Subgroups — C₃₁×#2448 #16842381

Order 3968.

512 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#399	C₂	1	2	yes	maximal, complemented
4	C₃₁×#397	C₂	1	2	yes	maximal, complemented
5	C₃₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#390	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#390	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#388	C₂	1	2	yes	maximal, complemented
10	C₃₁×#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₃₁	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	C₃₁×#124	—	2	4	no	—
15	C₄×Q₈×C₃₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
37	C₂⁴×C₃₁	C₂³	1	2	yes	—
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×Q₈×C₃₁	—	2	2	no	—
40	C₂×Q₈×C₃₁	D₄	1	4	yes	—
41	C₂×D₄×C₃₁	—	2	2	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂²⋊C₄×C₃₁	—	2	4	no	—
63	C₂²⋊C₄×C₃₁	—	2	4	no	—
64	C₂²⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	2	no	—
66	C₄²×C₃₁	—	2	4	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	8	no	—
79	D₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	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	2	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₃₁	#41	1	2	yes	—
107	C₂×C₄×C₃₁	#41	1	2	yes	—
108	C₂×C₄×C₃₁	#41	1	2	yes	—
109	C₂×C₄×C₃₁	#41	1	2	yes	—
110	#2448	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₁	—	4	2	no	—
112	C₂²×C₃₁	—	4	2	no	—
113	C₂²×C₃₁	—	4	2	no	—
114	C₂²×C₃₁	—	4	2	no	—
115	C₂²×C₃₁	—	4	2	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
122	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	4	1	no	—
129	C₄×C₃₁	—	4	1	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	#123	1	4	yes	—
135	C₂×#124	C₂×C₃₁	1	1	yes	characteristic
136	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#399	C₂×C₃₁	1	2	yes	complemented
138	#397	C₂×C₃₁	1	2	yes	complemented
139	#391	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#390	C₂×C₃₁	1	1	yes	characteristic
141	#390	C₂×C₃₁	1	1	yes	characteristic
142	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#388	C₂×C₃₁	1	2	yes	complemented
144	#388	C₂×C₃₁	1	2	yes	complemented
145	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×C₃₁	—	8	2	no	—
147	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
148	C₂×C₃₁	#529	1	1	yes	characteristic
149	C₂×C₃₁	#542	1	1	yes	characteristic
150	C₂×C₃₁	#548	1	1	yes	characteristic
151	C₂×C₃₁	#550	1	1	yes	characteristic
152	C₂×C₃₁	#556	1	1	yes	characteristic
153	C₂×C₃₁	#559	1	1	yes	characteristic
154	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
156	#124	—	2	4	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
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	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
172	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
174	#95	C₂²×C₃₁	1	1	yes	characteristic
175	#95	C₂²×C₃₁	1	1	yes	characteristic
176	#95	C₂²×C₃₁	1	2	yes	—
177	#95	C₂²×C₃₁	1	2	yes	—
178	#95	C₂²×C₃₁	1	2	yes	—
179	C₃₁	#2448	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁴	C₂³×C₃₁	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	D₄×C₃₁	1	4	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	D₄	—	4	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
248	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂×C₄	#41×C₃₁	1	2	yes	—
250	C₂×C₄	#41×C₃₁	1	2	yes	—
251	C₂×C₄	#41×C₃₁	1	2	yes	—
252	C₂×C₄	#41×C₃₁	1	2	yes	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
261	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
262	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
263	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
264	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
265	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	2	no	—
276	C₄	C₃₁×#123	1	4	yes	—
277	C₂	—	8	2	no	—
278	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
279	C₂	C₃₁×#529	1	1	yes	characteristic
280	C₂	C₃₁×#542	1	1	yes	characteristic
281	C₂	C₃₁×#548	1	1	yes	characteristic
282	C₂	C₃₁×#550	1	1	yes	characteristic
283	C₂	C₃₁×#556	1	1	yes	characteristic
284	C₂	C₃₁×#559	1	1	yes	characteristic

	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
2. C₂×C₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#399	C₂	semidirect
4. C₃₁×#397	C₂	semidirect
5. C₃₁×#391	C₂	semidirect
8. C₃₁×#389	C₂	semidirect
9. C₃₁×#388	C₂	semidirect
10. C₃₁×#388	C₂	semidirect
11. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
110. #2448	C₃₁	direct (N × H)
136. C₂×C₄×Q₈	C₂×C₃₁	semidirect
137. #399	C₂×C₃₁	semidirect
138. #397	C₂×C₃₁	semidirect
139. #391	C₂×C₃₁	semidirect
142. #389	C₂×C₃₁	semidirect
143. #388	C₂×C₃₁	semidirect
144. #388	C₂×C₃₁	semidirect
145. C₄×C₄⋊C₄	C₂×C₃₁	semidirect
179. C₃₁	#2448	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×#124	2
C₂×C₄×Q₈×C₃₁	2
C₃₁×#399	2
C₃₁×#397	2
C₃₁×#391	2
C₃₁×#390	2
C₃₁×#389	2
C₃₁×#388	2
C₄×C₄⋊C₄×C₃₁	2
#2448	31

Subgroups — C31×#2448 #16842381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2448 #16842381