Subgroups — C₃₁×#8608 #21465367

Order 7936.

550 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1807	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1799	C₂	1	2	yes	maximal
4	C₃₁×#1289	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₃₁	—	2	2	no	—
9	C₈×D₄×C₃₁	—	2	2	no	—
10	C₂×C₃₁×#106	—	2	2	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃₁×#99	—	2	2	no	—
13	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#380	C₂²	1	4	yes	—
17	C₃₁×#342	—	2	2	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	—	2	1	no	—
24	C₂²×C₈×C₃₁	—	4	2	no	—
25	C₂²×C₈×C₃₁	—	4	2	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	2	yes	—
31	C₂²×C₈×C₃₁	D₄	1	2	yes	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₄×D₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₃₁×#106	—	4	4	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×#99	—	4	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	2	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₈×C₃₁	—	2	1	no	—
46	C₄×C₈×C₃₁	—	2	1	no	—
47	C₃₁×#95	—	2	4	no	—
48	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	SD₁₆	1	4	yes	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	4	1	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	#41	1	2	yes	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	—	4	1	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	4	4	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₃₁	—	4	4	no	—
86	C₂²⋊C₄×C₃₁	—	4	4	no	—
87	C₄²×C₃₁	—	2	1	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	#8608	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	4	1	no	—
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	#106	1	2	yes	complemented
97	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₃₁	#121	1	2	yes	—
99	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
100	C₂³×C₃₁	#135	1	2	yes	—
101	D₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	1	no	—
117	C₂×C₄×C₃₁	—	8	4	no	—
118	C₂×C₄×C₃₁	—	8	4	no	—
119	C₂×C₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	#106	1	2	yes	complemented
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
124	C₈×C₃₁	—	2	1	no	—
125	C₈×C₃₁	—	2	2	no	—
126	C₈×C₃₁	—	4	1	no	—
127	C₈×C₃₁	—	4	1	no	—
128	C₈×C₃₁	#99	1	2	yes	complemented
129	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
130	#1807	C₂×C₃₁	1	2	yes	complemented
131	#1799	C₂×C₃₁	1	2	yes	—
132	#1289	C₂×C₃₁	1	2	yes	complemented
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	4	2	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	4	2	no	—
141	C₂²×C₃₁	—	4	2	no	—
142	C₂²×C₃₁	—	4	1	no	—
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	#380	1	1	yes	characteristic
145	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
146	C₂²×C₃₁	#427	1	1	yes	characteristic
147	C₂²×C₃₁	#465	1	4	yes	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	1	no	—
153	C₄×C₃₁	—	16	4	no	—
154	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
155	C₄×C₃₁	#410	1	1	yes	characteristic
156	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
157	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
159	C₈×D₄	—	2	2	no	—
160	C₈×D₄	—	2	2	no	—
161	C₂×#106	—	2	2	no	—
162	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
163	C₂×#99	—	2	2	no	—
164	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
165	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
167	#380	C₂²×C₃₁	1	4	yes	—
168	#342	—	2	2	no	—
169	#340	—	2	2	no	—
170	C₂×C₃₁	—	2	4	no	—
171	C₂×C₃₁	—	2	1	no	—
172	C₂×C₃₁	—	2	1	no	—
173	C₂×C₃₁	—	4	2	no	—
174	C₂×C₃₁	#1799	1	1	yes	characteristic
175	C₂×C₃₁	#1807	1	1	yes	characteristic
176	C₂×C₃₁	#1812	1	1	yes	characteristic
177	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
178	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	4	2	no	—
183	C₂²×C₈	—	4	2	no	—
184	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₃₁	1	2	yes	—
189	C₂²×C₈	D₄×C₃₁	1	2	yes	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
198	#106	—	4	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	#99	—	4	4	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	#95	—	2	4	no	—
206	C₃₁	#8608	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	SD₁₆×C₃₁	1	4	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	#41×C₃₁	1	2	yes	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
238	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₃₁×#106	1	2	yes	complemented
255	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂³	C₃₁×#121	1	2	yes	—
257	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
258	C₂³	C₃₁×#135	1	2	yes	—
259	D₄	—	2	8	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₃₁×#106	1	2	yes	complemented
279	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
281	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
282	C₈	—	2	1	no	—
283	C₈	—	2	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	C₃₁×#99	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₃₁×#380	1	1	yes	characteristic
299	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
300	C₂²	C₃₁×#427	1	1	yes	characteristic
301	C₂²	C₃₁×#465	1	4	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	16	4	no	—
308	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
309	C₄	C₃₁×#410	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	4	2	no	—
314	C₂	C₃₁×#1799	1	1	yes	characteristic
315	C₂	C₃₁×#1807	1	1	yes	characteristic
316	C₂	C₃₁×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1807	C₂	semidirect
4. C₃₁×#1289	C₂	semidirect
5. C₂³×C₈×C₃₁	C₄	semidirect
11. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
13. C₂×C₃₁×C₂²⋊C₈	C₄	semidirect
15. C₂×C₄×C₈×C₃₁	C₄	semidirect
89. #8608	C₃₁	direct (N × H)
96. C₂³×C₃₁	#106	semidirect
120. C₂×C₄×C₃₁	#106	semidirect
128. C₈×C₃₁	#99	semidirect
130. #1807	C₂×C₃₁	semidirect
132. #1289	C₂×C₃₁	semidirect
156. C₂³×C₈	C₄×C₃₁	semidirect
162. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
164. C₂×C₂²⋊C₈	C₄×C₃₁	semidirect
166. C₂×C₄×C₈	C₄×C₃₁	semidirect
206. C₃₁	#8608	direct (N × H)
254. C₂³	C₃₁×#106	semidirect
278. C₂×C₄	C₃₁×#106	semidirect
286. C₈	C₃₁×#99	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×D₄×C₃₁	2
C₃₁×#1807	2
C₃₁×#1799	2
C₃₁×#1289	2
#8608	31

Subgroups — C31×#8608 #21465367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8608 #21465367