Subgroups — C₃₁×#61883 #21518642

Order 7936.

1958 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	C₃₁×#3497	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3392	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3359	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3206	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3103	C₂	1	6	yes	maximal, complemented
7	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
9	C₃₁×#557	—	2	4	no	—
10	C₃₁×#557	C₂²	1	8	yes	—
11	C₃₁×#550	C₂²	1	4	yes	—
12	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#548	—	2	4	no	—
14	C₃₁×#537	—	2	3	no	—
15	C₃₁×#532	C₂²	1	6	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₂×C₃₁×#125	—	2	3	no	—
18	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#492	C₂²	1	12	yes	—
22	C₃₁×#487	C₂²	1	12	yes	—
23	C₃₁×#481	C₂²	1	4	yes	—
24	C₃₁×#475	C₂²	1	12	yes	—
25	C₃₁×#461	C₂²	1	24	yes	—
26	C₃₁×#460	C₂²	1	12	yes	—
27	C₃₁×#448	C₂²	1	12	yes	—
28	C₈×Q₈×C₃₁	C₂²	1	8	yes	—
29	C₄×SD₁₆×C₃₁	C₂²	1	24	yes	—
30	C₃₁×#431	C₂²	1	6	yes	—
31	C₃₁×2₋¹⁺⁴	—	4	8	no	—
32	C₂×#41×C₃₁	—	2	4	no	—
33	C₂×#41×C₃₁	—	2	6	no	—
34	C₂²×Q₈×C₃₁	—	2	4	no	—
35	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
37	C₂×SD₁₆×C₃₁	C₂³	1	24	yes	—
38	C₃₁×C₄⋊Q₈	—	2	6	no	—
39	C₃₁×C₄⋊Q₈	C₂³	1	12	yes	—
40	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₃₁×#127	C₂³	1	6	yes	—
42	C₃₁×#125	—	2	24	no	—
43	C₃₁×#124	—	2	6	no	—
44	C₃₁×C₂²⋊Q₈	—	2	12	no	—
45	C₃₁×#121	—	2	6	no	—
46	C₃₁×#121	—	2	12	no	—
47	C₃₁×#121	C₂³	1	4	yes	—
48	C₄×Q₈×C₃₁	—	2	8	no	—
49	C₄×Q₈×C₃₁	—	2	6	no	—
50	C₄×Q₈×C₃₁	D₄	1	4	yes	—
51	C₄×Q₈×C₃₁	C₂³	1	12	yes	—
52	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
53	C₄×D₄×C₃₁	—	2	6	no	—
54	C₄×D₄×C₃₁	C₂³	1	12	yes	—
55	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	—	2	3	no	—
61	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
62	C₃₁×#106	C₂³	1	8	yes	—
63	C₃₁×C₄⋊C₈	C₂³	1	12	yes	—
64	C₃₁×#103	C₂³	1	24	yes	—
65	C₃₁×#102	C₂³	1	8	yes	—
66	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₃₁	C₂³	1	12	yes	—
68	#41×C₃₁	—	4	16	no	—
69	#41×C₃₁	—	4	24	no	—
70	C₂×Q₈×C₃₁	—	2	3	no	—
71	C₂×Q₈×C₃₁	—	2	6	no	—
72	C₂×Q₈×C₃₁	—	4	6	no	—
73	C₂×Q₈×C₃₁	—	4	16	no	—
74	C₂×Q₈×C₃₁	SD₁₆	1	8	yes	complemented
75	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
76	C₂×Q₈×C₃₁	C₂⁴	1	6	yes	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	2	6	no	—
79	C₂×D₄×C₃₁	—	2	6	no	—
80	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₃₁	—	2	1	no	—
82	C₂²×C₄×C₃₁	—	2	3	no	—
83	C₂²×C₄×C₃₁	—	2	12	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	SD₁₆×C₃₁	—	4	24	no	—
88	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
89	C₄⋊C₄×C₃₁	—	2	12	no	—
90	C₄⋊C₄×C₃₁	—	2	12	no	—
91	C₄⋊C₄×C₃₁	—	2	12	no	—
92	C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₄⋊C₄×C₃₁	—	2	6	no	—
94	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
95	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
