Subgroups — C₃₁×#19699 #21476458

Order 7936.

744 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3229	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3226	C₂	1	2	yes	maximal
3	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1723	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1720	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	—	2	2	no	—
9	C₃₁×#533	C₂²	1	1	yes	characteristic
10	C₃₁×#532	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	4	yes	—
13	C₃₁×#498	C₂²	1	4	yes	—
14	C₃₁×#494	—	2	2	no	—
15	C₃₁×#494	C₂²	1	4	yes	—
16	C₃₁×#488	—	2	2	no	—
17	C₃₁×#487	C₂²	1	2	yes	—
18	C₃₁×#487	C₂²	1	4	yes	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#486	C₂²	1	4	yes	—
21	C₃₁×#483	—	2	4	no	—
22	C₃₁×#480	—	2	4	no	—
23	C₃₁×#475	—	2	4	no	—
24	C₃₁×#474	—	2	4	no	—
25	C₃₁×#431	C₂²	1	2	yes	—
26	C₃₁×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#322	C₂²	1	4	yes	—
29	C₂×#41×C₃₁	—	2	1	no	—
30	C₂²×Q₈×C₃₁	—	2	2	no	—
31	C₂×Q₁₆×C₃₁	—	2	8	no	—
32	C₂×Q₁₆×C₃₁	—	4	8	no	—
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₃₁×C₄⋊Q₈	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
38	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
40	C₃₁×#127	C₂³	1	1	yes	characteristic
41	C₃₁×#125	—	2	1	no	—
42	C₃₁×#124	—	2	1	no	—
43	C₃₁×C₂²⋊Q₈	—	2	2	no	—
44	C₃₁×#121	—	2	2	no	—
45	C₄×Q₈×C₃₁	—	2	2	no	—
46	C₄×D₄×C₃₁	—	2	2	no	—
47	C₃₁×#117	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	8	no	—
52	C₃₁×#106	—	2	8	no	—
53	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#103	—	2	8	no	—
56	C₃₁×#102	—	2	4	no	—
57	C₃₁×#102	—	2	8	no	—
58	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
59	C₄×C₈×C₃₁	D₄	1	2	yes	—
60	C₄×C₈×C₃₁	C₂³	1	2	yes	—
61	C₄×C₈×C₃₁	C₂³	1	4	yes	—
62	#41×C₃₁	—	4	4	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	—	2	1	no	—
65	C₂×Q₈×C₃₁	—	2	4	no	—
66	C₂×Q₈×C₃₁	—	4	2	no	—
67	C₂×Q₈×C₃₁	—	4	8	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	—	4	1	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₃₁	—	4	16	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	1	no	—
88	C₂²⋊C₄×C₃₁	—	4	2	no	—
89	C₄²×C₃₁	—	4	1	no	—
90	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
93	#19699	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	#127	1	1	yes	characteristic
96	Q₈×C₃₁	—	4	8	no	—
97	Q₈×C₃₁	—	4	2	no	—
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	8	4	no	—
100	D₄×C₃₁	—	4	2	no	—
101	D₄×C₃₁	—	4	4	no	—
102	D₄×C₃₁	—	8	2	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	1	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	8	4	no	—
108	C₂×C₄×C₃₁	—	8	2	no	—
109	C₂×C₄×C₃₁	#127	1	2	yes	—
110	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
111	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
114	C₈×C₃₁	—	2	4	no	—
115	C₈×C₃₁	—	2	8	no	—
116	#3229	C₂×C₃₁	1	2	yes	complemented
117	#3226	C₂×C₃₁	1	2	yes	—
118	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#3131	C₂×C₃₁	1	1	yes	characteristic
120	#1723	C₂×C₃₁	1	4	yes	complemented
121	#1720	C₂×C₃₁	1	4	yes	complemented
122	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	8	1	no	—
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	#496	1	2	yes	—
127	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	8	1	no	—
130	C₄×C₃₁	—	8	4	no	—
131	C₄×C₃₁	—	8	2	no	—
132	C₄×C₃₁	#496	1	2	yes	—
