Subgroups — C₃₁×#19003 #21475762

Order 7936.

808 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	C₂×C₃₁×#564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3234	C₂	1	2	yes	maximal
3	C₃₁×#3142	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1663	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1658	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1534	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1528	C₂	1	2	yes	maximal, complemented
8	C₃₁×#564	—	2	4	no	—
9	C₃₁×#564	—	2	2	no	—
10	C₃₁×#564	C₂²	1	4	yes	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	—	2	2	no	—
13	C₂×C₃₁×#125	—	2	2	no	—
14	C₂×C₃₁×#125	—	2	2	no	—
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₂²	1	2	yes	—
17	C₃₁×#501	C₂²	1	4	yes	—
18	C₃₁×#491	C₂²	1	2	yes	—
19	C₃₁×#490	C₂²	1	2	yes	—
20	C₃₁×#490	C₂²	1	4	yes	—
21	C₃₁×#488	—	2	2	no	—
22	C₃₁×#481	—	2	2	no	—
23	C₃₁×#480	—	2	2	no	—
24	C₃₁×#478	—	2	4	no	—
25	C₃₁×#432	C₂²	1	2	yes	—
26	C₃₁×#417	—	2	2	no	—
27	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#330	C₂²	1	8	yes	—
29	C₃₁×#330	C₂²	1	4	yes	—
30	C₂×#41×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
33	C₃₁×#127	C₂³	1	1	yes	characteristic
34	C₃₁×#125	—	2	8	no	—
35	C₃₁×#125	—	2	8	no	—
36	C₃₁×#125	—	2	8	no	—
37	C₃₁×#125	—	2	8	no	—
38	C₃₁×#125	—	2	8	no	—
39	C₃₁×#125	—	2	8	no	—
40	C₃₁×#125	—	2	2	no	—
41	C₃₁×#125	—	2	2	no	—
42	C₃₁×#125	—	2	2	no	—
43	C₃₁×#125	D₄	1	4	yes	—
44	C₃₁×#125	D₄	1	2	yes	—
45	C₃₁×#125	C₂³	1	2	yes	—
46	C₃₁×#125	C₂³	1	4	yes	—
47	C₃₁×#124	—	2	1	no	—
48	C₃₁×#121	—	2	2	no	—
49	C₄×D₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	—	2	4	no	—
62	C₃₁×#106	—	2	4	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×#103	—	2	4	no	—
65	C₃₁×#102	—	2	4	no	—
66	C₃₁×#102	—	2	4	no	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×C₂²⋊C₈	—	2	2	no	—
69	C₃₁×#97	C₂³	1	2	yes	—
70	C₃₁×#97	C₂³	1	4	yes	—
71	C₄×C₈×C₃₁	—	2	1	no	—
72	#41×C₃₁	—	4	4	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	M₁₆×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	8	no	—
87	C₄⋊C₄×C₃₁	—	2	8	no	—
88	C₄⋊C₄×C₃₁	—	2	8	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₄²×C₃₁	—	2	8	no	—
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	2	yes	—
106	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
107	#19003	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	4	1	no	—
109	C₂³×C₃₁	#120	1	1	yes	characteristic
110	Q₈×C₃₁	—	4	2	no	—
111	D₄×C₃₁	—	4	4	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	8	2	no	—
124	C₂×C₄×C₃₁	#120	1	2	yes	—
125	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₃₁	—	4	4	no	—
129	C₈×C₃₁	—	4	2	no	—
130	C₂×#564	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#3234	C₂×C₃₁	1	2	yes	—
132	#3142	C₂×C₃₁	1	1	yes	characteristic
133	#1663	C₂×C₃₁	1	4	yes	complemented
134	#1658	C₂×C₃₁	1	4	yes	complemented
135	#1534	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#1528	C₂×C₃₁	1	2	yes	complemented
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	8	1	no	—
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	#454	1	2	yes	—
141	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	4	4	no	—