96	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
97	C₂²⋊C₄×C₃₁	—	2	4	no	—
98	C₂²⋊C₄×C₃₁	—	2	12	no	—
99	C₄²×C₃₁	—	2	12	no	—
100	C₄²×C₃₁	C₂×D₄	1	6	yes	—
101	C₄²×C₃₁	C₂⁴	1	6	yes	—
102	#61883	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	4	no	—
104	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	2	12	no	—
106	Q₈×C₃₁	—	4	6	no	—
107	Q₈×C₃₁	—	4	12	no	—
108	Q₈×C₃₁	C₂×SD₁₆	1	16	yes	—
109	D₄×C₃₁	—	2	8	no	—
110	D₄×C₃₁	—	4	8	no	—
111	D₄×C₃₁	—	4	12	no	—
112	D₄×C₃₁	—	8	6	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	6	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	4	3	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	6	no	—
120	C₂×C₄×C₃₁	—	4	24	no	—
121	C₂×C₄×C₃₁	C₂×SD₁₆	1	12	yes	—
122	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
126	C₈×C₃₁	—	2	8	no	—
127	#3497	C₂×C₃₁	1	1	yes	characteristic
128	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#3392	C₂×C₃₁	1	8	yes	complemented
130	#3359	C₂×C₃₁	1	8	yes	complemented
131	#3206	C₂×C₃₁	1	6	yes	complemented
132	#3103	C₂×C₃₁	1	6	yes	complemented
133	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	4	4	no	—
137	C₂²×C₃₁	—	4	4	no	—
138	C₂²×C₃₁	—	4	4	no	—
139	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
140	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₃₁	—	2	6	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	4	6	no	—
145	C₄×C₃₁	—	4	4	no	—
146	C₄×C₃₁	C₂²×SD₁₆	1	12	yes	—
147	C₄×C₃₁	#585	1	2	yes	—
148	C₂×2₋¹⁺⁴	—	2	1	no	—
149	#557	—	2	4	no	—
150	#557	C₂²×C₃₁	1	8	yes	—
151	#550	C₂²×C₃₁	1	4	yes	—
152	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
153	#548	—	2	4	no	—
154	#537	—	2	3	no	—
155	#532	C₂²×C₃₁	1	6	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
157	C₂×#125	—	2	3	no	—
158	C₂×C₄×Q₈	—	2	1	no	—
159	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
160	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	#492	C₂²×C₃₁	1	12	yes	—
162	#487	C₂²×C₃₁	1	12	yes	—
163	#481	C₂²×C₃₁	1	4	yes	—
164	#475	C₂²×C₃₁	1	12	yes	—
165	#461	C₂²×C₃₁	1	24	yes	—
166	#460	C₂²×C₃₁	1	12	yes	—
167	#448	C₂²×C₃₁	1	12	yes	—
168	C₈×Q₈	C₂²×C₃₁	1	8	yes	—
169	C₄×SD₁₆	C₂²×C₃₁	1	24	yes	—
170	#431	C₂²×C₃₁	1	6	yes	—
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	—	8	4	no	—
173	C₂×C₃₁	#3487	1	1	yes	characteristic
174	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
175	C₂×C₃₁	#3586	1	1	yes	characteristic
176	2₋¹⁺⁴	—	4	8	no	—
177	C₂×#41	—	2	4	no	—
178	C₂×#41	—	2	6	no	—
179	C₂²×Q₈	—	2	4	no	—
180	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
182	C₂×SD₁₆	C₂³×C₃₁	1	24	yes	—
183	C₄⋊Q₈	—	2	6	no	—
184	C₄⋊Q₈	C₂³×C₃₁	1	12	yes	—
185	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
186	#127	C₂³×C₃₁	1	6	yes	—
187	#125	—	2	24	no	—
188	#124	—	2	6	no	—
189	C₂²⋊Q₈	—	2	12	no	—
190	#121	—	2	6	no	—
191	#121	—	2	12	no	—
192	#121	C₂³×C₃₁	1	4	yes	—
193	C₄×Q₈	—	2	8	no	—
194	C₄×Q₈	—	2	6	no	—
195	C₄×Q₈	D₄×C₃₁	1	4	yes	—
196	C₄×Q₈	C₂³×C₃₁	1	12	yes	—
197	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
