Subgroups — C₃₁×#21363 #21478122

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1949	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1853	C₂	1	2	yes	maximal
6	C₃₁×#1852	C₂	1	2	yes	maximal
7	C₃₁×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₃₁	—	2	1	no	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#432	C₄	1	8	yes	complemented
12	C₂×C₃₁×#107	C₂²	1	4	yes	—
13	C₂×C₃₁×#106	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₃₁×#383	—	2	1	no	—
18	C₃₁×#382	C₂²	1	2	yes	—
19	C₃₁×#381	—	2	1	no	—
20	C₃₁×#380	C₂²	1	1	yes	characteristic
21	C₃₁×#380	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
23	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
28	C₃₁×#376	—	2	1	no	—
29	C₂×C₃₁×#95	—	2	1	no	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₂³×C₄×C₃₁	—	2	2	no	—
33	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	—	4	4	no	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	—	2	8	no	—
55	C₃₁×#106	—	2	8	no	—
56	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
57	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
58	C₃₁×#97	C₂×C₄	1	8	yes	complemented
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	1	no	—
62	C₃₁×#95	—	2	1	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	8	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	4	no	—
76	C₂²×C₄×C₃₁	—	4	4	no	—
77	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
90	C₄⋊C₄×C₃₁	—	2	8	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	#41	1	8	yes	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	—	4	4	no	—
98	C₄²×C₃₁	—	4	4	no	—
99	C₄²×C₃₁	—	4	4	no	—
100	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
101	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
102	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
105	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
106	#21363	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	—	2	2	no	—
109	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₂³×C₃₁	#137	1	2	yes	—
112	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	8	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	4	no	—
123	C₂×C₄×C₃₁	—	2	8	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	16	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
142	C₈×C₃₁	—	2	8	no	—
143	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
144	#2283	C₂×C₃₁	1	1	yes	characteristic
145	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
146	#1949	C₂×C₃₁	1	4	yes	complemented
147	#1853	C₂×C₃₁	1	2	yes	—
148	#1852	C₂×C₃₁	1	2	yes	—
149	#1776	C₂×C₃₁	1	4	yes	complemented
150	C₂²×C₃₁	—	2	4	no	—
151	C₂²×C₃₁	—	2	4	no	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	#428	1	4	yes	—
155	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₃₁	#525	1	2	yes	—
157	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
159	C₂²×C₃₁	#575	1	2	yes	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	2	no	—
165	C₄×C₃₁	—	4	8	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	C₄×#41	1	2	yes	—
169	C₄×C₃₁	#533	1	2	yes	—
170	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
171	C₂²×C₄²	—	2	1	no	—
172	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
173	#432	C₄×C₃₁	1	8	yes	complemented
174	C₂×#107	C₂²×C₃₁	1	4	yes	—
175	C₂×#106	C₂²×C₃₁	1	4	yes	—
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
178	C₂×#97	C₂²×C₃₁	1	2	yes	—
179	#383	—	2	1	no	—
180	#382	C₂²×C₃₁	1	2	yes	—
181	#381	—	2	1	no	—
182	#380	C₂²×C₃₁	1	1	yes	characteristic
183	#380	C₂²×C₃₁	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	2	1	no	—
185	C₄×C₄⋊C₄	—	2	1	no	—
186	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
189	C₄×C₂²⋊C₄	—	2	1	no	—
190	#376	—	2	1	no	—
191	C₂×#95	—	2	1	no	—
192	#340	C₂²×C₃₁	1	4	yes	—
193	#340	C₂²×C₃₁	1	4	yes	—
194	C₂×C₃₁	—	2	4	no	—
195	C₂×C₃₁	#2296	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
197	C₂×C₃₁	#2911	1	1	yes	characteristic
198	C₂×C₃₁	#3114	1	2	yes	—
199	C₂×C₃₁	#3117	1	2	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
203	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	8	no	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	4	4	no	—
216	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
217	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	#107	—	2	8	no	—
223	#106	—	2	8	no	—
224	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
225	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
226	#97	C₂×C₄×C₃₁	1	8	yes	complemented
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	C₃₁	#21363	1	1	yes	characteristic, direct product factor, Sylow
235	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
247	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	#41×C₃₁	1	8	yes	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	4	4	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
270	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
271	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
272	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
273	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
274	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
278	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
279	C₂³	C₃₁×#137	1	2	yes	—
280	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	16	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
305	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
307	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
310	C₈	—	2	8	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	C₃₁×#428	1	4	yes	—
316	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
317	C₂²	C₃₁×#525	1	2	yes	—
318	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
320	C₂²	C₃₁×#575	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	C₄×#41×C₃₁	1	2	yes	—
330	C₄	C₃₁×#533	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₃₁×#2296	1	1	yes	characteristic
333	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
334	C₂	C₃₁×#2911	1	1	yes	characteristic
335	C₂	C₃₁×#3114	1	2	yes	—
336	C₂	C₃₁×#3117	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 ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1949	C₂	semidirect
7. C₃₁×#1776	C₂	semidirect
11. C₃₁×#432	C₄	semidirect
56. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
58. C₃₁×#97	C₂×C₄	semidirect
106. #21363	C₃₁	direct (N × H)
146. #1949	C₂×C₃₁	semidirect
149. #1776	C₂×C₃₁	semidirect
173. #432	C₄×C₃₁	semidirect
224. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
226. #97	C₂×C₄×C₃₁	semidirect
234. C₃₁	#21363	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#432	2
C₃₁×#2283	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1949	2
C₃₁×#1853	2
C₃₁×#1852	2
C₃₁×#1776	2
#21363	31

Subgroups — C31×#21363 #21478122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21363 #21478122