Subgroups — C₃₁×#22703 #21479462

Order 7936.

1054 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×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1853	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1807	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#340	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
13	C₈×D₄×C₃₁	C₄	1	8	yes	complemented
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×#106	C₂²	1	4	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#389	—	2	2	no	—
20	C₃₁×#385	—	2	4	no	—
21	C₃₁×#384	C₂²	1	2	yes	—
22	C₃₁×#380	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#340	C₂²	1	1	yes	characteristic
25	C₃₁×#340	C₂²	1	2	yes	—
26	C₃₁×#340	C₂²	1	2	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	1	yes	characteristic
29	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	—	2	4	no	—
31	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₁	—	4	4	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₄×D₄×C₃₁	—	4	8	no	—
38	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₃₁	Q₈	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#106	—	2	4	no	—
54	C₃₁×#106	—	4	4	no	—
55	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
58	C₃₁×#95	—	2	2	no	—
59	C₃₁×#95	—	2	4	no	—
60	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂×D₄×C₃₁	SD₁₆	1	8	yes	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	4	8	no	—
69	C₂²×C₄×C₃₁	—	4	8	no	—
70	C₂²×C₄×C₃₁	—	8	4	no	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	8	no	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	8	no	—
90	C₄⋊C₄×C₃₁	—	4	4	no	—
91	C₄⋊C₄×C₃₁	—	8	2	no	—
92	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
93	C₄⋊C₄×C₃₁	#41	1	4	yes	—
94	C₂²⋊C₄×C₃₁	—	4	8	no	—
95	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
96	C₄²×C₃₁	—	4	4	no	—
97	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
99	#22703	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	4	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	—	2	2	no	—
103	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
104	C₂³×C₃₁	C₂×SD₁₆	1	8	yes	—
105	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₃₁	#106	1	16	yes	complemented
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	8	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	8	16	no	—
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
127	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
131	C₈×C₃₁	—	2	2	no	—
132	C₈×C₃₁	—	2	2	no	—
133	C₈×C₃₁	—	2	4	no	—
134	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
135	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
136	#1948	C₂×C₃₁	1	2	yes	complemented
137	#1853	C₂×C₃₁	1	4	yes	complemented
138	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#1807	C₂×C₃₁	1	2	yes	complemented
140	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂×#340	C₂×C₃₁	1	2	yes	complemented
142	C₂²×C₃₁	—	2	4	no	—
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	—	2	8	no	—
145	C₂²×C₃₁	—	2	8	no	—
146	C₂²×C₃₁	C₂×#106	1	8	yes	—
147	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₃₁	#546	1	2	yes	—
149	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
150	C₂²×C₃₁	#576	1	1	yes	characteristic
151	C₂²×C₃₁	#578	1	1	yes	characteristic
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	—	8	4	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	C₂×#106	1	4	yes	—
158	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
159	C₄×C₃₁	#518	1	1	yes	characteristic
160	C₄×C₃₁	#554	1	2	yes	—
161	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
162	C₂×C₄×D₄	—	2	2	no	—
163	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
165	C₈×D₄	C₄×C₃₁	1	8	yes	complemented
166	C₂×#106	C₂²×C₃₁	1	2	yes	—
167	C₂×#106	C₂²×C₃₁	1	4	yes	—
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
171	#389	—	2	2	no	—
172	#385	—	2	4	no	—
173	#384	C₂²×C₃₁	1	2	yes	—
174	#380	C₂²×C₃₁	1	4	yes	—
175	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
176	#340	C₂²×C₃₁	1	1	yes	characteristic
177	#340	C₂²×C₃₁	1	2	yes	—
178	#340	C₂²×C₃₁	1	2	yes	—
179	#340	C₂²×C₃₁	1	4	yes	—
180	#340	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×C₃₁	—	2	8	no	—
182	C₂×C₃₁	#2349	1	1	yes	characteristic
183	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
184	C₂×C₃₁	#2913	1	1	yes	characteristic
185	C₂×C₃₁	#3295	1	2	yes	—
186	C₂×C₃₁	#3298	1	2	yes	—
187	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
188	C₂³×C₄	—	2	4	no	—
189	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₈	—	4	4	no	—
191	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
192	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₄×D₄	—	4	8	no	—
196	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
197	C₄×D₄	Q₈×C₃₁	1	4	yes	—
198	C₂×C₄⋊C₄	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	4	4	no	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
205	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	4	no	—
208	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
211	#106	—	2	4	no	—
212	#106	—	4	4	no	—
213	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
214	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
215	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
216	#95	—	2	2	no	—
217	#95	—	2	4	no	—
218	C₃₁	#22703	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
220	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
221	C₂×D₄	SD₁₆×C₃₁	1	8	yes	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	8	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	8	4	no	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
252	C₄⋊C₄	#41×C₃₁	1	4	yes	—
253	C₂²⋊C₄	—	4	8	no	—
254	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
255	C₄²	—	4	4	no	—
256	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
257	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
262	C₂³	C₂×SD₁₆×C₃₁	1	8	yes	—
263	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
264	D₄	C₃₁×#106	1	16	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	16	no	—
281	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
283	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
284	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
288	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	8	no	—
295	C₂²	—	2	8	no	—
296	C₂²	C₂×C₃₁×#106	1	8	yes	—
297	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
298	C₂²	C₃₁×#546	1	2	yes	—
299	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
300	C₂²	C₃₁×#576	1	1	yes	characteristic
301	C₂²	C₃₁×#578	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	4	no	—
306	C₄	—	8	2	no	—
307	C₄	C₂×C₃₁×#106	1	4	yes	—
308	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
309	C₄	C₃₁×#518	1	1	yes	characteristic
310	C₄	C₃₁×#554	1	2	yes	—
311	C₂	—	2	8	no	—
312	C₂	C₃₁×#2349	1	1	yes	characteristic
313	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
314	C₂	C₃₁×#2913	1	1	yes	characteristic
315	C₂	C₃₁×#3295	1	2	yes	—
316	C₂	C₃₁×#3298	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	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1948	C₂	semidirect
4. C₃₁×#1853	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
6. C₃₁×#1807	C₂	semidirect
7. C₄×C₃₁×#106	C₂	semidirect
8. C₂×C₃₁×#340	C₂	semidirect
13. C₈×D₄×C₃₁	C₄	semidirect
33. C₂²×C₈×C₃₁	C₂×C₄	semidirect
55. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
56. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
57. C₄×C₈×C₃₁	C₂×C₄	semidirect
99. #22703	C₃₁	direct (N × H)
106. D₄×C₃₁	#106	semidirect
136. #1948	C₂×C₃₁	semidirect
137. #1853	C₂×C₃₁	semidirect
138. #1834	C₂×C₃₁	semidirect
139. #1807	C₂×C₃₁	semidirect
140. C₄×#106	C₂×C₃₁	semidirect
141. C₂×#340	C₂×C₃₁	semidirect
165. C₈×D₄	C₄×C₃₁	semidirect
191. C₂²×C₈	C₂×C₄×C₃₁	semidirect
213. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
214. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
215. C₄×C₈	C₂×C₄×C₃₁	semidirect
218. C₃₁	#22703	direct (N × H)
264. D₄	C₃₁×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₁	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#1948	2
C₃₁×#1853	2
C₃₁×#1834	2
C₃₁×#1807	2
C₄×C₃₁×#106	2
C₂×C₃₁×#340	2
#22703	31

Subgroups — C31×#22703 #21479462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22703 #21479462