Subgroups — C₃₁×#21158 #21477917

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2316	C₂	1	2	yes	maximal
5	C₃₁×#1943	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1942	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	2	yes	—
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#427	—	2	4	no	—
14	C₂×C₃₁×#107	C₄	1	8	yes	complemented
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#389	—	2	4	no	—
23	C₃₁×#389	C₂²	1	8	yes	—
24	C₃₁×#380	—	2	4	no	—
25	C₃₁×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₃₁	D₄	1	4	yes	—
33	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	—	2	4	no	—
40	C₃₁×#117	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₃₁×#107	—	2	4	no	—
53	C₃₁×#107	C₂×C₄	1	8	yes	—
54	C₃₁×#107	C₂×C₄	1	8	yes	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×#97	—	2	4	no	—
57	C₄×C₈×C₃₁	—	2	4	no	—
58	C₃₁×#95	—	2	8	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	8	4	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	8	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	8	8	no	—
90	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
91	C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	4	4	no	—
96	#21158	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	C₄×D₄	1	2	yes	—
100	C₂³×C₃₁	#121	1	2	yes	—
101	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	4	8	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	8	8	no	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₃₁	#121	1	2	yes	—
116	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
117	C₂×C₄×C₃₁	#125	1	4	yes	—
118	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
125	C₈×C₃₁	—	2	4	no	—
126	C₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
127	C₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
129	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
130	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
131	#2316	C₂×C₃₁	1	2	yes	—
132	#1943	C₂×C₃₁	1	4	yes	complemented
133	#1942	C₂×C₃₁	1	4	yes	complemented
134	#1736	C₂×C₃₁	1	2	yes	—
135	C₂²×C₃₁	—	2	4	no	—
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	4	no	—
139	C₂²×C₃₁	#389	1	4	yes	—
140	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
142	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₂²×C₃₁	#576	1	2	yes	—
144	C₂²×C₃₁	#578	1	2	yes	—
145	C₄×C₃₁	—	4	8	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	#389	1	4	yes	—
149	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×#117	C₂²×C₃₁	1	2	yes	—
156	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
158	#427	—	2	4	no	—
159	C₂×#107	C₄×C₃₁	1	8	yes	complemented
160	C₂×#107	C₂²×C₃₁	1	2	yes	—
161	C₂×#107	C₂²×C₃₁	1	2	yes	—
162	C₂×#107	C₂²×C₃₁	1	2	yes	—
163	C₂×#106	C₂²×C₃₁	1	2	yes	—
164	#405	—	2	4	no	—
165	C₂×#97	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
167	#389	—	2	4	no	—
168	#389	C₂²×C₃₁	1	8	yes	—
169	#380	—	2	4	no	—
170	#340	C₂²×C₃₁	1	8	yes	—
171	C₂×C₃₁	—	2	4	no	—
172	C₂×C₃₁	C₂×#389	1	1	yes	characteristic
173	C₂×C₃₁	#2948	1	1	yes	characteristic
174	C₂×C₃₁	#2950	1	1	yes	characteristic
175	C₂×C₃₁	#3061	1	2	yes	—
176	C₂×C₃₁	#3164	1	2	yes	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
183	C₂²×C₈	D₄×C₃₁	1	4	yes	—
184	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
185	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	4	no	—
191	#117	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
196	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
203	#107	—	2	4	no	—
204	#107	C₂×C₄×C₃₁	1	8	yes	—
205	#107	C₂×C₄×C₃₁	1	8	yes	—
206	#106	—	2	4	no	—
207	#97	—	2	4	no	—
208	C₄×C₈	—	2	4	no	—
209	#95	—	2	8	no	—
210	C₃₁	#21158	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
222	C₂²×C₄	#41×C₃₁	1	2	yes	—
223	C₂²×C₄	#41×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₄⋊C₄×C₃₁	1	16	yes	complemented
229	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
233	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
234	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
235	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	8	8	no	—
242	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₄×D₄×C₃₁	1	2	yes	—
251	C₂³	C₃₁×#121	1	2	yes	—
252	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
253	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	8	no	—
263	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
264	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
266	C₂×C₄	C₃₁×#121	1	2	yes	—
267	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
268	C₂×C₄	C₃₁×#125	1	4	yes	—
269	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
270	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
278	C₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₃₁×#389	1	4	yes	—
284	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
285	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
286	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
287	C₂²	C₃₁×#576	1	2	yes	—
288	C₂²	C₃₁×#578	1	2	yes	—
289	C₄	—	4	8	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	4	no	—
292	C₄	C₃₁×#389	1	4	yes	—
293	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
294	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
295	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
296	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×C₃₁×#389	1	1	yes	characteristic
299	C₂	C₃₁×#2948	1	1	yes	characteristic
300	C₂	C₃₁×#2950	1	1	yes	characteristic
301	C₂	C₃₁×#3061	1	2	yes	—
302	C₂	C₃₁×#3164	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1943	C₂	semidirect
6. C₃₁×#1942	C₂	semidirect
14. C₂×C₃₁×#107	C₄	semidirect
76. C₂×C₈×C₃₁	C₄⋊C₄	semidirect
96. #21158	C₃₁	direct (N × H)
132. #1943	C₂×C₃₁	semidirect
133. #1942	C₂×C₃₁	semidirect
159. C₂×#107	C₄×C₃₁	semidirect
210. C₃₁	#21158	direct (N × H)
228. C₂×C₈	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#427	2
C₂²×C₃₁×#107	2
C₂×C₃₁×#405	2
C₃₁×#2316	2
C₃₁×#1943	2
C₃₁×#1942	2
C₃₁×#1736	2
#21158	31

Subgroups — C31×#21158 #21477917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21158 #21477917