Subgroups — C₃₁×#33190 #21489949

Order 7936.

1082 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#472	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1978	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1965	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
9	C₂²×D₈×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	C₂²	1	2	yes	—
12	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#493	—	2	1	no	—
14	C₃₁×#493	—	2	1	no	—
15	C₃₁×#493	—	2	2	no	—
16	C₃₁×#472	—	2	16	no	—
17	C₃₁×#460	—	2	4	no	—
18	C₃₁×#459	—	2	4	no	—
19	C₂×C₃₁×#108	C₂²	1	4	yes	—
20	C₃₁×#423	C₂²	1	2	yes	—
21	C₃₁×#418	C₂²	1	4	yes	—
22	C₂×C₃₁×#100	C₂²	1	8	yes	—
23	C₃₁×#405	—	2	2	no	—
24	C₃₁×#405	C₂²	1	2	yes	—
25	C₄×M₁₆×C₃₁	—	2	1	no	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₃₁	—	2	4	no	—
31	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
32	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
33	C₂×SD₁₆×C₃₁	—	2	4	no	—
34	C₂×D₈×C₃₁	—	2	4	no	—
35	C₂×D₈×C₃₁	—	2	8	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂×D₈×C₃₁	—	2	4	no	—
38	C₂×D₈×C₃₁	—	2	8	no	—
39	C₃₁×#131	—	2	4	no	—
40	C₃₁×#131	D₄	1	8	yes	complemented
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
43	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#127	—	2	4	no	—
47	C₃₁×#127	—	2	2	no	—
48	C₃₁×#127	—	2	2	no	—
49	C₄×Q₈×C₃₁	D₄	1	2	yes	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	D₄	1	2	yes	—
52	C₃₁×#117	—	2	1	no	—
53	C₃₁×#117	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	—	2	1	no	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#108	—	2	8	no	—
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₃₁×#102	—	2	8	no	—
59	C₃₁×#100	—	2	16	no	—
60	C₃₁×#97	—	2	2	no	—
61	C₃₁×#97	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	2	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	2	yes	—
67	C₂⁴×C₃₁	—	2	4	no	—
68	#41×C₃₁	—	2	2	no	—
69	#41×C₃₁	C₂×D₄	1	4	yes	—
70	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×D₄×C₃₁	—	2	1	no	—
72	C₂×D₄×C₃₁	—	2	8	no	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	2	4	no	—
75	C₂×D₄×C₃₁	—	2	8	no	—
76	C₂×D₄×C₃₁	—	4	4	no	—
77	C₂×D₄×C₃₁	—	4	4	no	—
78	C₂×D₄×C₃₁	—	8	8	no	—
79	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	—	2	1	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	#41	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₃₁	—	4	8	no	—
85	D₈×C₃₁	—	4	8	no	—
86	D₈×C₃₁	—	4	8	no	—
87	D₈×C₃₁	—	4	8	no	—
88	M₁₆×C₃₁	—	2	2	no	—
89	M₁₆×C₃₁	—	2	8	no	—
90	M₁₆×C₃₁	—	2	8	no	—
91	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	8	no	—
94	C₂×C₈×C₃₁	—	2	4	no	—
95	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₃₁	#41	1	4	yes	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²⋊C₄×C₃₁	—	2	1	no	—
102	C₂²⋊C₄×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	2	yes	—
108	#33190	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	2	1	no	—
110	C₂³×C₃₁	—	4	4	no	—
111	C₂³×C₃₁	—	4	8	no	—
112	C₂³×C₃₁	—	8	8	no	—
113	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	1	no	—
115	Q₈×C₃₁	#127	1	2	yes	—
116	D₄×C₃₁	—	2	4	no	—
117	D₄×C₃₁	—	2	1	no	—
118	D₄×C₃₁	—	4	8	no	—
119	D₄×C₃₁	—	4	4	no	—
120	D₄×C₃₁	—	4	4	no	—
121	D₄×C₃₁	—	8	8	no	—
122	D₄×C₃₁	—	8	8	no	—
123	D₄×C₃₁	#127	1	2	yes	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	4	no	—
126	C₂×C₄×C₃₁	—	2	4	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	#121	1	2	yes	—
130	C₂×C₄×C₃₁	#121	1	2	yes	—
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
135	C₈×C₃₁	—	2	4	no	—
136	C₈×C₃₁	—	4	4	no	—
137	C₈×C₃₁	#121	1	4	yes	—
138	C₈×C₃₁	#121	1	4	yes	—
139	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂×#472	C₂×C₃₁	1	4	yes	complemented
141	#3038	C₂×C₃₁	1	2	yes	complemented
142	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1978	C₂×C₃₁	1	4	yes	complemented
144	#1965	C₂×C₃₁	1	2	yes	complemented
145	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	—	8	4	no	—
150	C₂²×C₃₁	—	8	8	no	—
151	C₂²×C₃₁	—	16	8	no	—
152	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
153	C₂²×C₃₁	#532	1	2	yes	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	4	2	no	—
157	C₄×C₃₁	#472	1	4	yes	complemented
158	C₄×C₃₁	C₂×#121	1	2	yes	—