133	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₄×C₃₁	#578	1	4	yes	—
135	C₂²×Q₁₆	—	2	2	no	—
136	#533	C₂²×C₃₁	1	1	yes	characteristic
137	#532	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
139	#500	C₂²×C₃₁	1	4	yes	—
140	#498	C₂²×C₃₁	1	4	yes	—
141	#494	—	2	2	no	—
142	#494	C₂²×C₃₁	1	4	yes	—
143	#488	—	2	2	no	—
144	#487	C₂²×C₃₁	1	2	yes	—
145	#487	C₂²×C₃₁	1	4	yes	—
146	#486	C₂²×C₃₁	1	2	yes	—
147	#486	C₂²×C₃₁	1	4	yes	—
148	#483	—	2	4	no	—
149	#480	—	2	4	no	—
150	#475	—	2	4	no	—
151	#474	—	2	4	no	—
152	#431	C₂²×C₃₁	1	2	yes	—
153	#416	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	#322	C₂²×C₃₁	1	4	yes	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	—	16	1	no	—
158	C₂×C₃₁	C₂×#496	1	1	yes	characteristic
159	C₂×C₃₁	#3156	1	2	yes	—
160	C₂×#41	—	2	1	no	—
161	C₂²×Q₈	—	2	2	no	—
162	C₂×Q₁₆	—	2	8	no	—
163	C₂×Q₁₆	—	4	8	no	—
164	C₂²×C₈	D₄×C₃₁	1	2	yes	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
169	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
171	#127	C₂³×C₃₁	1	1	yes	characteristic
172	#125	—	2	1	no	—
173	#124	—	2	1	no	—
174	C₂²⋊Q₈	—	2	2	no	—
175	#121	—	2	2	no	—
176	C₄×Q₈	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	#117	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
182	#107	—	2	8	no	—
183	#106	—	2	8	no	—
184	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
185	#103	—	2	4	no	—
186	#103	—	2	8	no	—
187	#102	—	2	4	no	—
188	#102	—	2	8	no	—
189	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
190	C₄×C₈	D₄×C₃₁	1	2	yes	—
191	C₄×C₈	C₂³×C₃₁	1	2	yes	—
192	C₄×C₈	C₂³×C₃₁	1	4	yes	—
193	C₃₁	#19699	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	Q₁₆	—	4	16	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
210	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₃₁	1	2	yes	—
224	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
225	C₂³	—	4	1	no	—
226	C₂³	C₃₁×#127	1	1	yes	characteristic
227	Q₈	—	4	8	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	8	4	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₃₁×#127	1	2	yes	—
241	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
242	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
243	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
245	C₈	—	2	4	no	—
246	C₈	—	2	8	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₃₁×#496	1	2	yes	—
251	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	1	no	—
254	C₄	—	8	4	no	—
255	C₄	—	8	2	no	—
256	C₄	C₃₁×#496	1	2	yes	—
257	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
258	C₄	C₃₁×#578	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	16	1	no	—
261	C₂	C₂×C₃₁×#496	1	1	yes	characteristic
262	C₂	C₃₁×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
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. C₃₁×#3229	C₂	semidirect
3. C₂×C₃₁×#494	C₂	semidirect
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1720	C₂	semidirect
7. C₃₁×#1452	C₂	semidirect
93. #19699	C₃₁	direct (N × H)
116. #3229	C₂×C₃₁	semidirect
118. C₂×#494	C₂×C₃₁	semidirect
120. #1723	C₂×C₃₁	semidirect
121. #1720	C₂×C₃₁	semidirect
122. #1452	C₂×C₃₁	semidirect
193. C₃₁	#19699	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×#3229	2
C₃₁×#3226	2
C₂×C₃₁×#494	2
C₃₁×#3131	2
C₃₁×#1723	2
C₃₁×#1720	2
C₃₁×#1452	2
#19699	31

Subgroups — C31×#19699 #21476458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19699 #21476458