Subgroups — C₃₁×#21284 #21478043

Order 7936.

862 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×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1972	C₂	1	2	yes	maximal
4	C₃₁×#1943	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1934	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
11	C₄×D₈×C₃₁	C₄	1	8	yes	complemented
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×#102	C₂²	1	4	yes	—
15	C₂×C₃₁×#102	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#390	C₂²	1	4	yes	—
21	C₃₁×#389	C₂²	1	4	yes	—
22	C₃₁×#388	—	2	4	no	—
23	C₃₁×#381	—	2	2	no	—
24	C₃₁×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#376	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₃₁	—	2	2	no	—
31	C₂×D₈×C₃₁	—	2	4	no	—
32	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₃₁	—	2	4	no	—
37	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#107	C₂×C₄	1	4	yes	—
48	C₃₁×#106	—	2	4	no	—
49	C₃₁×#102	—	2	8	no	—
50	C₃₁×#102	C₂×C₄	1	8	yes	—
51	C₃₁×#97	—	2	4	no	—
52	C₃₁×#97	—	2	2	no	—
53	C₄×C₈×C₃₁	—	2	4	no	—
54	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
55	C₃₁×#95	—	2	2	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₃₁×#95	D₄	1	2	yes	—
58	C₃₁×#95	D₄	1	2	yes	—
59	C₂⁴×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	4	no	—
61	C₂×D₄×C₃₁	—	2	4	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	8	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	8	4	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	D₈×C₃₁	—	2	8	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₁	#41	1	4	yes	—
79	C₂×C₈×C₃₁	#41	1	4	yes	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	8	4	no	—
84	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₃₁	—	2	8	no	—
86	C₂²⋊C₄×C₃₁	—	8	4	no	—
87	C₂²⋊C₄×C₃₁	—	8	4	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	8	2	no	—
91	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
92	C₄²×C₃₁	#41	1	2	yes	—
93	#21284	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	4	no	—
95	C₂³×C₃₁	—	4	4	no	—
96	C₂³×C₃₁	—	4	4	no	—
97	C₂³×C₃₁	—	4	2	no	—
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	D₄×C₃₁	—	2	4	no	—
101	D₄×C₃₁	—	2	4	no	—
102	D₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	8	8	no	—
114	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	2	4	no	—
122	C₈×C₃₁	#117	1	4	yes	—
123	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
124	#2319	C₂×C₃₁	1	2	yes	complemented
125	#1972	C₂×C₃₁	1	2	yes	—
126	#1943	C₂×C₃₁	1	2	yes	complemented
127	#1934	C₂×C₃₁	1	4	yes	complemented
128	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1762	C₂×C₃₁	1	2	yes	complemented
130	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂²×C₃₁	—	4	4	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	—	8	4	no	—
135	C₂²×C₃₁	—	8	4	no	—
136	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₃₁	#532	1	2	yes	—
138	C₂²×C₃₁	#576	1	2	yes	—
139	C₂²×C₃₁	#577	1	2	yes	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	4	4	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
146	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
147	C₄×C₃₁	#529	1	2	yes	—
148	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
150	C₄×D₈	C₄×C₃₁	1	8	yes	complemented
151	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#106	C₂²×C₃₁	1	2	yes	—
153	C₂×#102	C₂²×C₃₁	1	4	yes	—
154	C₂×#102	C₂²×C₃₁	1	2	yes	—
155	C₂×#97	C₂²×C₃₁	1	2	yes	—
156	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	#390	C₂²×C₃₁	1	4	yes	—
160	#389	C₂²×C₃₁	1	4	yes	—
161	#388	—	2	4	no	—
162	#381	—	2	2	no	—
163	#380	—	2	2	no	—
164	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
165	#376	C₂²×C₃₁	1	1	yes	characteristic
166	#340	C₂²×C₃₁	1	2	yes	—
167	#340	C₂²×C₃₁	1	4	yes	—
168	C₂×C₃₁	—	8	4	no	—
169	C₂×C₃₁	#2293	1	1	yes	characteristic
170	C₂×C₃₁	#2948	1	1	yes	characteristic
171	C₂×C₃₁	#2952	1	1	yes	characteristic
172	C₂×C₃₁	#3179	1	4	yes	—
173	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
177	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	C₄×D₄	—	2	4	no	—
181	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	#107	C₂×C₄×C₃₁	1	4	yes	—
192	#106	—	2	4	no	—
193	#102	—	2	8	no	—
194	#102	C₂×C₄×C₃₁	1	8	yes	—
195	#97	—	2	4	no	—
196	#97	—	2	2	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	D₄×C₃₁	1	2	yes	—
202	#95	D₄×C₃₁	1	2	yes	—
203	C₃₁	#21284	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	D₈	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
223	C₂×C₈	#41×C₃₁	1	4	yes	—
224	C₂×C₈	#41×C₃₁	1	4	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	8	4	no	—
229	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	8	4	no	—
232	C₂²⋊C₄	—	8	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	8	2	no	—
236	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
237	C₄²	#41×C₃₁	1	2	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
244	D₄	—	2	4	no	—
245	D₄	—	2	4	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
263	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	C₃₁×#117	1	4	yes	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
273	C₂²	C₃₁×#532	1	2	yes	—
274	C₂²	C₃₁×#576	1	2	yes	—
275	C₂²	C₃₁×#577	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
282	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
283	C₄	C₃₁×#529	1	2	yes	—
284	C₂	—	8	4	no	—
285	C₂	C₃₁×#2293	1	1	yes	characteristic
286	C₂	C₃₁×#2948	1	1	yes	characteristic
287	C₂	C₃₁×#2952	1	1	yes	characteristic
288	C₂	C₃₁×#3179	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2319	C₂	semidirect
4. C₃₁×#1943	C₂	semidirect
5. C₃₁×#1934	C₂	semidirect
6. C₄×C₃₁×#106	C₂	semidirect
7. C₃₁×#1762	C₂	semidirect
8. C₃₁×#1726	C₂	semidirect
11. C₄×D₈×C₃₁	C₄	semidirect
93. #21284	C₃₁	direct (N × H)
124. #2319	C₂×C₃₁	semidirect
126. #1943	C₂×C₃₁	semidirect
127. #1934	C₂×C₃₁	semidirect
128. C₄×#106	C₂×C₃₁	semidirect
129. #1762	C₂×C₃₁	semidirect
130. #1726	C₂×C₃₁	semidirect
150. C₄×D₈	C₄×C₃₁	semidirect
203. C₃₁	#21284	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×C₄×D₈×C₃₁	2
C₃₁×#2319	2
C₃₁×#1972	2
C₃₁×#1943	2
C₃₁×#1934	2
C₄×C₃₁×#106	2
C₃₁×#1762	2
C₃₁×#1726	2
#21284	31

Subgroups — C31×#21284 #21478043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21284 #21478043