Subgroups — C₃₁×#33068 #21489827

Order 7936.

870 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1920	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1864	C₂	1	4	yes	maximal
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1802	C₂	1	2	yes	maximal
8	C₂×C₃₁×#125	—	2	1	no	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₃₁	—	2	1	no	—
12	C₂×C₃₁×#106	C₂²	1	4	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×#102	C₂²	1	4	yes	—
15	C₃₁×#389	—	2	2	no	—
16	C₃₁×#389	C₂²	1	4	yes	—
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₃₁×#382	—	2	1	no	—
19	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
22	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#340	C₂²	1	4	yes	—
25	C₃₁×#340	C₂²	1	4	yes	—
26	C₃₁×#336	C₂²	1	4	yes	—
27	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	—	2	2	no	—
29	C₂³×C₄×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
31	C₃₁×#125	—	4	4	no	—
32	C₄×D₄×C₃₁	—	2	4	no	—
33	C₄×D₄×C₃₁	—	2	4	no	—
34	C₄×D₄×C₃₁	—	2	2	no	—
35	C₄×D₄×C₃₁	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	4	4	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#106	—	2	8	no	—
53	C₃₁×C₄⋊C₈	—	2	4	no	—
54	C₃₁×#102	—	2	8	no	—
55	C₃₁×#95	—	2	2	no	—
56	C₃₁×#95	—	2	1	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₂⁴×C₃₁	—	2	1	no	—
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
62	C₂×D₄×C₃₁	#41	1	4	yes	—
63	C₂²×C₄×C₃₁	—	2	8	no	—
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	4	4	no	—
69	C₂²×C₄×C₃₁	—	4	4	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₄⋊C₄×C₃₁	—	4	8	no	—
86	C₄⋊C₄×C₃₁	—	4	4	no	—
87	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₃₁	#41	1	4	yes	—
89	C₂²⋊C₄×C₃₁	—	2	8	no	—
90	C₂²⋊C₄×C₃₁	—	2	4	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	4	2	no	—
95	C₄²×C₃₁	—	4	4	no	—
96	C₄²×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	SD₁₆	1	4	yes	—
98	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
99	C₄²×C₃₁	#41	1	2	yes	—
100	#33068	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	—	4	1	no	—
104	C₂³×C₃₁	—	4	1	no	—
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	D₄×C₃₁	—	2	4	no	—
108	D₄×C₃₁	—	2	2	no	—
109	D₄×C₃₁	#125	1	4	yes	complemented
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	16	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	1	no	—
121	C₂×C₄×C₃₁	—	4	1	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	8	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
130	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
136	C₈×C₃₁	—	4	4	no	—
137	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
139	#1948	C₂×C₃₁	1	2	yes	complemented
140	#1920	C₂×C₃₁	1	4	yes	complemented
141	#1864	C₂×C₃₁	1	4	yes	—
142	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1802	C₂×C₃₁	1	2	yes	—
144	C₂²×C₃₁	—	2	4	no	—
145	C₂²×C₃₁	—	4	2	no	—
146	C₂²×C₃₁	—	4	4	no	—
147	C₂²×C₃₁	—	4	4	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₂²×C₃₁	#525	1	2	yes	—
151	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×C₃₁	#575	1	2	yes	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	4	no	—
156	C₄×C₃₁	—	4	4	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	8	4	no	—
160	C₄×C₃₁	#476	1	4	yes	—
161	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
162	C₄×C₃₁	#529	1	2	yes	—
163	C₄×C₃₁	#533	1	1	yes	characteristic
164	C₂×#125	—	2	1	no	—
165	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
167	C₂²×C₄²	—	2	1	no	—
168	C₂×#106	C₂²×C₃₁	1	4	yes	—
169	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂×#102	C₂²×C₃₁	1	4	yes	—
171	#389	—	2	2	no	—
172	#389	C₂²×C₃₁	1	4	yes	—
173	#384	C₂²×C₃₁	1	2	yes	—
174	#382	—	2	1	no	—
175	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
177	C₄×C₂²⋊C₄	—	2	1	no	—
178	C₄×C₂²⋊C₄	—	2	2	no	—
179	C₄³	C₂²×C₃₁	1	1	yes	characteristic
180	#340	C₂²×C₃₁	1	4	yes	—
181	#340	C₂²×C₃₁	1	4	yes	—
182	#336	C₂²×C₃₁	1	4	yes	—
183	C₂×C₃₁	—	4	4	no	—
184	C₂×C₃₁	#2851	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
186	C₂×C₃₁	#3080	1	1	yes	characteristic
187	C₂×C₃₁	#3101	1	2	yes	—
188	C₂×C₃₁	#3115	1	2	yes	—
189	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂³×C₄	—	2	2	no	—
191	C₂³×C₄	—	2	1	no	—
192	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
193	#125	—	4	4	no	—
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	Q₈×C₃₁	1	4	yes	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	4	4	no	—
209	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₃₁	1	2	yes	—
211	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
214	#106	—	2	8	no	—
215	C₄⋊C₈	—	2	4	no	—
216	#102	—	2	8	no	—
217	#95	—	2	2	no	—
218	#95	—	2	1	no	—
219	#95	—	2	4	no	—
220	C₃₁	#33068	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
225	C₂×D₄	#41×C₃₁	1	4	yes	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
251	C₄⋊C₄	#41×C₃₁	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	4	no	—
259	C₄²	—	4	2	no	—
260	C₄²	SD₁₆×C₃₁	1	4	yes	—
261	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
262	C₄²	#41×C₃₁	1	2	yes	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	C₃₁×#125	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	16	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
290	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
291	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
292	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
293	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
294	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
296	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂²	C₃₁×#525	1	2	yes	—
306	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
307	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂²	C₃₁×#575	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	8	4	no	—
315	C₄	C₃₁×#476	1	4	yes	—
316	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
317	C₄	C₃₁×#529	1	2	yes	—
318	C₄	C₃₁×#533	1	1	yes	characteristic
319	C₂	—	4	4	no	—
320	C₂	C₃₁×#2851	1	1	yes	characteristic
321	C₂	C₂×C₃₁×#476	1	1	yes	characteristic
322	C₂	C₃₁×#3080	1	1	yes	characteristic
323	C₂	C₃₁×#3101	1	2	yes	—
324	C₂	C₃₁×#3115	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2859	C₂	semidirect
3. C₃₁×#1948	C₂	semidirect
4. C₃₁×#1920	C₂	semidirect
6. C₃₁×#1831	C₂	semidirect
100. #33068	C₃₁	direct (N × H)
109. D₄×C₃₁	#125	semidirect
137. #2859	C₂×C₃₁	semidirect
139. #1948	C₂×C₃₁	semidirect
140. #1920	C₂×C₃₁	semidirect
142. #1831	C₂×C₃₁	semidirect
220. C₃₁	#33068	direct (N × H)
271. D₄	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2859	2
C₄²×D₄×C₃₁	2
C₃₁×#1948	2
C₃₁×#1920	2
C₃₁×#1864	2
C₃₁×#1831	2
C₃₁×#1802	2
#33068	31

Subgroups — C31×#33068 #21489827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33068 #21489827