Subgroups — C₃₁×#19703 #21476462

Order 7936.

704 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	2	yes	maximal
2	C₃₁×#3209	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1720	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1716	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#533	C₂²	1	1	yes	characteristic
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃₁×#498	—	2	2	no	—
12	C₃₁×#498	C₂²	1	4	yes	—
13	C₃₁×#498	C₂²	1	4	yes	—
14	C₃₁×#494	C₂²	1	4	yes	—
15	C₃₁×#488	—	2	2	no	—
16	C₃₁×#487	C₂²	1	2	yes	—
17	C₃₁×#487	C₂²	1	4	yes	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#486	C₂²	1	4	yes	—
20	C₃₁×#481	—	2	4	no	—
21	C₃₁×#476	—	2	4	no	—
22	C₃₁×#462	—	2	4	no	—
23	C₃₁×#451	—	2	4	no	—
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₂×C₃₁×#106	—	2	2	no	—
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	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₃₁×C₄⋊Q₈	—	2	2	no	—
34	C₃₁×C₄⋊Q₈	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
37	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
39	C₃₁×#127	C₂³	1	1	yes	characteristic
40	C₃₁×#125	—	2	1	no	—
41	C₃₁×#124	—	2	1	no	—
42	C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₃₁×#121	—	2	2	no	—
44	C₄×Q₈×C₃₁	—	2	2	no	—
45	C₄×D₄×C₃₁	—	2	2	no	—
46	C₃₁×#117	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#106	—	2	8	no	—
51	C₃₁×#106	—	2	8	no	—
52	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
53	C₃₁×#103	—	2	4	no	—
54	C₃₁×#103	—	2	8	no	—
55	C₃₁×#102	—	2	4	no	—
56	C₃₁×#102	—	2	8	no	—
57	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
58	C₄×C₈×C₃₁	D₄	1	2	yes	—
59	C₄×C₈×C₃₁	C₂³	1	2	yes	—
60	C₄×C₈×C₃₁	C₂³	1	4	yes	—
61	#41×C₃₁	—	4	4	no	—
62	C₂×Q₈×C₃₁	—	2	1	no	—
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	4	2	no	—
66	C₂×Q₈×C₃₁	—	8	4	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	2	1	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	1	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	8	8	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	1	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	—	4	1	no	—
89	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂×D₄	1	2	yes	—
91	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
92	#19703	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	#127	1	1	yes	characteristic
95	Q₈×C₃₁	—	4	2	no	—
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	—	4	8	no	—
98	Q₈×C₃₁	—	8	4	no	—
99	D₄×C₃₁	—	4	2	no	—
100	D₄×C₃₁	—	4	4	no	—
101	D₄×C₃₁	—	8	2	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	—	8	2	no	—
108	C₂×C₄×C₃₁	#127	1	2	yes	—
109	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#137	1	4	yes	—
111	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
113	C₈×C₃₁	—	2	4	no	—
114	C₈×C₃₁	—	2	8	no	—
115	#3227	C₂×C₃₁	1	2	yes	—
116	#3209	C₂×C₃₁	1	2	yes	complemented
117	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#3131	C₂×C₃₁	1	1	yes	characteristic
119	#1720	C₂×C₃₁	1	4	yes	complemented
120	#1716	C₂×C₃₁	1	4	yes	complemented
121	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	—	8	1	no	—
124	C₂²×C₃₁	—	8	2	no	—
125	C₂²×C₃₁	#497	1	2	yes	—
126	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	8	1	no	—
129	C₄×C₃₁	—	8	4	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	#497	1	2	yes	—
132	C₄×C₃₁	C₂×#137	1	2	yes	—
