Subgroups — C₃₁×#20754 #21477513

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	C₂	1	2	yes	maximal
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1832	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1831	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1729	C₂	1	2	yes	maximal
6	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#424	—	2	8	no	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
17	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
18	C₃₁×#376	C₂²	1	2	yes	—
19	C₄³×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#336	C₂²	1	2	yes	—
21	C₃₁×#336	C₂²	1	2	yes	—
22	C₃₁×#336	C₂²	1	8	yes	—
23	C₃₁×#336	C₂²	1	2	yes	—
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	—	2	4	no	—
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₃₁×#117	—	2	4	no	—
36	C₃₁×#117	D₄	1	4	yes	—
37	C₃₁×#117	Q₈	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
51	C₃₁×C₄⋊C₈	—	2	16	no	—
52	C₃₁×#95	C₂×C₄	1	4	yes	—
53	C₃₁×#95	C₂×C₄	1	4	yes	—
54	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
66	C₂²×C₄×C₃₁	#41	1	4	yes	—
67	C₂²×C₄×C₃₁	#41	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₃₁	—	4	8	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	4	8	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
80	C₂²⋊C₄×C₃₁	—	2	4	no	—
81	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
82	C₄²×C₃₁	—	2	4	no	—
83	C₄²×C₃₁	—	2	8	no	—
84	C₄²×C₃₁	—	2	2	no	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	C₂×D₄	1	4	yes	—
89	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
90	#20754	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₃₁	#117	1	2	yes	—
95	C₂³×C₃₁	#117	1	2	yes	—
96	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	8	no	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	#117	1	2	yes	—
112	C₂×C₄×C₃₁	#117	1	2	yes	—
113	C₂×C₄×C₃₁	#124	1	8	yes	—
114	C₂×C₄×C₃₁	#125	1	8	yes	—
115	C₂×C₄×C₃₁	#131	1	4	yes	—
116	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	—	4	4	no	—
122	C₂×#424	C₂×C₃₁	1	2	yes	—
123	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
124	#1832	C₂×C₃₁	1	4	yes	complemented
125	#1831	C₂×C₃₁	1	4	yes	complemented
126	#1729	C₂×C₃₁	1	2	yes	—
127	#1728	C₂×C₃₁	1	1	yes	characteristic
128	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	#382	1	4	yes	—
134	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
137	C₂²×C₃₁	C₂×#131	1	2	yes	—
138	C₂²×C₃₁	#568	1	2	yes	—
139	C₄×C₃₁	—	2	8	no	—
140	C₄×C₃₁	—	2	4	no	—
141	C₄×C₃₁	—	4	2	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	#382	1	4	yes	—
145	C₄×C₃₁	C₂×#124	1	2	yes	—
146	C₄×C₃₁	C₂×#125	1	2	yes	—
147	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
148	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
151	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
152	#424	—	2	8	no	—
153	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
154	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
156	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
157	#376	C₂²×C₃₁	1	2	yes	—
158	C₄³	C₂²×C₃₁	1	2	yes	—
159	#336	C₂²×C₃₁	1	2	yes	—
160	#336	C₂²×C₃₁	1	2	yes	—
161	#336	C₂²×C₃₁	1	8	yes	—
162	#336	C₂²×C₃₁	1	2	yes	—
163	#336	C₂²×C₃₁	1	2	yes	—
164	C₂×C₃₁	—	2	4	no	—
165	C₂×C₃₁	C₂×#382	1	1	yes	characteristic
166	C₂×C₃₁	#2906	1	2	yes	—
167	C₂×C₃₁	#2922	1	1	yes	characteristic
168	C₂×C₃₁	#2923	1	1	yes	characteristic
169	C₂×C₃₁	#2924	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
171	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
180	#117	—	2	4	no	—
181	#117	D₄×C₃₁	1	4	yes	—
182	#117	Q₈×C₃₁	1	4	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
192	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
193	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
196	C₄⋊C₈	—	2	16	no	—
197	#95	C₂×C₄×C₃₁	1	4	yes	—
198	#95	C₂×C₄×C₃₁	1	4	yes	—
199	C₃₁	#20754	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
212	C₂²×C₄	#41×C₃₁	1	4	yes	—
213	C₂²×C₄	#41×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
215	M₁₆	—	4	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	8	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂×D₄×C₃₁	1	4	yes	—
235	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
239	C₂³	C₃₁×#117	1	2	yes	—
240	C₂³	C₃₁×#117	1	2	yes	—
241	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
256	C₂×C₄	C₃₁×#117	1	2	yes	—
257	C₂×C₄	C₃₁×#117	1	2	yes	—
258	C₂×C₄	C₃₁×#124	1	8	yes	—
259	C₂×C₄	C₃₁×#125	1	8	yes	—
260	C₂×C₄	C₃₁×#131	1	4	yes	—
261	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
264	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
265	C₈	—	4	4	no	—
266	C₈	—	4	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₃₁×#382	1	4	yes	—
272	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
274	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
275	C₂²	C₂×C₃₁×#131	1	2	yes	—
276	C₂²	C₃₁×#568	1	2	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	C₃₁×#382	1	4	yes	—
283	C₄	C₂×C₃₁×#124	1	2	yes	—
284	C₄	C₂×C₃₁×#125	1	2	yes	—
285	C₂	—	2	4	no	—
286	C₂	C₂×C₃₁×#382	1	1	yes	characteristic
287	C₂	C₃₁×#2906	1	2	yes	—
288	C₂	C₃₁×#2922	1	1	yes	characteristic
289	C₂	C₃₁×#2923	1	1	yes	characteristic
290	C₂	C₃₁×#2924	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1832	C₂	semidirect
4. C₃₁×#1831	C₂	semidirect
90. #20754	C₃₁	direct (N × H)
124. #1832	C₂×C₃₁	semidirect
125. #1831	C₂×C₃₁	semidirect
199. C₃₁	#20754	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#424	2
C₄×C₃₁×#117	2
C₃₁×#1832	2
C₃₁×#1831	2
C₃₁×#1729	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#20754	31

Subgroups — C31×#20754 #21477513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20754 #21477513