148	C₄×C₃₁	—	4	4	no	—
149	C₄×C₃₁	—	8	1	no	—
150	#564	—	2	4	no	—
151	#564	—	2	2	no	—
152	#564	C₂²×C₃₁	1	4	yes	—
153	#532	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×#125	—	2	2	no	—
155	C₂×#125	—	2	2	no	—
156	C₂×#125	—	2	2	no	—
157	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×#125	C₂²×C₃₁	1	2	yes	—
159	#501	C₂²×C₃₁	1	4	yes	—
160	#491	C₂²×C₃₁	1	2	yes	—
161	#490	C₂²×C₃₁	1	2	yes	—
162	#490	C₂²×C₃₁	1	4	yes	—
163	#488	—	2	2	no	—
164	#481	—	2	2	no	—
165	#480	—	2	2	no	—
166	#478	—	2	4	no	—
167	#432	C₂²×C₃₁	1	2	yes	—
168	#417	—	2	2	no	—
169	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
170	#330	C₂²×C₃₁	1	8	yes	—
171	#330	C₂²×C₃₁	1	4	yes	—
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	—	16	1	no	—
174	C₂×C₃₁	C₂×#454	1	1	yes	characteristic
175	C₂×C₃₁	#3020	1	2	yes	—
176	C₂×#41	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
179	#127	C₂³×C₃₁	1	1	yes	characteristic
180	#125	—	2	8	no	—
181	#125	—	2	8	no	—
182	#125	—	2	8	no	—
183	#125	—	2	8	no	—
184	#125	—	2	8	no	—
185	#125	—	2	8	no	—
186	#125	—	2	2	no	—
187	#125	—	2	2	no	—
188	#125	—	2	2	no	—
189	#125	D₄×C₃₁	1	4	yes	—
190	#125	D₄×C₃₁	1	2	yes	—
191	#125	C₂³×C₃₁	1	2	yes	—
192	#125	C₂³×C₃₁	1	4	yes	—
193	#124	—	2	1	no	—
194	#121	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#106	—	2	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	#103	—	2	4	no	—
211	#102	—	2	4	no	—
212	#102	—	2	4	no	—
213	#102	—	2	4	no	—
214	C₂²⋊C₈	—	2	2	no	—
215	#97	C₂³×C₃₁	1	2	yes	—
216	#97	C₂³×C₃₁	1	4	yes	—
217	C₄×C₈	—	2	1	no	—
218	C₃₁	#19003	1	1	yes	characteristic, direct product factor, Sylow
219	#41	—	4	4	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
247	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
248	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	2	8	no	—
251	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₃₁	1	2	yes	—
253	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₂³	—	4	1	no	—
255	C₂³	C₃₁×#120	1	1	yes	characteristic
256	Q₈	—	4	2	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	8	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	C₃₁×#120	1	2	yes	—
271	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
273	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
274	C₈	—	4	4	no	—
275	C₈	—	4	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	8	1	no	—
278	C₂²	—	8	2	no	—
279	C₂²	C₃₁×#454	1	2	yes	—
280	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	16	1	no	—
291	C₂	C₂×C₃₁×#454	1	1	yes	characteristic
292	C₂	C₃₁×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#564	C₂	semidirect
4. C₃₁×#1663	C₂	semidirect
5. C₃₁×#1658	C₂	semidirect
6. C₃₁×#1534	C₂	semidirect
7. C₃₁×#1528	C₂	semidirect
107. #19003	C₃₁	direct (N × H)
130. C₂×#564	C₂×C₃₁	semidirect
133. #1663	C₂×C₃₁	semidirect
134. #1658	C₂×C₃₁	semidirect
135. #1534	C₂×C₃₁	semidirect
136. #1528	C₂×C₃₁	semidirect
218. C₃₁	#19003	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×#564	2
C₃₁×#3234	2
C₃₁×#3142	2
C₃₁×#1663	2
C₃₁×#1658	2
C₃₁×#1534	2
C₃₁×#1528	2
#19003	31

Subgroups — C31×#19003 #21475762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19003 #21475762