes	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
236	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	D₄×C₃₁	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	D₄×C₃₁	1	2	yes	—
245	C₂²⋊C₈	—	2	2	no	—
246	#97	—	2	2	no	—
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₃₁	1	2	yes	—
249	C₃₁	#20040	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	4	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	8	2	no	—
260	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	Q₁₆	—	8	2	no	—
264	SD₁₆	—	2	4	no	—
265	SD₁₆	—	8	2	no	—
266	SD₁₆	—	8	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
281	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	1	no	—
287	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	8	4	no	—
291	C₂³	—	16	1	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	8	1	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	16	1	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	4	no	—
302	D₄	C₃₁×#137	1	4	yes	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	16	2	no	—
311	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
312	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	16	4	no	—
324	C₂²	D₄²×C₃₁	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₂×C₃₁×#137	1	2	yes	—
332	C₄	C₃₁×#577	1	2	yes	—
333	C₂	—	4	4	no	—
334	C₂	—	16	1	no	—
335	C₂	C₃₁×#3266	1	2	yes	—
336	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	D₄²×C₃₁	C₂²
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×#3385	C₂	semidirect
3. C₃₁×#1722	C₂	semidirect
4. C₃₁×#1698	C₂	semidirect
5. C₃₁×#1675	C₂	semidirect
6. C₃₁×#1632	C₂	semidirect
8. C₃₁×#1623	C₂	semidirect
9. C₃₁×#1585	C₂	semidirect
10. Q₈²×C₃₁	C₂²	semidirect
14. C₃₁×#500	C₂²	semidirect
15. C₃₁×#494	C₂²	semidirect
20. C₃₁×#477	C₂²	semidirect
27. C₃₁×#446	C₂²	semidirect
32. C₃₁×#335	C₂²	semidirect
33. C₃₁×#333	C₂²	semidirect
34. C₃₁×#333	C₂²	semidirect
36. C₃₁×Q₈⋊C₈	C₂²	semidirect
38. C₃₁×#322	C₂²	semidirect
119. #20040	C₃₁	direct (N × H)
150. #3385	C₂×C₃₁	semidirect
152. #1722	C₂×C₃₁	semidirect
153. #1698	C₂×C₃₁	semidirect
154. #1675	C₂×C₃₁	semidirect
155. #1632	C₂×C₃₁	semidirect
157. #1623	C₂×C₃₁	semidirect
158. #1585	C₂×C₃₁	semidirect
174. Q₈²	C₂²×C₃₁	semidirect
178. #500	C₂²×C₃₁	semidirect
179. #494	C₂²×C₃₁	semidirect
184. #477	C₂²×C₃₁	semidirect
191. #446	C₂²×C₃₁	semidirect
196. #335	C₂²×C₃₁	semidirect
197. #333	C₂²×C₃₁	semidirect
198. #333	C₂²×C₃₁	semidirect
200. Q₈⋊C₈	C₂²×C₃₁	semidirect
202. #322	C₂²×C₃₁	semidirect
249. C₃₁	#20040	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×#3385	2
C₃₁×#3295	2
C₃₁×#1722	2
C₃₁×#1698	2
C₃₁×#1675	2
C₃₁×#1632	2
C₃₁×#1629	2
C₃₁×#1623	2
C₃₁×#1585	2
#20040	31

Subgroups — C31×#20040 #21476799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20040 #21476799