Subgroups — C₃₁×#8001 #21464760

Order 7936.

634 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	2	yes	maximal
3	C₃₁×#1292	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	—	2	2	no	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#431	—	2	2	no	—
7	C₃₁×#431	—	2	2	no	—
8	C₃₁×#423	—	2	2	no	—
9	C₃₁×#423	—	2	2	no	—
10	C₃₁×#423	C₄	1	4	yes	—
11	C₃₁×#423	C₂²	1	4	yes	—
12	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	4	no	—
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	4	4	no	—
23	C₂×M₁₆×C₃₁	—	4	4	no	—
24	C₂²×C₈×C₃₁	—	2	4	no	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×C₄⋊C₈	—	2	4	no	—
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₃₁×C₂²⋊C₈	—	2	4	no	—
40	C₄×C₈×C₃₁	—	2	2	no	—
41	C₄×C₈×C₃₁	—	2	2	no	—
42	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	—	4	2	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	D₈	1	2	yes	—
55	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
60	M₁₆×C₃₁	—	4	8	no	—
61	M₁₆×C₃₁	—	4	4	no	—
62	M₁₆×C₃₁	—	4	4	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	8	no	—
65	C₂×C₈×C₃₁	—	4	8	no	—
66	C₂×C₈×C₃₁	—	4	4	no	—
67	C₄²×C₃₁	C₄²	1	1	yes	characteristic
68	C₄²×C₃₁	C₄²	1	1	yes	characteristic
69	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
75	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₃₁	C₂×D₄	1	2	yes	—
77	#8001	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	4	1	no	—
79	C₂³×C₃₁	—	4	1	no	—
80	C₂³×C₃₁	#95	1	2	yes	—
81	C₂³×C₃₁	#106	1	2	yes	—
82	C₂³×C₃₁	#107	1	2	yes	—
83	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	1	no	—
91	C₂×C₄×C₃₁	—	4	1	no	—
92	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#95	1	2	yes	—
94	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
96	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
97	C₂×C₄×C₃₁	#102	1	4	yes	—
98	C₂×C₄×C₃₁	#102	1	4	yes	—
99	C₂×C₄×C₃₁	#103	1	4	yes	—
100	C₂×C₄×C₃₁	#103	1	4	yes	—
101	C₂×C₄×C₃₁	#106	1	2	yes	—
102	C₂×C₄×C₃₁	#107	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
110	C₈×C₃₁	—	4	4	no	—
111	C₈×C₃₁	—	4	8	no	—
112	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
113	C₂×#423	C₂×C₃₁	1	2	yes	—
114	#1292	C₂×C₃₁	1	4	yes	complemented
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	—	4	2	no	—
121	C₂²×C₃₁	#340	1	4	yes	—
122	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
123	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
124	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	#338	1	2	yes	—
130	C₄×C₃₁	#340	1	2	yes	—
131	C₄×C₃₁	#340	1	2	yes	—
132	C₄×C₃₁	#413	1	2	yes	—
133	C₄×C₃₁	C₂×#102	1	2	yes	—
134	C₄×C₃₁	C₂×#103	1	2	yes	—
135	C₂²×M₁₆	—	2	2	no	—
136	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
137	#431	—	2	2	no	—
138	#431	—	2	2	no	—
139	#423	—	2	2	no	—
140	#423	—	2	2	no	—
141	#423	C₄×C₃₁	1	4	yes	—
142	#423	C₂²×C₃₁	1	4	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
145	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
147	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
148	C₂×C₃₁	—	2	4	no	—
149	C₂×C₃₁	—	2	1	no	—
150	C₂×C₃₁	—	2	1	no	—
151	C₂×C₃₁	C₂×#338	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
153	C₂×C₃₁	#1742	1	1	yes	characteristic
154	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
156	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	4	4	no	—
160	C₂×M₁₆	—	4	4	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₃₁	1	2	yes	—
168	C₂×C₄²	D₄×C₃₁	1	2	yes	—
169	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
176	C₂²⋊C₈	—	2	4	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₃₁	#8001	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
191	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	D₈×C₃₁	1	2	yes	—
193	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
194	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
198	M₁₆	—	4	8	no	—
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	4	8	no	—
204	C₂×C₈	—	4	4	no	—
205	C₄²	C₄²×C₃₁	1	1	yes	characteristic
206	C₄²	C₄²×C₃₁	1	1	yes	characteristic
207	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
209	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
210	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
211	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
212	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
213	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
214	C₄²	C₂×D₄×C₃₁	1	2	yes	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₃₁×#95	1	2	yes	—
218	C₂³	C₃₁×#106	1	2	yes	—
219	C₂³	C₃₁×#107	1	2	yes	—
220	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
230	C₂×C₄	C₃₁×#95	1	2	yes	—
231	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
232	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
234	C₂×C₄	C₃₁×#102	1	4	yes	—
235	C₂×C₄	C₃₁×#102	1	4	yes	—
236	C₂×C₄	C₃₁×#103	1	4	yes	—
237	C₂×C₄	C₃₁×#103	1	4	yes	—
238	C₂×C₄	C₃₁×#106	1	2	yes	—
239	C₂×C₄	C₃₁×#107	1	2	yes	—
240	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
246	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
247	C₈	—	4	4	no	—
248	C₈	—	4	8	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	C₃₁×#340	1	4	yes	—
256	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
257	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
258	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	C₃₁×#338	1	2	yes	—
264	C₄	C₃₁×#340	1	2	yes	—
265	C₄	C₃₁×#340	1	2	yes	—
266	C₄	C₃₁×#413	1	2	yes	—
267	C₄	C₂×C₃₁×#102	1	2	yes	—
268	C₄	C₂×C₃₁×#103	1	2	yes	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	—	2	4	no	—
272	C₂	C₂×C₃₁×#338	1	1	yes	characteristic
273	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
274	C₂	C₃₁×#1742	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#95
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1292	C₂	semidirect
77. #8001	C₃₁	direct (N × H)
114. #1292	C₂×C₃₁	semidirect
179. C₃₁	#8001	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×#431	2
C₂×C₃₁×#423	2
C₃₁×#1292	2
#8001	31

Subgroups — C31×#8001 #21464760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8001 #21464760