133	C₄×C₃₁	#577	1	4	yes	—
134	#533	C₂²×C₃₁	1	1	yes	characteristic
135	#532	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
137	#498	—	2	2	no	—
138	#498	C₂²×C₃₁	1	4	yes	—
139	#498	C₂²×C₃₁	1	4	yes	—
140	#494	C₂²×C₃₁	1	4	yes	—
141	#488	—	2	2	no	—
142	#487	C₂²×C₃₁	1	2	yes	—
143	#487	C₂²×C₃₁	1	4	yes	—
144	#486	C₂²×C₃₁	1	2	yes	—
145	#486	C₂²×C₃₁	1	4	yes	—
146	#481	—	2	4	no	—
147	#476	—	2	4	no	—
148	#462	—	2	4	no	—
149	#451	—	2	4	no	—
150	#431	C₂²×C₃₁	1	2	yes	—
151	C₂×#106	—	2	2	no	—
152	#416	—	2	2	no	—
153	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	#322	C₂²×C₃₁	1	4	yes	—
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	—	16	1	no	—
157	C₂×C₃₁	C₂×#497	1	1	yes	characteristic
158	C₂×C₃₁	#3154	1	2	yes	—
159	C₂×#41	—	2	1	no	—
160	C₂²×Q₈	—	2	2	no	—
161	C₂×Q₁₆	—	2	8	no	—
162	C₂²×C₈	D₄×C₃₁	1	2	yes	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
167	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
169	#127	C₂³×C₃₁	1	1	yes	characteristic
170	#125	—	2	1	no	—
171	#124	—	2	1	no	—
172	C₂²⋊Q₈	—	2	2	no	—
173	#121	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	#117	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
180	#106	—	2	8	no	—
181	#106	—	2	8	no	—
182	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
183	#103	—	2	4	no	—
184	#103	—	2	8	no	—
185	#102	—	2	4	no	—
186	#102	—	2	8	no	—
187	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
188	C₄×C₈	D₄×C₃₁	1	2	yes	—
189	C₄×C₈	C₂³×C₃₁	1	2	yes	—
190	C₄×C₈	C₂³×C₃₁	1	4	yes	—
191	C₃₁	#19703	1	1	yes	characteristic, direct product factor, Sylow
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	8	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	Q₁₆	—	8	8	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
208	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₃₁	1	2	yes	—
222	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
223	C₂³	—	4	1	no	—
224	C₂³	C₃₁×#127	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	4	no	—
227	Q₈	—	4	8	no	—
228	Q₈	—	8	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	D₄	—	8	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	C₃₁×#127	1	2	yes	—
239	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
240	C₂×C₄	C₃₁×#137	1	4	yes	—
241	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
243	C₈	—	2	4	no	—
244	C₈	—	2	8	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₃₁×#497	1	2	yes	—
249	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	8	1	no	—
252	C₄	—	8	4	no	—
253	C₄	—	8	2	no	—
254	C₄	C₃₁×#497	1	2	yes	—
255	C₄	C₂×C₃₁×#137	1	2	yes	—
256	C₄	C₃₁×#577	1	4	yes	—
257	C₂	—	2	2	no	—
258	C₂	—	16	1	no	—
259	C₂	C₂×C₃₁×#497	1	1	yes	characteristic
260	C₂	C₃₁×#3154	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
2. C₃₁×#3209	C₂	semidirect
3. C₂×C₃₁×#498	C₂	semidirect
5. C₃₁×#1720	C₂	semidirect
6. C₃₁×#1716	C₂	semidirect
7. C₃₁×#1452	C₂	semidirect
92. #19703	C₃₁	direct (N × H)
116. #3209	C₂×C₃₁	semidirect
117. C₂×#498	C₂×C₃₁	semidirect
119. #1720	C₂×C₃₁	semidirect
120. #1716	C₂×C₃₁	semidirect
121. #1452	C₂×C₃₁	semidirect
191. C₃₁	#19703	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×#3227	2
C₃₁×#3209	2
C₂×C₃₁×#498	2
C₃₁×#3131	2
C₃₁×#1720	2
C₃₁×#1716	2
C₃₁×#1452	2
#19703	31

Subgroups — C31×#19703 #21476462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19703 #21476462