198	C₄×D₄	—	2	6	no	—
199	C₄×D₄	C₂³×C₃₁	1	12	yes	—
200	C₂×C₄⋊C₄	—	2	3	no	—
201	C₂×C₄⋊C₄	—	2	3	no	—
202	C₂×C₄⋊C₄	—	2	3	no	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	—	2	3	no	—
206	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
207	#106	C₂³×C₃₁	1	8	yes	—
208	C₄⋊C₈	C₂³×C₃₁	1	12	yes	—
209	#103	C₂³×C₃₁	1	24	yes	—
210	#102	C₂³×C₃₁	1	8	yes	—
211	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
212	C₄×C₈	C₂³×C₃₁	1	12	yes	—
213	C₃₁	#61883	1	1	yes	characteristic, direct product factor, Sylow
214	#41	—	4	16	no	—
215	#41	—	4	24	no	—
216	C₂×Q₈	—	2	3	no	—
217	C₂×Q₈	—	2	6	no	—
218	C₂×Q₈	—	4	6	no	—
219	C₂×Q₈	—	4	16	no	—
220	C₂×Q₈	SD₁₆×C₃₁	1	8	yes	complemented
221	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
222	C₂×Q₈	C₂⁴×C₃₁	1	6	yes	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	6	no	—
225	C₂×D₄	—	2	6	no	—
226	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	3	no	—
229	C₂²×C₄	—	2	12	no	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
232	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
233	SD₁₆	—	4	24	no	—
234	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
241	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
242	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	2	12	no	—
245	C₄²	—	2	12	no	—
246	C₄²	C₂×D₄×C₃₁	1	6	yes	—
247	C₄²	C₂⁴×C₃₁	1	6	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	Q₈	—	2	12	no	—
251	Q₈	—	4	6	no	—
252	Q₈	—	4	12	no	—
253	Q₈	C₂×SD₁₆×C₃₁	1	16	yes	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	12	no	—
257	D₄	—	8	6	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	6	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	C₂×SD₁₆×C₃₁	1	12	yes	—
267	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
269	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
270	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
278	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	6	no	—
281	C₄	—	4	6	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	C₂²×SD₁₆×C₃₁	1	12	yes	—
285	C₄	C₃₁×#585	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	8	4	no	—
288	C₂	C₃₁×#3487	1	1	yes	characteristic
289	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
290	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3392	C₂	semidirect
4. C₃₁×#3359	C₂	semidirect
5. C₃₁×#3206	C₂	semidirect
6. C₃₁×#3103	C₂	semidirect
7. C₃₁×#2982	C₂	semidirect
74. C₂×Q₈×C₃₁	SD₁₆	semidirect
102. #61883	C₃₁	direct (N × H)
128. C₂×#557	C₂×C₃₁	semidirect
129. #3392	C₂×C₃₁	semidirect
130. #3359	C₂×C₃₁	semidirect
131. #3206	C₂×C₃₁	semidirect
132. #3103	C₂×C₃₁	semidirect
133. #2982	C₂×C₃₁	semidirect
213. C₃₁	#61883	direct (N × H)
220. C₂×Q₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3497	2
C₂×C₃₁×#557	2
C₃₁×#3392	2
C₃₁×#3359	2
C₃₁×#3206	2
C₃₁×#3103	2
C₃₁×#2982	2
#61883	31

Subgroups — C31×#61883 #21518642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61883 #21518642