Subgroups — C₃₁×#33210 #21489969

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2857	C₂	1	2	yes	maximal
4	C₃₁×#1990	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1943	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#498	—	2	4	no	—
13	C₂×C₃₁×#107	C₂²	1	4	yes	—
14	C₂×C₃₁×#106	C₂²	1	4	yes	—
15	C₄×M₁₆×C₃₁	—	2	4	no	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#391	—	2	2	no	—
20	C₃₁×#389	—	2	2	no	—
21	C₃₁×#389	C₂²	1	4	yes	—
22	C₃₁×#384	C₂²	1	2	yes	—
23	C₃₁×#380	—	2	2	no	—
24	C₃₁×#380	C₂²	1	4	yes	—
25	C₃₁×#340	C₂²	1	8	yes	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₃₁×C₄⋊Q₈	—	2	4	no	—
35	C₃₁×C₄⋊Q₈	—	2	4	no	—
36	C₃₁×C₂²⋊Q₈	—	4	8	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#107	—	2	8	no	—
50	C₃₁×#106	—	2	8	no	—
51	C₃₁×#97	—	2	4	no	—
52	C₄×C₈×C₃₁	—	2	2	no	—
53	C₄×C₈×C₃₁	D₄	1	4	yes	—
54	C₃₁×#95	—	2	4	no	—
55	C₃₁×#95	—	2	4	no	—
56	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₃₁	—	4	4	no	—
58	C₂×Q₈×C₃₁	—	4	4	no	—
59	C₂×Q₈×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	16	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₃₁	#41	1	4	yes	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	4	8	no	—
84	C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	—	4	4	no	—
87	C₂²⋊C₄×C₃₁	—	4	8	no	—
88	C₄²×C₃₁	—	2	4	no	—
89	C₄²×C₃₁	—	2	1	no	—
90	C₄²×C₃₁	—	2	1	no	—
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	C₄²×C₃₁	#41	1	2	yes	—
93	#33210	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	1	no	—
95	C₂³×C₃₁	—	2	1	no	—
96	C₂³×C₃₁	—	2	2	no	—
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₃₁	—	8	8	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	8	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	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	#121	1	4	yes	—
116	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₄×C₃₁	#137	1	4	yes	—
119	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₈×C₃₁	—	2	4	no	—
124	C₈×C₃₁	C₂²⋊Q₈	1	8	yes	complemented
125	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
127	#2857	C₂×C₃₁	1	2	yes	—
128	#1990	C₂×C₃₁	1	2	yes	complemented
129	#1943	C₂×C₃₁	1	4	yes	complemented
130	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1809	C₂×C₃₁	1	4	yes	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	4	1	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₃₁	#532	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₃₁	C₂×#137	1	2	yes	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	—	8	4	no	—
146	C₄×C₃₁	#468	1	8	yes	—
147	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
148	C₄×C₃₁	#533	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊Q₈	—	2	2	no	—
152	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
153	#498	—	2	4	no	—
154	C₂×#107	C₂²×C₃₁	1	4	yes	—
155	C₂×#106	C₂²×C₃₁	1	4	yes	—
156	C₄×M₁₆	—	2	4	no	—
157	C₂×#97	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#391	—	2	2	no	—
161	#389	—	2	2	no	—
162	#389	C₂²×C₃₁	1	4	yes	—
163	#384	C₂²×C₃₁	1	2	yes	—
164	#380	—	2	2	no	—
165	#380	C₂²×C₃₁	1	4	yes	—
166	#340	C₂²×C₃₁	1	8	yes	—
167	C₂×C₃₁	—	4	2	no	—
168	C₂×C₃₁	#2857	1	1	yes	characteristic
169	C₂×C₃₁	C₂×#468	1	2	yes	—
170	C₂×C₃₁	#3165	1	2	yes	—
171	C₂×C₃₁	#3184	1	1	yes	characteristic
172	C₂×C₃₁	#3185	1	1	yes	characteristic
173	C₂²×Q₈	—	2	2	no	—
174	C₂³×C₄	D₄×C₃₁	1	2	yes	—
175	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
179	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
180	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₂²⋊Q₈	—	4	8	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	4	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	D₄×C₃₁	1	2	yes	—
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	#107	—	2	8	no	—
197	#106	—	2	8	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	D₄×C₃₁	1	4	yes	—
201	#95	—	2	4	no	—
202	#95	—	2	4	no	—
203	C₃₁	#33210	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
221	M₁₆	—	2	16	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
225	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
226	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
227	C₂×C₈	#41×C₃₁	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	C₂×D₄×C₃₁	1	2	yes	—
240	C₄²	#41×C₃₁	1	2	yes	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
246	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
247	Q₈	—	8	8	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₃₁×#121	1	4	yes	—
263	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
264	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
265	C₂×C₄	C₃₁×#137	1	4	yes	—
266	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
268	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
269	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	C₃₁×C₂²⋊Q₈	1	8	yes	complemented
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
278	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
279	C₂²	C₃₁×#532	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
281	C₂²	C₂×C₃₁×#137	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	4	no	—
286	C₄	C₃₁×#468	1	8	yes	—
287	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
288	C₄	C₃₁×#533	1	2	yes	—
289	C₂	—	4	2	no	—
290	C₂	C₃₁×#2857	1	1	yes	characteristic
291	C₂	C₂×C₃₁×#468	1	2	yes	—
292	C₂	C₃₁×#3165	1	2	yes	—
293	C₂	C₃₁×#3184	1	1	yes	characteristic
294	C₂	C₃₁×#3185	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#498	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1943	C₂	semidirect
6. C₃₁×#1845	C₂	semidirect
31. C₂×M₁₆×C₃₁	Q₈	semidirect
93. #33210	C₃₁	direct (N × H)
124. C₈×C₃₁	C₂²⋊Q₈	semidirect
125. C₂×#498	C₂×C₃₁	semidirect
128. #1990	C₂×C₃₁	semidirect
129. #1943	C₂×C₃₁	semidirect
130. #1845	C₂×C₃₁	semidirect
178. C₂×M₁₆	Q₈×C₃₁	semidirect
203. C₃₁	#33210	direct (N × H)
271. C₈	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#498	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#1990	2
C₃₁×#1943	2
C₃₁×#1845	2
C₃₁×#1809	2
#33210	31

Subgroups — C31×#33210 #21489969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33210 #21489969