Subgroups — C₃₁×#13251 #21470010

Order 7936.

478 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#359	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#135	—	2	1	no	—
6	C₂×C₃₁×#135	C₂²	1	2	yes	—
7	C₃₁×#532	—	2	1	no	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#495	—	2	1	no	—
10	C₃₁×#495	C₄	1	2	yes	—
11	C₃₁×#494	C₄	1	1	yes	characteristic
12	C₃₁×#493	C₄	1	1	yes	characteristic
13	C₃₁×#492	—	2	1	no	—
14	C₂×C₃₁×#108	C₂²	1	2	yes	—
15	C₄×M₁₆×C₃₁	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×C₄⋊C₁₆	—	2	1	no	—
19	C₃₁×#359	—	2	8	no	—
20	C₃₁×#341	—	2	1	no	—
21	C₂×#41×C₃₁	—	2	2	no	—
22	C₃₁×#135	—	2	4	no	—
23	C₃₁×#135	—	2	4	no	—
24	C₃₁×#135	—	4	4	no	—
25	C₂×Q₁₆×C₃₁	—	2	1	no	—
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₃₁	—	2	2	no	—
28	C₂×SD₁₆×C₃₁	—	2	2	no	—
29	C₂×D₈×C₃₁	—	2	1	no	—
30	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×#127	—	2	1	no	—
36	C₃₁×#124	—	2	2	no	—
37	C₃₁×#121	—	4	1	no	—
38	C₃₁×#121	—	4	1	no	—
39	C₄×D₄×C₃₁	—	4	1	no	—
40	C₄×D₄×C₃₁	—	4	1	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₃₁	—	2	4	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₃₁×#108	—	2	2	no	—
45	C₃₁×#108	D₄	1	4	yes	complemented
46	C₃₁×#97	—	4	1	no	—
47	C₄×C₈×C₃₁	—	4	1	no	—
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
52	#41×C₃₁	—	4	4	no	—
53	#41×C₃₁	—	4	4	no	—
54	C₂×Q₈×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	4	2	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	4	no	—
64	Q₁₆×C₃₁	—	4	2	no	—
65	SD₁₆×C₃₁	—	4	4	no	—
66	SD₁₆×C₃₁	—	4	2	no	—
67	SD₁₆×C₃₁	—	4	2	no	—
68	D₈×C₃₁	—	2	4	no	—
69	D₈×C₃₁	—	4	2	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	#41	1	2	yes	—
80	C₄⋊C₄×C₃₁	—	4	1	no	—
81	C₄⋊C₄×C₃₁	—	4	1	no	—
82	C₂²⋊C₄×C₃₁	—	4	2	no	—
83	C₂²⋊C₄×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₃₁	D₈	1	1	yes	characteristic
87	C₄²×C₃₁	D₈	1	1	yes	characteristic
88	C₁₆×C₃₁	—	4	2	no	—
89	#13251	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
92	Q₈×C₃₁	—	4	2	no	—
93	Q₈×C₃₁	—	4	2	no	—
94	D₄×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	4	2	no	—
96	D₄×C₃₁	—	8	2	no	—
97	D₄×C₃₁	—	8	4	no	—
98	D₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	1	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	8	4	no	—
103	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#121	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	2	1	no	—
113	C₈×C₃₁	—	4	4	no	—
114	C₈×C₃₁	C₄×D₄	1	2	yes	—
115	C₈×C₃₁	#124	1	1	yes	characteristic
116	C₈×C₃₁	#127	1	1	yes	characteristic
117	#3147	C₂×C₃₁	1	1	yes	characteristic
118	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂×#359	C₂×C₃₁	1	4	yes	complemented
120	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	8	2	no	—
124	C₂²×C₃₁	—	8	2	no	—
125	C₂²×C₃₁	—	8	2	no	—
126	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
127	C₂²×C₃₁	#460	1	2	yes	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	8	2	no	—
132	C₄×C₃₁	#359	1	2	yes	complemented
133	C₄×C₃₁	#390	1	1	yes	characteristic
134	C₄×C₃₁	#418	1	1	yes	characteristic
135	C₄×C₃₁	#459	1	2	yes	—
136	C₂×#135	—	2	1	no	—
137	C₂×#135	C₂²×C₃₁	1	2	yes	—
138	#532	—	2	1	no	—
139	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
140	#495	—	2	1	no	—
141	#495	C₄×C₃₁	1	2	yes	—
142	#494	C₄×C₃₁	1	1	yes	characteristic
143	#493	C₄×C₃₁	1	1	yes	characteristic
144	#492	—	2	1	no	—
145	C₂×#108	C₂²×C₃₁	1	2	yes	—
146	C₄×M₁₆	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	—	2	1	no	—
150	#359	—	2	8	no	—
151	#341	—	2	1	no	—
152	C₂×C₃₁	—	2	2	no	—
153	C₂×C₃₁	—	16	2	no	—
154	C₂×C₃₁	#1982	1	1	yes	characteristic
155	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
156	C₂×C₃₁	#2143	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	#135	—	2	4	no	—
159	#135	—	2	4	no	—
160	#135	—	4	4	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	1	no	—
171	#127	—	2	1	no	—
172	#124	—	2	2	no	—
173	#121	—	4	1	no	—
174	#121	—	4	1	no	—
175	C₄×D₄	—	4	1	no	—
176	C₄×D₄	—	4	1	no	—
177	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	—	2	2	no	—
180	#108	—	2	2	no	—
181	#108	D₄×C₃₁	1	4	yes	complemented
182	#97	—	4	1	no	—
183	C₄×C₈	—	4	1	no	—
184	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
188	C₃₁	#13251	1	1	yes	characteristic, direct product factor, Sylow
189	#41	—	4	4	no	—
190	#41	—	4	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₃₁	1	2	yes	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
222	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
223	C₄²	D₈×C₃₁	1	1	yes	characteristic
224	C₄²	D₈×C₃₁	1	1	yes	characteristic
225	C₁₆	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	4	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
240	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
242	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₃₁×#121	1	2	yes	—
245	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	4	no	—
250	C₈	C₄×D₄×C₃₁	1	2	yes	—
251	C₈	C₃₁×#124	1	1	yes	characteristic
252	C₈	C₃₁×#127	1	1	yes	characteristic
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
259	C₂²	C₃₁×#460	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₃₁×#359	1	2	yes	complemented
265	C₄	C₃₁×#390	1	1	yes	characteristic
266	C₄	C₃₁×#418	1	1	yes	characteristic
267	C₄	C₃₁×#459	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	16	2	no	—
270	C₂	C₃₁×#1982	1	1	yes	characteristic
271	C₂	C₂×C₃₁×#359	1	1	yes	characteristic
272	C₂	C₃₁×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₃₁×#359	C₂	semidirect
4. C₄×C₃₁×#108	C₂	semidirect
45. C₃₁×#108	D₄	semidirect
89. #13251	C₃₁	direct (N × H)
118. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
119. C₂×#359	C₂×C₃₁	semidirect
120. C₄×#108	C₂×C₃₁	semidirect
132. C₄×C₃₁	#359	semidirect
181. #108	D₄×C₃₁	semidirect
188. C₃₁	#13251	direct (N × H)
264. C₄	C₃₁×#359	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3147	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#359	2
C₄×C₃₁×#108	2
#13251	31

Subgroups — C31×#13251 #21470010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13251 #21470010