Subgroups — C₃₁×#22280 #21479039

Order 7936.

846 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	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2931	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1962	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1876	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#135	C₂²	1	4	yes	—
8	C₂×C₃₁×#131	C₂²	1	4	yes	—
9	C₃₁×#532	C₂²	1	2	yes	—
10	C₃₁×#495	C₄	1	4	yes	complemented
11	C₃₁×#494	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#492	C₄	1	2	yes	complemented
14	C₃₁×#436	—	2	2	no	—
15	C₃₁×#436	—	2	2	no	—
16	C₈×D₄×C₃₁	—	2	2	no	—
17	C₈×D₄×C₃₁	—	2	2	no	—
18	C₃₁×#423	C₂²	1	2	yes	—
19	C₂×C₃₁×#104	C₂²	1	8	yes	—
20	C₃₁×#408	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#341	C₂²	1	4	yes	—
25	C₃₁×#336	C₂²	1	4	yes	—
26	C₂×#41×C₃₁	C₂³	1	4	yes	—
27	C₃₁×#135	—	4	8	no	—
28	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	—
30	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
31	C₃₁×#131	—	4	8	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₃₁×#127	C₂×C₄	1	2	yes	—
39	C₃₁×#124	C₂×C₄	1	4	yes	—
40	C₃₁×#121	—	2	2	no	—
41	C₃₁×#121	—	2	2	no	—
42	C₄×D₄×C₃₁	—	2	2	no	—
43	C₄×D₄×C₃₁	—	2	2	no	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	—	2	2	no	—
49	C₃₁×C₄⋊C₈	—	2	2	no	—
50	C₃₁×#104	—	4	8	no	—
51	C₃₁×C₂²⋊C₈	—	2	4	no	—
52	C₃₁×C₂²⋊C₈	—	2	4	no	—
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₄×C₈×C₃₁	D₄	1	2	yes	—
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	#41×C₃₁	—	2	8	no	—
60	#41×C₃₁	—	2	8	no	—
61	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
71	Q₁₆×C₃₁	—	4	4	no	—
72	SD₁₆×C₃₁	—	4	8	no	—
73	D₈×C₃₁	—	4	4	no	—
74	M₁₆×C₃₁	—	2	8	no	—
75	M₁₆×C₃₁	—	4	8	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	4	8	no	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂²⋊C₄×C₃₁	—	2	4	no	—
88	C₄²×C₃₁	—	2	4	no	—
89	C₄²×C₃₁	—	2	4	no	—
90	C₄²×C₃₁	—	4	2	no	—
91	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	#22280	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	4	no	—
97	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
98	Q₈×C₃₁	—	2	8	no	—
99	D₄×C₃₁	—	2	8	no	—
100	D₄×C₃₁	—	4	8	no	—
101	D₄×C₃₁	—	4	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	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
118	C₈×C₃₁	—	2	2	no	—
119	C₈×C₃₁	—	2	2	no	—
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	2	2	no	—
122	C₈×C₃₁	—	4	4	no	—
123	#3147	C₂×C₃₁	1	1	yes	characteristic
124	#2931	C₂×C₃₁	1	2	yes	complemented
125	#1962	C₂×C₃₁	1	2	yes	complemented
126	#1876	C₂×C₃₁	1	8	yes	complemented
127	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂²×C₃₁	—	2	1	no	—
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	4	4	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	4	4	no	—
134	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₃₁	#540	1	2	yes	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	—	4	4	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	#443	1	2	yes	—
141	C₄×C₃₁	#444	1	2	yes	—
142	C₄×C₃₁	#518	1	2	yes	—
143	C₄×C₃₁	#535	1	2	yes	—
144	C₂×#135	C₂²×C₃₁	1	4	yes	—
145	C₂×#131	C₂²×C₃₁	1	4	yes	—
146	#532	C₂²×C₃₁	1	2	yes	—
147	#495	C₄×C₃₁	1	4	yes	complemented
148	#494	C₄×C₃₁	1	1	yes	characteristic, complemented
149	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
150	#492	C₄×C₃₁	1	2	yes	complemented
151	#436	—	2	2	no	—
152	#436	—	2	2	no	—
153	C₈×D₄	—	2	2	no	—
154	C₈×D₄	—	2	2	no	—
155	#423	C₂²×C₃₁	1	2	yes	—
156	C₂×#104	C₂²×C₃₁	1	8	yes	—
157	#408	C₂²×C₃₁	1	4	yes	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₄³	C₂²×C₃₁	1	1	yes	characteristic
161	#341	C₂²×C₃₁	1	4	yes	—
162	#336	C₂²×C₃₁	1	4	yes	—
163	C₂×C₃₁	—	2	2	no	—
164	C₂×C₃₁	—	8	4	no	—
165	C₂×C₃₁	#2328	1	1	yes	characteristic
166	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
167	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
168	C₂×#41	C₂³×C₃₁	1	4	yes	—
169	#135	—	4	8	no	—
170	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
171	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	—
172	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
173	#131	—	4	8	no	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
179	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
180	#127	C₂×C₄×C₃₁	1	2	yes	—
181	#124	C₂×C₄×C₃₁	1	4	yes	—
182	#121	—	2	2	no	—
183	#121	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	#104	—	4	8	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	C₂²⋊C₈	—	2	4	no	—
195	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
197	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
198	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
199	C₄×C₈	D₄×C₃₁	1	2	yes	—
200	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
201	C₃₁	#22280	1	1	yes	characteristic, direct product factor, Sylow
202	#41	—	2	8	no	—
203	#41	—	2	8	no	—
204	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	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	4	no	—
211	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
214	Q₁₆	—	4	4	no	—
215	SD₁₆	—	4	8	no	—
216	D₈	—	4	4	no	—
217	M₁₆	—	2	8	no	—
218	M₁₆	—	4	8	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	8	no	—
223	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
224	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	4	2	no	—
234	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂³	—	2	4	no	—
239	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	D₄	—	2	8	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
271	C₂²	C₃₁×#540	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	4	no	—
276	C₄	C₃₁×#443	1	2	yes	—
277	C₄	C₃₁×#444	1	2	yes	—
278	C₄	C₃₁×#518	1	2	yes	—
279	C₄	C₃₁×#535	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	8	4	no	—
282	C₂	C₃₁×#2328	1	1	yes	characteristic
283	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
284	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2931	C₂	semidirect
3. C₃₁×#1962	C₂	semidirect
4. C₃₁×#1876	C₂	semidirect
5. C₃₁×#1837	C₂	semidirect
6. C₃₁×#1744	C₂	semidirect
10. C₃₁×#495	C₄	semidirect
11. C₃₁×#494	C₄	semidirect
12. C₃₁×#493	C₄	semidirect
13. C₃₁×#492	C₄	semidirect
95. #22280	C₃₁	direct (N × H)
124. #2931	C₂×C₃₁	semidirect
125. #1962	C₂×C₃₁	semidirect
126. #1876	C₂×C₃₁	semidirect
127. #1837	C₂×C₃₁	semidirect
128. #1744	C₂×C₃₁	semidirect
147. #495	C₄×C₃₁	semidirect
148. #494	C₄×C₃₁	semidirect
149. #493	C₄×C₃₁	semidirect
150. #492	C₄×C₃₁	semidirect
201. C₃₁	#22280	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3147	2
C₃₁×#2931	2
C₃₁×#1962	2
C₃₁×#1876	2
C₃₁×#1837	2
C₃₁×#1744	2
#22280	31

Subgroups — C31×#22280 #21479039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22280 #21479039