Subgroups — C₃₁×#9900 #21466659

Order 7936.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1296	C₂	1	2	yes	maximal
5	C₃₁×#1296	C₂	1	2	yes	maximal
6	C₂×C₃₁×#125	—	2	1	no	—
7	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
8	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	—	2	1	no	—
12	C₃₁×#389	—	2	1	no	—
13	C₃₁×#389	—	2	1	no	—
14	C₃₁×#389	C₄	1	2	yes	—
15	C₃₁×#389	C₄	1	2	yes	—
16	C₃₁×#384	C₂²	1	1	yes	characteristic
17	C₃₁×#384	C₂²	1	1	yes	characteristic
18	C₃₁×#382	—	2	1	no	—
19	C₄³×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#336	—	2	1	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₃₁×#125	—	4	2	no	—
24	C₃₁×#125	—	4	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	4	no	—
42	C₃₁×C₄⋊C₈	—	4	2	no	—
43	C₃₁×C₄⋊C₈	—	4	2	no	—
44	C₃₁×C₄⋊C₈	Q₈	1	4	yes	—
45	C₄×C₈×C₃₁	—	4	2	no	—
46	C₃₁×#95	—	2	2	no	—
47	C₃₁×#95	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	—	4	2	no	—
60	C₂×C₈×C₃₁	—	4	2	no	—
61	C₂×C₈×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	—	4	4	no	—
67	C₄⋊C₄×C₃₁	—	4	1	no	—
68	C₄⋊C₄×C₃₁	—	4	1	no	—
69	C₄⋊C₄×C₃₁	—	4	1	no	—
70	C₄⋊C₄×C₃₁	—	4	1	no	—
71	C₄⋊C₄×C₃₁	—	4	2	no	—
72	C₄⋊C₄×C₃₁	—	4	1	no	—
73	C₄⋊C₄×C₃₁	—	4	1	no	—
74	C₄²×C₃₁	—	2	2	no	—
75	C₄²×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	—	4	1	no	—
78	C₄²×C₃₁	—	4	1	no	—
79	C₄²×C₃₁	D₈	1	2	yes	—
80	C₄²×C₃₁	SD₁₆	1	2	yes	—
81	C₄²×C₃₁	SD₁₆	1	2	yes	—
82	C₄²×C₃₁	Q₁₆	1	2	yes	—
83	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₃₁	#41	1	2	yes	—
85	#9900	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	#102	1	2	yes	—
104	C₂×C₄×C₃₁	#102	1	2	yes	—
105	C₂×C₄×C₃₁	#103	1	2	yes	—
106	C₂×C₄×C₃₁	#103	1	2	yes	—
107	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#135	1	2	yes	—
116	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
118	C₈×C₃₁	—	4	4	no	—
119	C₈×C₃₁	—	8	2	no	—
120	#2859	C₂×C₃₁	1	1	yes	characteristic
121	#1842	C₂×C₃₁	1	1	yes	characteristic
122	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
123	#1296	C₂×C₃₁	1	2	yes	—
124	#1296	C₂×C₃₁	1	2	yes	—
125	C₂²×C₃₁	#391	1	1	yes	characteristic
126	C₂²×C₃₁	#413	1	1	yes	characteristic
127	C₂²×C₃₁	#413	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
129	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
130	C₂²×C₃₁	#416	1	1	yes	characteristic
131	C₂²×C₃₁	#417	1	1	yes	characteristic
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	2	2	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	—	8	2	no	—
139	C₄×C₃₁	#478	1	2	yes	—
140	C₄×C₃₁	#479	1	2	yes	—
141	C₄×C₃₁	#486	1	1	yes	characteristic
142	C₄×C₃₁	#487	1	1	yes	characteristic
143	C₄×C₃₁	#498	1	1	yes	characteristic
144	C₄×C₃₁	#500	1	1	yes	characteristic
145	C₂×#125	—	2	1	no	—
146	C₂×C₄⋊C₈	—	2	1	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
150	C₂×C₄×C₈	—	2	1	no	—
151	#389	—	2	1	no	—
152	#389	—	2	1	no	—
153	#389	C₄×C₃₁	1	2	yes	—
154	#389	C₄×C₃₁	1	2	yes	—
155	#384	C₂²×C₃₁	1	1	yes	characteristic
156	#384	C₂²×C₃₁	1	1	yes	characteristic
157	#382	—	2	1	no	—
158	C₄³	C₂²×C₃₁	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂×C₃₁	#1540	1	1	yes	characteristic
161	C₂×C₃₁	#1542	1	1	yes	characteristic
162	C₂×C₃₁	#1547	1	1	yes	characteristic
163	C₂×C₃₁	#1550	1	1	yes	characteristic
164	C₂×C₃₁	#1989	1	1	yes	characteristic
165	C₂×C₃₁	#1990	1	1	yes	characteristic
166	C₂×C₃₁	#1998	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	#125	—	4	2	no	—
170	#125	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	4	2	no	—
189	C₄⋊C₈	—	4	2	no	—
190	C₄⋊C₈	Q₈×C₃₁	1	4	yes	—
191	C₄×C₈	—	4	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₃₁	#9900	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	D₈×C₃₁	1	2	yes	—
227	C₄²	SD₁₆×C₃₁	1	2	yes	—
228	C₄²	SD₁₆×C₃₁	1	2	yes	—
229	C₄²	Q₁₆×C₃₁	1	2	yes	—
230	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
231	C₄²	#41×C₃₁	1	2	yes	—
232	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₃₁×#102	1	2	yes	—
250	C₂×C₄	C₃₁×#102	1	2	yes	—
251	C₂×C₄	C₃₁×#103	1	2	yes	—
252	C₂×C₄	C₃₁×#103	1	2	yes	—
253	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
256	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
257	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₃₁×#135	1	2	yes	—
262	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	8	2	no	—
266	C₂²	C₃₁×#391	1	1	yes	characteristic
267	C₂²	C₃₁×#413	1	1	yes	characteristic
268	C₂²	C₃₁×#413	1	1	yes	characteristic
269	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
270	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
271	C₂²	C₃₁×#416	1	1	yes	characteristic
272	C₂²	C₃₁×#417	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	C₃₁×#478	1	2	yes	—
281	C₄	C₃₁×#479	1	2	yes	—
282	C₄	C₃₁×#486	1	1	yes	characteristic
283	C₄	C₃₁×#487	1	1	yes	characteristic
284	C₄	C₃₁×#498	1	1	yes	characteristic
285	C₄	C₃₁×#500	1	1	yes	characteristic
286	C₂	C₃₁×#1540	1	1	yes	characteristic
287	C₂	C₃₁×#1542	1	1	yes	characteristic
288	C₂	C₃₁×#1547	1	1	yes	characteristic
289	C₂	C₃₁×#1550	1	1	yes	characteristic
290	C₂	C₃₁×#1989	1	1	yes	characteristic
291	C₂	C₃₁×#1990	1	1	yes	characteristic
292	C₂	C₃₁×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
85. #9900	C₃₁	direct (N × H)
194. C₃₁	#9900	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×#2859	2
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1296	2
#9900	31

Subgroups — C31×#9900 #21466659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9900 #21466659