Subgroups — C₃₁×#9063 #21465822

Order 7936.

654 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	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1982	C₂	1	2	yes	maximal
4	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂²×SD₁₆×C₃₁	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₃₁×#492	—	2	2	no	—
10	C₃₁×#492	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×#102	—	2	2	no	—
13	C₂×C₃₁×#102	—	2	2	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#391	—	2	2	no	—
16	C₃₁×#389	—	2	2	no	—
17	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#333	—	2	2	no	—
20	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
21	C₂²×D₄×C₃₁	—	2	2	no	—
22	C₂²×D₄×C₃₁	—	2	1	no	—
23	C₂×SD₁₆×C₃₁	—	2	8	no	—
24	C₂×SD₁₆×C₃₁	—	4	4	no	—
25	C₂×SD₁₆×C₃₁	—	4	4	no	—
26	C₂²×C₈×C₃₁	—	2	1	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₃₁×C₄⋊Q₈	—	2	2	no	—
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
31	C₃₁×#127	—	2	1	no	—
32	C₃₁×#127	—	2	2	no	—
33	C₃₁×#127	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	—	2	2	no	—
39	C₂×C₄²×C₃₁	—	2	2	no	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×#102	—	4	4	no	—
44	C₃₁×#102	—	4	4	no	—
45	C₄×C₈×C₃₁	—	2	1	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₃₁×#95	—	2	4	no	—
48	C₃₁×#95	—	2	2	no	—
49	C₂⁴×C₃₁	—	4	1	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	4	2	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	4	2	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	—	4	2	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	SD₁₆×C₃₁	—	4	16	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	8	2	no	—
82	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
83	C₄²×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	—	8	2	no	—
86	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₁	#41	1	2	yes	—
89	#9063	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	—	8	2	no	—
92	C₂³×C₃₁	—	8	2	no	—
93	C₂³×C₃₁	—	8	1	no	—
94	C₂³×C₃₁	—	8	1	no	—
95	C₂³×C₃₁	#120	1	1	yes	characteristic
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	—	8	2	no	—
98	D₄×C₃₁	—	4	4	no	—
99	D₄×C₃₁	—	4	4	no	—
100	D₄×C₃₁	—	4	4	no	—
101	D₄×C₃₁	—	8	2	no	—
102	D₄×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	8	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	#121	1	2	yes	—
118	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₃₁	#135	1	2	yes	—
121	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₃₁	#137	1	2	yes	—
123	C₈×C₃₁	—	2	4	no	—
124	C₈×C₃₁	—	8	2	no	—
125	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
126	#2369	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#1982	C₂×C₃₁	1	2	yes	—
128	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1279	C₂×C₃₁	1	2	yes	complemented
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	—	8	2	no	—
132	C₂²×C₃₁	—	16	2	no	—
133	C₂²×C₃₁	—	16	1	no	—
134	C₂²×C₃₁	—	16	1	no	—
135	C₂²×C₃₁	#386	1	1	yes	characteristic
136	C₂²×C₃₁	#448	1	2	yes	—
137	C₂²×C₃₁	#449	1	2	yes	—
138	C₂²×C₃₁	#453	1	1	yes	characteristic
139	C₂²×C₃₁	#453	1	1	yes	characteristic
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	4	4	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	8	2	no	—
145	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
146	C₄×C₃₁	#440	1	2	yes	—
147	C₄×C₃₁	#460	1	2	yes	—
148	C₄×C₃₁	#464	1	2	yes	—
149	C₂²×SD₁₆	—	2	2	no	—
150	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
152	#492	—	2	2	no	—
153	#492	C₄×C₃₁	1	4	yes	complemented
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×#102	—	2	2	no	—
156	C₂×#102	—	2	2	no	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	#391	—	2	2	no	—
159	#389	—	2	2	no	—
160	C₄×C₄⋊C₄	—	2	2	no	—
161	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	#333	—	2	2	no	—
163	C₂×C₃₁	—	16	2	no	—
164	C₂×C₃₁	#1622	1	2	yes	—
165	C₂×C₃₁	#1623	1	1	yes	characteristic
166	C₂×C₃₁	#1624	1	1	yes	characteristic
167	C₂×C₃₁	#1880	1	2	yes	—
168	C₂×C₃₁	#1887	1	1	yes	characteristic
169	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂×SD₁₆	—	2	8	no	—
173	C₂×SD₁₆	—	4	4	no	—
174	C₂×SD₁₆	—	4	4	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
179	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
180	#127	—	2	1	no	—
181	#127	—	2	2	no	—
182	#127	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
186	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	#102	—	4	4	no	—
193	#102	—	4	4	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
196	#95	—	2	4	no	—
197	#95	—	2	2	no	—
198	C₃₁	#9063	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	—	4	1	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
219	SD₁₆	—	4	16	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	8	2	no	—
236	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₄²	#41×C₃₁	1	2	yes	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	C₃₁×#120	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
266	C₂×C₄	C₃₁×#121	1	2	yes	—
267	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
269	C₂×C₄	C₃₁×#135	1	2	yes	—
270	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄	C₃₁×#137	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	1	no	—
278	C₂²	—	16	1	no	—
279	C₂²	C₃₁×#386	1	1	yes	characteristic
280	C₂²	C₃₁×#448	1	2	yes	—
281	C₂²	C₃₁×#449	1	2	yes	—
282	C₂²	C₃₁×#453	1	1	yes	characteristic
283	C₂²	C₃₁×#453	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
290	C₄	C₃₁×#440	1	2	yes	—
291	C₄	C₃₁×#460	1	2	yes	—
292	C₄	C₃₁×#464	1	2	yes	—
293	C₂	—	16	2	no	—
294	C₂	C₃₁×#1622	1	2	yes	—
295	C₂	C₃₁×#1623	1	1	yes	characteristic
296	C₂	C₃₁×#1624	1	1	yes	characteristic
297	C₂	C₃₁×#1880	1	2	yes	—
298	C₂	C₃₁×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2369	C₂	semidirect
4. C₂×C₃₁×#333	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
10. C₃₁×#492	C₄	semidirect
82. C₄⋊C₄×C₃₁	SD₁₆	semidirect
89. #9063	C₃₁	direct (N × H)
126. #2369	C₂×C₃₁	semidirect
128. C₂×#333	C₂×C₃₁	semidirect
129. #1279	C₂×C₃₁	semidirect
153. #492	C₄×C₃₁	semidirect
198. C₃₁	#9063	direct (N × H)
232. C₄⋊C₄	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#492	2
C₃₁×#2369	2
C₃₁×#1982	2
C₂×C₃₁×#333	2
C₃₁×#1279	2
#9063	31

Subgroups — C31×#9063 #21465822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9063 #21465822