Subgroups — C₃₁×#13064 #21469823

Order 7936.

488 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#359	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#525	—	2	1	no	—
9	C₂×C₃₁×M₃₂	—	2	2	no	—
10	C₂×C₃₁×M₃₂	—	2	4	no	—
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#467	C₄	1	2	yes	—
13	C₃₁×#466	C₄	1	2	yes	—
14	C₃₁×#465	—	2	2	no	—
15	C₂×C₃₁×#108	C₂²	1	2	yes	—
16	C₃₁×#427	C₂²	1	1	yes	characteristic
17	C₃₁×#416	—	2	1	no	—
18	C₃₁×#407	—	2	1	no	—
19	C₃₁×#359	—	2	4	no	—
20	C₃₁×#341	—	2	1	no	—
21	C₂×#41×C₃₁	—	2	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₃₁×#135	—	2	2	no	—
24	C₃₁×#135	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#123	—	4	1	no	—
36	C₃₁×C₂²⋊Q₈	—	2	2	no	—
37	C₃₁×#121	—	2	2	no	—
38	C₃₁×#120	—	4	1	no	—
39	C₄×D₄×C₃₁	—	4	2	no	—
40	C₃₁×#117	—	2	1	no	—
41	C₃₁×M₃₂	—	2	8	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
45	C₃₁×#108	—	2	2	no	—
46	C₃₁×#108	—	2	2	no	—
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×#106	—	2	1	no	—
49	C₃₁×#103	—	2	2	no	—
50	C₃₁×#102	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	4	2	no	—
52	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
53	#41×C₃₁	—	4	2	no	—
54	#41×C₃₁	—	4	2	no	—
55	C₂×Q₈×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	—	2	1	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	D₈	1	2	yes	—
64	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₃₁	—	2	2	no	—
67	SD₁₆×C₃₁	—	4	1	no	—
68	SD₁₆×C₃₁	—	4	1	no	—
69	D₈×C₃₁	—	2	2	no	—
70	M₁₆×C₃₁	—	4	2	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	#41	1	2	yes	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₂²⋊C₄×C₃₁	—	4	2	no	—
85	C₂²⋊C₄×C₃₁	—	4	1	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	1	no	—
88	C₁₆×C₃₁	—	2	4	no	—
89	#13064	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	—	4	1	no	—
93	C₂³×C₃₁	#102	1	2	yes	—
94	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
95	Q₈×C₃₁	—	4	1	no	—
96	Q₈×C₃₁	—	4	1	no	—
97	D₄×C₃₁	—	4	1	no	—
98	D₄×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	8	4	no	—
100	D₄×C₃₁	—	8	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	8	2	no	—
108	C₂×C₄×C₃₁	#102	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#121	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#135	1	2	yes	—
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	C₈×C₃₁	C₄×D₄	1	2	yes	—
116	C₈×C₃₁	#120	1	1	yes	characteristic
117	C₈×C₃₁	#123	1	1	yes	characteristic
118	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#3050	C₂×C₃₁	1	1	yes	characteristic
120	C₂×#359	C₂×C₃₁	1	2	yes	complemented
121	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1382	C₂×C₃₁	1	2	yes	complemented
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	2	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	4	4	no	—
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	8	1	no	—
130	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
131	C₂²×C₃₁	#465	1	2	yes	—
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	8	2	no	—
134	C₄×C₃₁	—	8	1	no	—
135	C₄×C₃₁	#386	1	1	yes	characteristic
136	C₄×C₃₁	#416	1	1	yes	characteristic
137	C₄×C₃₁	#466	1	2	yes	—
138	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	#525	—	2	1	no	—
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	—	2	4	no	—
143	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
144	#467	C₄×C₃₁	1	2	yes	—
145	#466	C₄×C₃₁	1	2	yes	—
146	#465	—	2	2	no	—
147	C₂×#108	C₂²×C₃₁	1	2	yes	—
148	#427	C₂²×C₃₁	1	1	yes	characteristic
149	#416	—	2	1	no	—
150	#407	—	2	1	no	—
151	#359	—	2	4	no	—
152	#341	—	2	1	no	—
153	C₂×C₃₁	—	2	2	no	—
154	C₂×C₃₁	—	4	2	no	—
155	C₂×C₃₁	—	16	1	no	—
156	C₂×C₃₁	#1893	1	1	yes	characteristic
157	C₂×C₃₁	#2143	1	2	yes	—
158	C₂×#41	—	2	1	no	—
159	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
160	#135	—	2	2	no	—
161	#135	—	2	2	no	—
162	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂×SD₁₆	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
169	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₃₁	1	2	yes	—
171	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	#123	—	4	1	no	—
173	C₂²⋊Q₈	—	2	2	no	—
174	#121	—	2	2	no	—
175	#120	—	4	1	no	—
176	C₄×D₄	—	4	2	no	—
177	#117	—	2	1	no	—
178	M₃₂	—	2	8	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
182	#108	—	2	2	no	—
183	#108	—	2	2	no	—
184	#107	C₂×C₄×C₃₁	1	2	yes	—
185	#106	—	2	1	no	—
186	#103	—	2	2	no	—
187	#102	—	2	2	no	—
188	C₂²⋊C₈	—	4	2	no	—
189	C₃₁	#13064	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
201	C₂²×C₄	D₈×C₃₁	1	2	yes	—
202	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	1	no	—
206	SD₁₆	—	4	1	no	—
207	D₈	—	2	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₈	#41×C₃₁	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₁₆	—	2	4	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₃₁×#102	1	2	yes	—
231	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	4	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	C₃₁×#102	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₃₁×#121	1	2	yes	—
248	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#135	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	C₄×D₄×C₃₁	1	2	yes	—
253	C₈	C₃₁×#120	1	1	yes	characteristic
254	C₈	C₃₁×#123	1	1	yes	characteristic
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
263	C₂²	C₃₁×#465	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	1	no	—
267	C₄	C₃₁×#386	1	1	yes	characteristic
268	C₄	C₃₁×#416	1	1	yes	characteristic
269	C₄	C₃₁×#466	1	2	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	2	no	—
272	C₂	—	16	1	no	—
273	C₂	C₃₁×#1893	1	1	yes	characteristic
274	C₂	C₃₁×#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×M₃₂	C₂	semidirect
3. C₂×C₃₁×#359	C₂	semidirect
4. C₃₁×#1810	C₂	semidirect
5. C₃₁×#1382	C₂	semidirect
44. C₂×C₁₆×C₃₁	D₄	semidirect
89. #13064	C₃₁	direct (N × H)
118. C₂²×M₃₂	C₂×C₃₁	semidirect
120. C₂×#359	C₂×C₃₁	semidirect
121. #1810	C₂×C₃₁	semidirect
122. #1382	C₂×C₃₁	semidirect
181. C₂×C₁₆	D₄×C₃₁	semidirect
189. C₃₁	#13064	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×M₃₂	2
C₃₁×#3050	2
C₂×C₃₁×#359	2
C₃₁×#1810	2
C₃₁×#1382	2
#13064	31

Subgroups — C31×#13064 #21469823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13064 #21469823