Subgroups — C₃₁×#2421 #16842354

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 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
2	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#400	C₂	1	2	yes	maximal, complemented
4	C₃₁×#397	C₂	1	2	yes	maximal
5	C₃₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#388	C₂	1	2	yes	maximal
8	C₃₁×#387	C₂	1	2	yes	maximal, complemented
9	C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₂²⋊Q₈	—	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	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
26	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#95	C₂²	1	2	yes	—
30	C₃₁×#95	C₂²	1	2	yes	—
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	1	yes	characteristic
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×Q₈×C₃₁	—	2	2	no	—
40	C₂×Q₈×C₃₁	D₄	1	4	yes	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×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	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
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	4	no	—
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂²⋊C₄×C₃₁	—	2	4	no	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₄²×C₃₁	—	2	4	no	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	—	2	1	no	—
68	C₂³×C₃₁	—	2	1	no	—
69	C₂³×C₃₁	—	2	1	no	—
70	C₂³×C₃₁	—	2	1	no	—
71	C₂³×C₃₁	—	2	1	no	—
72	C₂³×C₃₁	—	2	1	no	—
73	C₂³×C₃₁	#41	1	2	yes	—
74	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
75	Q₈×C₃₁	—	2	8	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	2	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	2	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	2	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₃₁	#41	1	2	yes	—
105	C₂×C₄×C₃₁	#41	1	2	yes	—
106	C₂×C₄×C₃₁	#41	1	2	yes	—
107	#2421	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	4	1	no	—
112	C₂²×C₃₁	—	4	1	no	—
113	C₂²×C₃₁	—	4	1	no	—
114	C₂²×C₃₁	—	4	1	no	—
115	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
118	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
119	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	4	1	no	—
125	C₄×C₃₁	—	4	2	no	—
126	C₄×C₃₁	—	4	2	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	—	4	1	no	—
129	C₄×C₃₁	—	4	1	no	—
130	C₄×C₃₁	—	4	1	no	—
131	C₄×C₃₁	#123	1	4	yes	—
132	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#400	C₂×C₃₁	1	2	yes	complemented
135	#397	C₂×C₃₁	1	2	yes	—
136	#391	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#388	C₂×C₃₁	1	2	yes	—
139	#387	C₂×C₃₁	1	2	yes	complemented
140	#382	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#380	C₂×C₃₁	1	1	yes	characteristic
142	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
143	C₂×C₃₁	—	4	2	no	—
144	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
145	C₂×C₃₁	#529	1	1	yes	characteristic
146	C₂×C₃₁	#541	1	1	yes	characteristic
147	C₂×C₃₁	#548	1	1	yes	characteristic
148	C₂×C₃₁	#548	1	1	yes	characteristic
149	C₂×C₃₁	#552	1	1	yes	characteristic
150	C₂×C₃₁	#556	1	1	yes	characteristic
151	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²⋊Q₈	—	2	4	no	—
154	C₄×Q₈	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
157	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
165	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
168	#95	C₂²×C₃₁	1	1	yes	characteristic
169	#95	C₂²×C₃₁	1	1	yes	characteristic
170	#95	C₂²×C₃₁	1	2	yes	—
171	#95	C₂²×C₃₁	1	2	yes	—
172	#95	C₂²×C₃₁	1	2	yes	—
173	#95	C₂²×C₃₁	1	2	yes	—
174	#95	C₂²×C₃₁	1	2	yes	—
175	#95	C₂²×C₃₁	1	2	yes	—
176	C₃₁	#2421	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	D₄×C₃₁	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
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	2	yes	—
191	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	#41×C₃₁	1	2	yes	—
214	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
215	Q₈	—	2	8	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	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	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
243	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂×C₄	#41×C₃₁	1	2	yes	—
245	C₂×C₄	#41×C₃₁	1	2	yes	—
246	C₂×C₄	#41×C₃₁	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
256	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
257	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
258	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
259	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
260	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	C₃₁×#123	1	4	yes	—
271	C₂	—	4	2	no	—
272	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
273	C₂	C₃₁×#529	1	1	yes	characteristic
274	C₂	C₃₁×#541	1	1	yes	characteristic
275	C₂	C₃₁×#548	1	1	yes	characteristic
276	C₂	C₃₁×#548	1	1	yes	characteristic
277	C₂	C₃₁×#552	1	1	yes	characteristic
278	C₂	C₃₁×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
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×#400	C₂	semidirect
5. C₃₁×#391	C₂	semidirect
6. C₃₁×#389	C₂	semidirect
8. C₃₁×#387	C₂	semidirect
9. C₃₁×#382	C₂	semidirect
107. #2421	C₃₁	direct (N × H)
133. C₂×C₄×Q₈	C₂×C₃₁	semidirect
134. #400	C₂×C₃₁	semidirect
136. #391	C₂×C₃₁	semidirect
137. #389	C₂×C₃₁	semidirect
139. #387	C₂×C₃₁	semidirect
140. #382	C₂×C₃₁	semidirect
176. C₃₁	#2421	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×C₂²⋊Q₈	2
C₂×C₄×Q₈×C₃₁	2
C₃₁×#400	2
C₃₁×#397	2
C₃₁×#391	2
C₃₁×#389	2
C₃₁×#388	2
C₃₁×#387	2
C₃₁×#382	2
C₃₁×#380	2
C₄×C₂²⋊C₄×C₃₁	2
#2421	31

Subgroups — C31×#2421 #16842354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2421 #16842354