Subgroups — C₃₁×#10919 #21467678

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₃₁×#1742	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#405	—	2	4	no	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₃₁×#389	—	2	2	no	—
14	C₃₁×#385	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
16	C₃₁×#344	C₄	1	8	yes	complemented
17	C₃₁×#343	—	2	2	no	—
18	C₃₁×#336	—	2	4	no	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	D₄	1	4	yes	—
35	C₃₁×#117	—	2	2	no	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	D₄	1	2	yes	—
50	C₃₁×#97	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	4	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₃₁	#41	1	2	yes	—
65	C₂²×C₄×C₃₁	#41	1	2	yes	—
66	C₂²×C₄×C₃₁	#41	1	2	yes	—
67	M₁₆×C₃₁	—	2	4	no	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	complemented
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	4	8	no	—
84	C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
87	C₄²×C₃₁	—	2	4	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	4	1	no	—
91	C₄²×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	—	4	1	no	—
93	#10919	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₃₁	#117	1	1	yes	characteristic
98	C₂³×C₃₁	C₄×D₄	1	2	yes	—
99	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₃₁	#117	1	2	yes	—
111	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	2	yes	—
117	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	#123	1	2	yes	—
120	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#125	1	2	yes	—
123	C₂×C₄×C₃₁	#126	1	2	yes	—
124	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
125	C₈×C₃₁	—	2	8	no	—
126	C₈×C₃₁	—	4	4	no	—
127	#2873	C₂×C₃₁	1	1	yes	characteristic
128	#2296	C₂×C₃₁	1	1	yes	characteristic
129	C₂×#344	C₂×C₃₁	1	2	yes	—
130	#1742	C₂×C₃₁	1	1	yes	characteristic
131	#1728	C₂×C₃₁	1	2	yes	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	4	1	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₃₁	#386	1	1	yes	characteristic
140	C₂²×C₃₁	#387	1	2	yes	—
141	C₂²×C₃₁	#391	1	1	yes	characteristic
142	C₄×C₃₁	—	4	4	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	4	4	no	—
145	C₄×C₃₁	#385	1	2	yes	—
146	C₄×C₃₁	#387	1	2	yes	—
147	C₄×C₃₁	#388	1	2	yes	—
148	C₄×C₃₁	#389	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
150	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
151	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
154	#405	—	2	4	no	—
155	C₄×M₁₆	—	2	2	no	—
156	#389	—	2	2	no	—
157	#385	C₄×C₃₁	1	4	yes	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	#344	C₄×C₃₁	1	8	yes	complemented
160	#343	—	2	2	no	—
161	#336	—	2	4	no	—
162	C₂×C₃₁	—	2	1	no	—
163	C₂×C₃₁	—	2	2	no	—
164	C₂×C₃₁	—	2	2	no	—
165	C₂×C₃₁	—	2	1	no	—
166	C₂×C₃₁	#1436	1	1	yes	characteristic
167	C₂×C₃₁	#1891	1	2	yes	—
168	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
169	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
180	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
181	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	D₄×C₃₁	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	C₂×C₄×C₃₁	1	4	yes	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	D₄×C₃₁	1	2	yes	—
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	4	no	—
201	#95	—	2	4	no	—
202	C₃₁	#10919	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
214	C₂²×C₄	#41×C₃₁	1	2	yes	—
215	C₂²×C₄	#41×C₃₁	1	2	yes	—
216	C₂²×C₄	#41×C₃₁	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	complemented
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	8	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
245	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
246	C₂³	C₃₁×#117	1	1	yes	characteristic
247	C₂³	C₄×D₄×C₃₁	1	2	yes	—
248	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
259	C₂×C₄	C₃₁×#117	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
261	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
262	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
263	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
264	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
265	C₂×C₄	C₃₁×#121	1	2	yes	—
266	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
267	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
268	C₂×C₄	C₃₁×#123	1	2	yes	—
269	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#125	1	2	yes	—
272	C₂×C₄	C₃₁×#126	1	2	yes	—
273	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₈	—	4	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
283	C₂²	C₃₁×#386	1	1	yes	characteristic
284	C₂²	C₃₁×#387	1	2	yes	—
285	C₂²	C₃₁×#391	1	1	yes	characteristic
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	C₃₁×#385	1	2	yes	—
290	C₄	C₃₁×#387	1	2	yes	—
291	C₄	C₃₁×#388	1	2	yes	—
292	C₄	C₃₁×#389	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	C₃₁×#1436	1	1	yes	characteristic
298	C₂	C₃₁×#1891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#386
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₃₁×#344	C₄	semidirect
80. C₂×C₈×C₃₁	C₄⋊C₄	semidirect
93. #10919	C₃₁	direct (N × H)
159. #344	C₄×C₃₁	semidirect
202. C₃₁	#10919	direct (N × H)
230. C₂×C₈	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2873	2
C₃₁×#2296	2
C₂×C₃₁×#344	2
C₃₁×#1742	2
C₃₁×#1728	2
#10919	31

Subgroups — C31×#10919 #21467678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10919 #21467678