159	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
160	C₄×C₃₁	#532	1	1	yes	characteristic
161	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
162	C₂²×D₈	C₂²×C₃₁	1	4	yes	—
163	C₂×#131	C₂²×C₃₁	1	2	yes	—
164	C₂×#127	C₂²×C₃₁	1	2	yes	—
165	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
166	#493	—	2	1	no	—
167	#493	—	2	1	no	—
168	#493	—	2	2	no	—
169	#472	—	2	16	no	—
170	#460	—	2	4	no	—
171	#459	—	2	4	no	—
172	C₂×#108	C₂²×C₃₁	1	4	yes	—
173	#423	C₂²×C₃₁	1	2	yes	—
174	#418	C₂²×C₃₁	1	4	yes	—
175	C₂×#100	C₂²×C₃₁	1	8	yes	—
176	#405	—	2	2	no	—
177	#405	C₂²×C₃₁	1	2	yes	—
178	C₄×M₁₆	—	2	1	no	—
179	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄×C₈	—	2	1	no	—
181	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₃₁	—	2	2	no	—
183	C₂×C₃₁	—	4	2	no	—
184	C₂×C₃₁	—	16	4	no	—
185	C₂×C₃₁	#2856	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#472	1	2	yes	—
187	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
188	C₂²×D₄	—	2	4	no	—
189	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
190	AGL(1,ℤ/8)	—	2	16	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×D₈	—	2	4	no	—
193	C₂×D₈	—	2	8	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	8	no	—
197	#131	—	2	4	no	—
198	#131	D₄×C₃₁	1	8	yes	complemented
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
201	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
204	#127	—	2	4	no	—
205	#127	—	2	2	no	—
206	#127	—	2	2	no	—
207	C₄×Q₈	D₄×C₃₁	1	2	yes	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	D₄×C₃₁	1	2	yes	—
210	#117	—	2	1	no	—
211	#117	C₂³×C₃₁	1	1	yes	characteristic
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
214	#108	—	2	8	no	—
215	C₄⋊C₈	—	2	4	no	—
216	#102	—	2	8	no	—
217	#100	—	2	16	no	—
218	#97	—	2	2	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₃₁	1	2	yes	—
225	C₃₁	#33190	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	—	2	4	no	—
227	#41	—	2	2	no	—
228	#41	C₂×D₄×C₃₁	1	4	yes	—
229	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	8	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	8	8	no	—
238	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	#41×C₃₁	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
243	SD₁₆	—	4	8	no	—
244	D₈	—	4	8	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
257	C₂×C₈	#41×C₃₁	1	4	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₃₁	1	2	yes	—
267	C₂³	—	2	1	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	8	8	no	—
271	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	C₃₁×#127	1	2	yes	—
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	8	8	no	—
280	D₄	—	8	8	no	—
281	D₄	C₃₁×#127	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₃₁×#121	1	2	yes	—
288	C₂×C₄	C₃₁×#121	1	2	yes	—
289	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	4	4	no	—
295	C₈	C₃₁×#121	1	4	yes	—
296	C₈	C₃₁×#121	1	4	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	8	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
304	C₂²	C₃₁×#532	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	C₃₁×#472	1	4	yes	complemented
309	C₄	C₂×C₃₁×#121	1	2	yes	—
310	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
311	C₄	C₃₁×#532	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	—	16	4	no	—
315	C₂	C₃₁×#2856	1	1	yes	characteristic
316	C₂	C₂×C₃₁×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#493	C₂	semidirect
2. C₂×C₃₁×#472	C₂	semidirect
3. C₃₁×#3038	C₂	semidirect
4. C₄×C₃₁×#131	C₂	semidirect
5. C₃₁×#1978	C₂	semidirect
6. C₃₁×#1965	C₂	semidirect
7. C₃₁×#1848	C₂	semidirect
40. C₃₁×#131	D₄	semidirect
108. #33190	C₃₁	direct (N × H)
139. C₂×#493	C₂×C₃₁	semidirect
140. C₂×#472	C₂×C₃₁	semidirect
141. #3038	C₂×C₃₁	semidirect
142. C₄×#131	C₂×C₃₁	semidirect
143. #1978	C₂×C₃₁	semidirect
144. #1965	C₂×C₃₁	semidirect
145. #1848	C₂×C₃₁	semidirect
157. C₄×C₃₁	#472	semidirect
198. #131	D₄×C₃₁	semidirect
225. C₃₁	#33190	direct (N × H)
308. C₄	C₃₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₂×C₃₁×#472	2
C₃₁×#3038	2
C₄×C₃₁×#131	2
C₃₁×#1978	2
C₃₁×#1965	2
C₃₁×#1848	2
#33190	31

Subgroups — C31×#33190 #21489949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33190 #21489949