Subgroups — C₃₁×#8600 #21465359

Order 7936.

478 conjugacy classes of subgroups.

The subgroup lattice has 290 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	C₃₁×#1797	C₂	1	2	yes	maximal
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1277	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#431	—	2	2	no	—
8	C₃₁×#431	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
13	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#336	—	2	2	no	—
17	C₃₁×#335	—	2	2	no	—
18	C₃₁×#334	—	2	2	no	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	—	2	1	no	—
27	C₂×C₄²×C₃₁	—	2	1	no	—
28	C₂×C₄²×C₃₁	—	2	1	no	—
29	C₂×C₄²×C₃₁	—	2	1	no	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
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	2	no	—
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₂²⋊C₈	—	2	4	no	—
40	C₃₁×C₂²⋊C₈	—	4	4	no	—
41	C₄×C₈×C₃₁	—	2	1	no	—
42	C₄×C₈×C₃₁	—	2	1	no	—
43	C₄×C₈×C₃₁	D₄	1	2	yes	—
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₃₁	—	2	1	no	—
74	C₄²×C₃₁	—	2	1	no	—
75	C₄²×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	—	2	1	no	—
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₃₁	#41	1	1	yes	characteristic
82	C₄²×C₃₁	#41	1	1	yes	characteristic
83	#8600	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	#108	1	2	yes	—
89	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	1	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	#108	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#131	1	4	yes	—
113	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₃₁	#137	1	2	yes	—
115	C₈×C₃₁	—	2	4	no	—
116	C₈×C₃₁	—	4	2	no	—
117	C₈×C₃₁	—	8	2	no	—
118	C₈×C₃₁	—	8	2	no	—
119	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
120	#1797	C₂×C₃₁	1	2	yes	—
121	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#1277	C₂×C₃₁	1	2	yes	complemented
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	4	1	no	—
127	C₂²×C₃₁	—	4	1	no	—
128	C₂²×C₃₁	—	4	2	no	—
129	C₂²×C₃₁	#380	1	1	yes	characteristic
130	C₂²×C₃₁	#424	1	2	yes	—
131	C₂²×C₃₁	#428	1	2	yes	—
132	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	4	1	no	—
138	C₄×C₃₁	—	4	1	no	—
139	C₄×C₃₁	#408	1	2	yes	—
140	C₄×C₃₁	#433	1	2	yes	—
141	C₄×C₃₁	#468	1	2	yes	—
142	C₄×C₃₁	#469	1	1	yes	characteristic
143	C₄×C₃₁	#470	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
145	#431	—	2	2	no	—
146	#431	—	2	2	no	—
147	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
150	C₂×C₂²⋊C₈	—	2	2	no	—
151	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	#336	—	2	2	no	—
155	#335	—	2	2	no	—
156	#334	—	2	2	no	—
157	C₂×C₃₁	—	4	2	no	—
158	C₂×C₃₁	#1573	1	2	yes	—
159	C₂×C₃₁	#1574	1	2	yes	—
160	C₂×C₃₁	#1798	1	1	yes	characteristic
161	C₂×C₃₁	#1809	1	1	yes	characteristic
162	C₂×C₃₁	#1810	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
165	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	C₂²⋊C₈	—	4	4	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	D₄×C₃₁	1	2	yes	—
188	C₄×C₈	D₄×C₃₁	1	2	yes	—
189	C₃₁	#8600	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₄²	#41×C₃₁	1	1	yes	characteristic
227	C₄²	#41×C₃₁	1	1	yes	characteristic
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	C₃₁×#108	1	2	yes	—
233	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₃₁×#108	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
252	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
253	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₃₁×#131	1	4	yes	—
257	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
258	C₂×C₄	C₃₁×#137	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	2	no	—
262	C₈	—	8	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₃₁×#380	1	1	yes	characteristic
269	C₂²	C₃₁×#424	1	2	yes	—
270	C₂²	C₃₁×#428	1	2	yes	—
271	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
272	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	C₃₁×#408	1	2	yes	—
279	C₄	C₃₁×#433	1	2	yes	—
280	C₄	C₃₁×#468	1	2	yes	—
281	C₄	C₃₁×#469	1	1	yes	characteristic
282	C₄	C₃₁×#470	1	1	yes	characteristic
283	C₂	—	4	2	no	—
284	C₂	C₃₁×#1573	1	2	yes	—
285	C₂	C₃₁×#1574	1	2	yes	—
286	C₂	C₃₁×#1798	1	1	yes	characteristic
287	C₂	C₃₁×#1809	1	1	yes	characteristic
288	C₂	C₃₁×#1810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#335	C₂	semidirect
4. C₂×C₃₁×#334	C₂	semidirect
5. C₃₁×#1277	C₂	semidirect
83. #8600	C₃₁	direct (N × H)
121. C₂×#335	C₂×C₃₁	semidirect
122. C₂×#334	C₂×C₃₁	semidirect
123. #1277	C₂×C₃₁	semidirect
189. C₃₁	#8600	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
C₃₁×#1797	2
C₂×C₃₁×#335	2
C₂×C₃₁×#334	2
C₃₁×#1277	2
#8600	31

Subgroups — C31×#8600 #21465359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8600 #21465359