Subgroups — C₃₁×#13210 #21469969

Order 7936.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#360	C₂	1	4	yes	maximal, complemented
3	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#135	—	2	1	no	—
6	C₂×C₃₁×#135	C₂²	1	2	yes	—
7	C₃₁×#532	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₃₁×#495	—	2	1	no	—
10	C₃₁×#495	C₄	1	2	yes	complemented
11	C₃₁×#494	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#492	—	2	1	no	—
14	C₂×C₃₁×#106	—	2	1	no	—
15	C₂×C₃₁×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#384	—	2	1	no	—
18	C₃₁×#360	—	2	8	no	—
19	C₃₁×#346	—	2	1	no	—
20	C₄×C₁₆×C₃₁	—	2	1	no	—
21	C₂×#41×C₃₁	—	2	2	no	—
22	C₃₁×#135	—	2	4	no	—
23	C₃₁×#135	—	2	4	no	—
24	C₃₁×#135	—	4	2	no	—
25	C₃₁×#135	—	4	2	no	—
26	C₂×Q₁₆×C₃₁	—	2	1	no	—
27	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×SD₁₆×C₃₁	—	2	2	no	—
29	C₂×SD₁₆×C₃₁	—	2	2	no	—
30	C₂×D₈×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×#127	—	2	1	no	—
36	C₃₁×#124	—	2	2	no	—
37	C₃₁×#121	—	4	2	no	—
38	C₄×D₄×C₃₁	—	4	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₃₁×M₃₂	—	2	2	no	—
43	C₃₁×M₃₂	D₄	1	4	yes	complemented
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₃₁×#106	—	2	4	no	—
46	C₃₁×#106	—	4	2	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	#41×C₃₁	—	4	2	no	—
52	#41×C₃₁	—	4	2	no	—
53	#41×C₃₁	—	4	2	no	—
54	#41×C₃₁	—	4	2	no	—
55	C₂×Q₈×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	D₈	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	4	no	—
64	Q₁₆×C₃₁	—	4	2	no	—
65	SD₁₆×C₃₁	—	4	4	no	—
66	SD₁₆×C₃₁	—	4	4	no	—
67	D₈×C₃₁	—	2	4	no	—
68	D₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	D₈	1	2	yes	—
72	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
73	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	2	yes	—
77	C₄⋊C₄×C₃₁	—	4	2	no	—
78	C₄⋊C₄×C₃₁	—	4	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	8	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	4	no	—
82	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₃₁	SD₁₆	1	2	yes	—
85	C₁₆×C₃₁	—	2	4	no	—
86	#13210	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	4	2	no	—
88	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₃₁	—	4	2	no	—
90	Q₈×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	D₄×C₃₁	—	4	2	no	—
93	D₄×C₃₁	—	8	2	no	—
94	D₄×C₃₁	—	8	2	no	—
95	D₄×C₃₁	—	8	4	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	—	8	4	no	—
101	C₂×C₄×C₃₁	—	8	2	no	—
102	C₂×C₄×C₃₁	#102	1	2	yes	—
103	C₂×C₄×C₃₁	#102	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₃₁	#102	1	2	yes	—
111	C₈×C₃₁	#102	1	2	yes	—
112	C₈×C₃₁	C₄×D₄	1	2	yes	—
113	C₈×C₃₁	#121	1	2	yes	—
114	#3147	C₂×C₃₁	1	1	yes	characteristic
115	C₂×#360	C₂×C₃₁	1	4	yes	complemented
116	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₂²×C₃₁	—	2	2	no	—
119	C₂²×C₃₁	—	8	4	no	—
120	C₂²×C₃₁	—	8	2	no	—
121	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
122	C₂²×C₃₁	#486	1	1	yes	characteristic
123	C₂²×C₃₁	#493	1	1	yes	characteristic
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	8	2	no	—
126	C₄×C₃₁	—	16	2	no	—
127	C₄×C₃₁	#360	1	4	yes	complemented
128	C₄×C₃₁	#390	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
130	C₄×C₃₁	#486	1	1	yes	characteristic
131	C₄×C₃₁	#492	1	1	yes	characteristic
132	C₂×#135	—	2	1	no	—
133	C₂×#135	C₂²×C₃₁	1	2	yes	—
134	#532	—	2	1	no	—
135	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
136	#495	—	2	1	no	—
137	#495	C₄×C₃₁	1	2	yes	complemented
138	#494	C₄×C₃₁	1	1	yes	characteristic, complemented
139	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
140	#492	—	2	1	no	—
141	C₂×#106	—	2	1	no	—
142	C₂×#106	C₂²×C₃₁	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	#384	—	2	1	no	—
145	#360	—	2	8	no	—
146	#346	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	C₂×C₃₁	—	2	1	no	—
149	C₂×C₃₁	—	2	1	no	—
150	C₂×C₃₁	—	16	2	no	—
151	C₂×C₃₁	#1957	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#360	1	2	yes	—
153	C₂×#41	—	2	2	no	—
154	#135	—	2	4	no	—
155	#135	—	2	4	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
164	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#124	—	2	2	no	—
169	#121	—	4	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	4	2	no	—
173	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
174	M₃₂	—	2	2	no	—
175	M₃₂	D₄×C₃₁	1	4	yes	complemented
176	C₂×C₁₆	—	2	2	no	—
177	#106	—	2	4	no	—
178	#106	—	4	2	no	—
179	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₃₁	#13210	1	1	yes	characteristic, direct product factor, Sylow
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	D₈×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	Q₁₆	—	2	4	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	2	4	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	D₈×C₃₁	1	2	yes	—
205	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₈	#41×C₃₁	1	2	yes	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	8	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₄²	SD₁₆×C₃₁	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₃₁×#102	1	2	yes	—
235	C₂×C₄	C₃₁×#102	1	2	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
238	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
242	C₈	C₃₁×#102	1	2	yes	—
243	C₈	C₃₁×#102	1	2	yes	—
244	C₈	C₄×D₄×C₃₁	1	2	yes	—
245	C₈	C₃₁×#121	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
250	C₂²	C₃₁×#486	1	1	yes	characteristic
251	C₂²	C₃₁×#493	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	16	2	no	—
255	C₄	C₃₁×#360	1	4	yes	complemented
256	C₄	C₃₁×#390	1	1	yes	characteristic
257	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
258	C₄	C₃₁×#486	1	1	yes	characteristic
259	C₄	C₃₁×#492	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	C₃₁×#1957	1	1	yes	characteristic
264	C₂	C₂×C₃₁×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#360	C₂	semidirect
3. C₄×C₃₁×M₃₂	C₂	semidirect
4. C₃₁×#1844	C₂	semidirect
10. C₃₁×#495	C₄	semidirect
11. C₃₁×#494	C₄	semidirect
12. C₃₁×#493	C₄	semidirect
43. C₃₁×M₃₂	D₄	semidirect
86. #13210	C₃₁	direct (N × H)
115. C₂×#360	C₂×C₃₁	semidirect
116. C₄×M₃₂	C₂×C₃₁	semidirect
117. #1844	C₂×C₃₁	semidirect
127. C₄×C₃₁	#360	semidirect
137. #495	C₄×C₃₁	semidirect
138. #494	C₄×C₃₁	semidirect
139. #493	C₄×C₃₁	semidirect
175. M₃₂	D₄×C₃₁	semidirect
183. C₃₁	#13210	direct (N × H)
255. C₄	C₃₁×#360	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3147	2
C₂×C₃₁×#360	2
C₄×C₃₁×M₃₂	2
C₃₁×#1844	2
#13210	31

Subgroups — C31×#13210 #21469969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13210 #21469969