Subgroups — C₃₁×#8664 #21465423

Order 7936.

434 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1486	C₂	1	2	yes	maximal
5	C₃₁×#1301	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	—	2	1	no	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	1	no	—
9	C₃₁×#436	—	2	1	no	—
10	C₃₁×#436	C₄	1	2	yes	—
11	C₃₁×#436	C₄	1	2	yes	—
12	C₃₁×#431	C₂²	1	2	yes	—
13	C₃₁×#423	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#341	—	2	1	no	—
18	C₃₁×#341	—	2	1	no	—
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×D₄⋊C₈	—	2	4	no	—
22	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	—	2	1	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	4	4	no	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₄×D₄×C₃₁	—	2	1	no	—
32	C₄×D₄×C₃₁	—	2	1	no	—
33	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₃₁	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₃₁×C₂²⋊C₈	—	2	2	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂⁴×C₃₁	—	2	1	no	—
52	C₂×D₄×C₃₁	—	2	1	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
63	M₁₆×C₃₁	—	4	4	no	—
64	M₁₆×C₃₁	—	4	4	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	#41	1	1	yes	characteristic
85	#8664	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	—	4	2	no	—
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	4	2	no	—
90	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
91	D₄×C₃₁	—	2	4	no	—
92	D₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	4	no	—
100	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#135	1	2	yes	—
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	8	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
116	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#1486	C₂×C₃₁	1	2	yes	—
119	#1301	C₂×C₃₁	1	2	yes	complemented
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	4	2	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	8	4	no	—
126	C₂²×C₃₁	#380	1	1	yes	characteristic
127	C₂²×C₃₁	#478	1	2	yes	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	#416	1	1	yes	characteristic
133	C₄×C₃₁	#418	1	1	yes	characteristic
134	C₄×C₃₁	#430	1	2	yes	—
135	C₄×C₃₁	#463	1	2	yes	—
136	C₂²×M₁₆	—	2	1	no	—
137	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
138	#436	—	2	1	no	—
139	#436	—	2	1	no	—
140	#436	C₄×C₃₁	1	2	yes	—
141	#436	C₄×C₃₁	1	2	yes	—
142	#431	C₂²×C₃₁	1	2	yes	—
143	#423	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×C₂²⋊C₈	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	#341	—	2	1	no	—
148	#341	—	2	1	no	—
149	#335	—	2	1	no	—
150	#334	—	2	1	no	—
151	D₄⋊C₈	—	2	4	no	—
152	C₂×C₃₁	—	2	2	no	—
153	C₂×C₃₁	—	8	2	no	—
154	C₂×C₃₁	#1804	1	1	yes	characteristic
155	C₂×C₃₁	#1811	1	1	yes	characteristic
156	C₂×C₃₁	#1814	1	1	yes	characteristic
157	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
158	C₂³×C₄	—	2	1	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
169	C₄×D₄	D₄×C₃₁	1	2	yes	—
170	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₃₁	#8664	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄²	#41×C₃₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
238	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
239	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
240	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#135	1	2	yes	—
242	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	—	8	2	no	—
249	C₈	—	8	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	C₃₁×#380	1	1	yes	characteristic
257	C₂²	C₃₁×#478	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	C₃₁×#416	1	1	yes	characteristic
263	C₄	C₃₁×#418	1	1	yes	characteristic
264	C₄	C₃₁×#430	1	2	yes	—
265	C₄	C₃₁×#463	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₃₁×#1804	1	1	yes	characteristic
269	C₂	C₃₁×#1811	1	1	yes	characteristic
270	C₂	C₃₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂²×D₄×C₃₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1837	C₂	semidirect
3. C₃₁×#1564	C₂	semidirect
5. C₃₁×#1301	C₂	semidirect
85. #8664	C₃₁	direct (N × H)
116. #1837	C₂×C₃₁	semidirect
117. #1564	C₂×C₃₁	semidirect
119. #1301	C₂×C₃₁	semidirect
186. C₃₁	#8664	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×#436	2
C₃₁×#1837	2
C₃₁×#1564	2
C₃₁×#1486	2
C₃₁×#1301	2
#8664	31

Subgroups — C31×#8664 #21465423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8664 #21465423