Subgroups — C₃₁×#2564 #16842497

Order 3968.

584 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#393	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#387	C₂	1	2	yes	maximal, complemented
8	C₃₁×#386	C₂	1	2	yes	maximal
9	C₃₁×#385	C₂	1	2	yes	maximal, complemented
10	C₃₁×#380	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₂²⋊Q₈	—	2	4	no	—
18	C₄×D₄×C₃₁	—	2	4	no	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#95	C₂²	1	1	yes	characteristic
31	C₃₁×#95	C₂²	1	2	yes	—
32	C₃₁×#95	C₂²	1	1	yes	characteristic
33	C₃₁×#95	C₂²	1	2	yes	—
34	C₃₁×#95	C₂²	1	2	yes	—
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₃₁	—	2	2	no	—
40	C₂×Q₈×C₃₁	—	4	2	no	—
41	C₂×D₄×C₃₁	—	2	2	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	Q₈	1	4	yes	complemented
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	4	2	no	—
48	C₂²×C₄×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	D₄	1	4	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	4	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	2	1	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	C₂×D₄	1	2	yes	—
78	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
79	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
80	C₂³×C₃₁	#41	1	2	yes	—
81	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
82	Q₈×C₃₁	—	4	4	no	—
83	D₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	8	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₃₁	#41	1	4	yes	—
110	#2564	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	4	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	4	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	C₂²⋊Q₈	1	4	yes	—
121	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
124	C₂²×C₃₁	C₂×#41	1	2	yes	—
125	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	2	4	no	—
130	C₄×C₃₁	—	4	1	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	—	4	1	no	—
133	C₄×C₃₁	—	4	2	no	—
134	C₄×C₃₁	—	4	2	no	—
135	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
137	#400	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#399	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#395	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#393	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#387	C₂×C₃₁	1	2	yes	complemented
142	#386	C₂×C₃₁	1	2	yes	—
143	#385	C₂×C₃₁	1	2	yes	complemented
144	#380	C₂×C₃₁	1	2	yes	complemented
145	C₂×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×C₃₁	—	2	4	no	—
147	C₂×C₃₁	—	4	2	no	—
148	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₃₁	#532	1	1	yes	characteristic
150	C₂×C₃₁	#534	1	1	yes	characteristic
151	C₂×C₃₁	#551	1	2	yes	—
152	C₂×C₃₁	#554	1	2	yes	—
153	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
156	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
158	C₂²⋊Q₈	—	2	4	no	—
159	C₄×D₄	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
170	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
171	#95	C₂²×C₃₁	1	2	yes	—
172	#95	C₂²×C₃₁	1	1	yes	characteristic
173	#95	C₂²×C₃₁	1	1	yes	characteristic
174	#95	C₂²×C₃₁	1	2	yes	—
175	#95	C₂²×C₃₁	1	2	yes	—
176	#95	C₂²×C₃₁	1	2	yes	—
177	#95	C₂²×C₃₁	1	2	yes	—
178	C₃₁	#2564	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
180	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	Q₈×C₃₁	1	4	yes	complemented
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	D₄×C₃₁	1	4	yes	—
193	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
196	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂×D₄×C₃₁	1	2	yes	—
220	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
221	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
222	C₂³	#41×C₃₁	1	2	yes	—
223	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	D₄	—	2	8	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
251	C₂×C₄	#41×C₃₁	1	4	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₃₁×C₂²⋊Q₈	1	4	yes	—
262	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
264	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
265	C₂²	C₂×#41×C₃₁	1	2	yes	—
266	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₂	—	2	4	no	—
277	C₂	—	4	2	no	—
278	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂	C₃₁×#532	1	1	yes	characteristic
280	C₂	C₃₁×#534	1	1	yes	characteristic
281	C₂	C₃₁×#551	1	2	yes	—
282	C₂	C₃₁×#554	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#400	C₂	semidirect
4. C₃₁×#399	C₂	semidirect
5. C₃₁×#395	C₂	semidirect
6. C₃₁×#393	C₂	semidirect
7. C₃₁×#387	C₂	semidirect
9. C₃₁×#385	C₂	semidirect
10. C₃₁×#380	C₂	semidirect
11. C₂×C₃₁×#95	C₂	semidirect
43. C₂×D₄×C₃₁	Q₈	semidirect
110. #2564	C₃₁	direct (N × H)
135. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
137. #400	C₂×C₃₁	semidirect
138. #399	C₂×C₃₁	semidirect
139. #395	C₂×C₃₁	semidirect
140. #393	C₂×C₃₁	semidirect
141. #387	C₂×C₃₁	semidirect
143. #385	C₂×C₃₁	semidirect
144. #380	C₂×C₃₁	semidirect
145. C₂×#95	C₂×C₃₁	semidirect
178. C₃₁	#2564	direct (N × H)
185. C₂×D₄	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₄×D₄×C₃₁	2
C₃₁×#400	2
C₃₁×#399	2
C₃₁×#395	2
C₃₁×#393	2
C₃₁×#387	2
C₃₁×#386	2
C₃₁×#385	2
C₃₁×#380	2
C₂×C₃₁×#95	2
#2564	31

Subgroups — C31×#2564 #16842497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2564 #16842497