Subgroups — C₃₁×#21330 #21478089

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3458	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1995	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1994	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1733	C₂	1	2	yes	maximal
7	C₂×C₃₁×#339	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#544	—	2	2	no	—
10	C₃₁×#544	C₄	1	8	yes	complemented
11	C₃₁×#543	—	2	2	no	—
12	C₃₁×#533	—	2	4	no	—
13	C₃₁×#533	C₂²	1	4	yes	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#424	—	2	4	no	—
19	C₃₁×#423	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
22	C₃₁×#376	C₂²	1	2	yes	—
23	C₄³×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#339	C₂²	1	4	yes	—
25	C₃₁×#339	C₂²	1	4	yes	—
26	C₃₁×#339	C₂²	1	8	yes	—
27	C₂³×C₄×C₃₁	—	2	1	no	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₃₁×C₄⋊Q₈	—	2	8	no	—
35	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
36	C₃₁×#125	—	2	8	no	—
37	C₃₁×#125	C₂×C₄	1	8	yes	complemented
38	C₃₁×C₂²⋊Q₈	—	2	8	no	—
39	C₃₁×C₂²⋊Q₈	—	2	16	no	—
40	C₄×Q₈×C₃₁	—	2	8	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	C₂×C₄	1	4	yes	—
45	C₃₁×#117	C₂×C₄	1	4	yes	—
46	C₃₁×#117	D₄	1	4	yes	—
47	C₃₁×#117	Q₈	1	4	yes	complemented
48	C₃₁×#117	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
55	C₂×C₄²×C₃₁	—	2	1	no	—
56	C₂×C₄²×C₃₁	—	2	2	no	—
57	C₂×C₄²×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	—	2	4	no	—
59	C₂×C₄²×C₃₁	Q₈	1	4	yes	complemented
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₃₁×C₄⋊C₈	—	2	8	no	—
65	C₃₁×#95	—	2	4	no	—
66	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₃₁	—	2	8	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	4	4	no	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₃₁	#41	1	2	yes	—
81	C₂²×C₄×C₃₁	#41	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	—	2	4	no	—
85	M₁₆×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	4	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₄⋊C₄×C₃₁	—	2	16	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
93	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄×C₃₁	—	2	4	no	—
95	C₂²⋊C₄×C₃₁	—	2	4	no	—
96	C₂²⋊C₄×C₃₁	—	2	4	no	—
97	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	2	4	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	4	8	no	—
105	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
106	C₄²×C₃₁	C₂×D₄	1	4	yes	—
107	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
108	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
109	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
110	#21330	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
115	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₃₁	—	8	8	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	8	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	8	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
128	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
130	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₃₁	#124	1	4	yes	—
134	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
135	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₃₁	—	4	4	no	—
142	#3458	C₂×C₃₁	1	1	yes	characteristic
143	#2905	C₂×C₃₁	1	1	yes	characteristic
144	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
145	#1995	C₂×C₃₁	1	4	yes	complemented
146	#1994	C₂×C₃₁	1	4	yes	complemented
147	#1733	C₂×C₃₁	1	2	yes	—
148	C₂×#339	C₂×C₃₁	1	2	yes	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	2	4	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	#391	1	4	yes	—
154	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₄×C₃₁	—	2	4	no	—
158	C₄×C₃₁	—	2	8	no	—
159	C₄×C₃₁	—	4	4	no	—
160	C₄×C₃₁	—	4	2	no	—
161	C₄×C₃₁	—	4	8	no	—
162	C₄×C₃₁	#391	1	4	yes	—
163	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
166	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
168	#544	—	2	2	no	—
169	#544	C₄×C₃₁	1	8	yes	complemented
170	#543	—	2	2	no	—
171	#533	—	2	4	no	—
172	#533	C₂²×C₃₁	1	4	yes	—
173	C₂×#117	C₂²×C₃₁	1	2	yes	—
174	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
177	#424	—	2	4	no	—
178	#423	C₂²×C₃₁	1	4	yes	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₂²⋊C₄	—	2	2	no	—
181	#376	C₂²×C₃₁	1	2	yes	—
182	C₄³	C₂²×C₃₁	1	2	yes	—
183	#339	C₂²×C₃₁	1	4	yes	—
184	#339	C₂²×C₃₁	1	4	yes	—
185	#339	C₂²×C₃₁	1	8	yes	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	4	no	—
189	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
190	C₂×C₃₁	#2901	1	2	yes	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
200	#125	—	2	8	no	—
201	#125	C₂×C₄×C₃₁	1	8	yes	complemented
202	C₂²⋊Q₈	—	2	8	no	—
203	C₂²⋊Q₈	—	2	16	no	—
204	C₄×Q₈	—	2	8	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	C₂×C₄×C₃₁	1	4	yes	—
209	#117	C₂×C₄×C₃₁	1	4	yes	—
210	#117	D₄×C₃₁	1	4	yes	—
211	#117	Q₈×C₃₁	1	4	yes	complemented
212	#117	C₂³×C₃₁	1	4	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
215	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	Q₈×C₃₁	1	4	yes	complemented
224	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
225	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
228	C₄⋊C₈	—	2	8	no	—
229	#95	—	2	4	no	—
230	C₃₁	#21330	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂×Q₈	—	2	8	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
245	C₂²×C₄	#41×C₃₁	1	2	yes	—
246	C₂²×C₄	#41×C₃₁	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
258	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	8	no	—
270	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
271	C₄²	C₂×D₄×C₃₁	1	4	yes	—
272	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
273	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
274	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
277	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
278	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
279	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
280	Q₈	—	8	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
292	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
293	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
294	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
295	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
296	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
297	C₂×C₄	C₃₁×#124	1	4	yes	—
298	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
299	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₈	—	4	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₃₁×#391	1	4	yes	—
311	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
312	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
313	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	8	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	8	no	—
319	C₄	C₃₁×#391	1	4	yes	—
320	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
321	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
322	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
323	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	4	no	—
327	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
328	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1995	C₂	semidirect
5. C₃₁×#1994	C₂	semidirect
10. C₃₁×#544	C₄	semidirect
35. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
37. C₃₁×#125	C₂×C₄	semidirect
47. C₃₁×#117	Q₈	semidirect
59. C₂×C₄²×C₃₁	Q₈	semidirect
110. #21330	C₃₁	direct (N × H)
145. #1995	C₂×C₃₁	semidirect
146. #1994	C₂×C₃₁	semidirect
169. #544	C₄×C₃₁	semidirect
199. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
201. #125	C₂×C₄×C₃₁	semidirect
211. #117	Q₈×C₃₁	semidirect
223. C₂×C₄²	Q₈×C₃₁	semidirect
230. C₃₁	#21330	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×#3458	2
C₃₁×#2905	2
C₄×C₃₁×#117	2
C₃₁×#1995	2
C₃₁×#1994	2
C₃₁×#1733	2
C₂×C₃₁×#339	2
#21330	31

Subgroups — C31×#21330 #21478089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21330 #21478089