Subgroups — C₃₁×#29427 #21486186

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2323	C₂	1	2	yes	maximal
2	C₃₁×#2075	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2036	C₂	1	2	yes	maximal
4	C₃₁×#2001	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1818	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#103	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
21	C₃₁×#391	C₂²	1	2	yes	—
22	C₃₁×#389	—	2	2	no	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#388	—	2	2	no	—
25	C₃₁×#388	—	2	2	no	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	2	yes	—
30	C₃₁×#340	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₃₁	—	2	4	no	—
38	C₂×Q₁₆×C₃₁	—	4	1	no	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₂²⋊Q₈	—	2	4	no	—
43	C₃₁×C₂²⋊Q₈	—	2	4	no	—
44	C₄×Q₈×C₃₁	—	4	2	no	—
45	C₄×Q₈×C₃₁	—	4	2	no	—
46	C₃₁×#117	—	4	1	no	—
47	C₃₁×#117	—	4	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	—	2	2	no	—
61	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
62	C₃₁×#107	—	2	4	no	—
63	C₃₁×#107	—	4	1	no	—
64	C₃₁×#103	—	2	4	no	—
65	C₃₁×#103	—	2	4	no	—
66	C₃₁×#103	—	4	2	no	—
67	C₃₁×C₂²⋊C₈	—	4	1	no	—
68	C₃₁×C₂²⋊C₈	—	4	1	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	—	2	2	no	—
71	C₃₁×#95	—	2	2	no	—
72	C₃₁×#95	—	2	2	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	#41	1	4	yes	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	2	1	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₃₁	—	4	4	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	8	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	4	no	—
110	C₄⋊C₄×C₃₁	—	8	2	no	—
111	C₄⋊C₄×C₃₁	#41	1	4	yes	—
112	C₂²⋊C₄×C₃₁	—	2	8	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	8	2	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	4	4	no	—
118	C₄²×C₃₁	—	8	2	no	—
119	#29427	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	2	2	no	—
121	C₂³×C₃₁	—	2	1	no	—
122	C₂³×C₃₁	—	2	1	no	—
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	1	no	—
125	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₃₁	—	2	4	no	—
127	Q₈×C₃₁	—	2	4	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	2	4	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	1	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
153	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
154	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈×C₃₁	—	4	1	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	—	4	1	no	—
159	#2323	C₂×C₃₁	1	2	yes	—
160	#2075	C₂×C₃₁	1	1	yes	characteristic
161	#2036	C₂×C₃₁	1	2	yes	—
162	#2001	C₂×C₃₁	1	1	yes	characteristic
163	#1928	C₂×C₃₁	1	2	yes	complemented
164	#1918	C₂×C₃₁	1	2	yes	complemented
165	#1878	C₂×C₃₁	1	2	yes	complemented
166	#1863	C₂×C₃₁	1	2	yes	complemented
167	#1818	C₂×C₃₁	1	1	yes	characteristic
168	C₂²×C₃₁	—	4	1	no	—
169	C₂²×C₃₁	—	4	2	no	—
170	C₂²×C₃₁	—	4	2	no	—
171	C₂²×C₃₁	—	4	1	no	—
172	C₂²×C₃₁	—	4	1	no	—
173	C₂²×C₃₁	#525	1	2	yes	—
174	C₂²×C₃₁	#534	1	1	yes	characteristic
175	C₂²×C₃₁	#576	1	1	yes	characteristic
176	C₂²×C₃₁	#578	1	2	yes	—
177	C₂²×C₃₁	#578	1	1	yes	characteristic
178	C₄×C₃₁	—	4	1	no	—
179	C₄×C₃₁	—	4	4	no	—
180	C₄×C₃₁	—	4	4	no	—
181	C₄×C₃₁	—	4	2	no	—
182	C₄×C₃₁	—	4	2	no	—
183	C₄×C₃₁	—	8	2	no	—
184	C₄×C₃₁	—	8	2	no	—
185	C₄×C₃₁	#551	1	1	yes	characteristic
186	C₄×C₃₁	#553	1	2	yes	—
187	C₄×C₃₁	#559	1	1	yes	characteristic
188	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
190	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
191	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×#107	C₂²×C₃₁	1	2	yes	—
193	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×#103	C₂²×C₃₁	1	2	yes	—
195	C₂×#103	C₂²×C₃₁	1	2	yes	—
196	C₂×#103	C₂²×C₃₁	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
199	#391	C₂²×C₃₁	1	2	yes	—
200	#389	—	2	2	no	—
201	#389	C₂²×C₃₁	1	2	yes	—
202	#388	—	2	2	no	—
203	#388	—	2	2	no	—
204	#387	—	2	2	no	—
205	#380	C₂²×C₃₁	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
207	#340	C₂²×C₃₁	1	2	yes	—
208	#340	C₂²×C₃₁	1	1	yes	characteristic
209	#340	C₂²×C₃₁	1	2	yes	—
210	#340	C₂²×C₃₁	1	1	yes	characteristic
211	#336	C₂²×C₃₁	1	2	yes	—
212	#336	C₂²×C₃₁	1	2	yes	—
213	C₂×C₃₁	—	8	1	no	—
214	C₂×C₃₁	#2600	1	1	yes	characteristic
215	C₂×C₃₁	#3124	1	2	yes	—
216	C₂×C₃₁	#3127	1	2	yes	—
217	C₂×C₃₁	#3202	1	1	yes	characteristic
218	C₂×C₃₁	#3203	1	1	yes	characteristic
219	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
220	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	4	1	no	—
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
226	C₂²⋊Q₈	—	2	4	no	—
227	C₂²⋊Q₈	—	2	4	no	—
228	C₄×Q₈	—	4	2	no	—
229	C₄×Q₈	—	4	2	no	—
230	#117	—	4	1	no	—
231	#117	—	4	1	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₂²⋊C₄	—	2	2	no	—
241	C₂×C₂²⋊C₄	—	2	2	no	—
242	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
243	C₂×C₄²	—	2	2	no	—
244	C₂×C₄²	—	2	2	no	—
245	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
246	#107	—	2	4	no	—
247	#107	—	4	1	no	—
248	#103	—	2	4	no	—
249	#103	—	2	4	no	—
250	#103	—	4	2	no	—
251	C₂²⋊C₈	—	4	1	no	—
252	C₂²⋊C₈	—	4	1	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	C₃₁	#29427	1	1	yes	characteristic, direct product factor, Sylow
259	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	#41×C₃₁	1	4	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
278	Q₁₆	—	4	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	4	no	—
295	C₄⋊C₄	—	8	2	no	—
296	C₄⋊C₄	#41×C₃₁	1	4	yes	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	8	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	4	no	—
303	C₄²	—	8	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	1	no	—
309	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
337	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
338	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	C₃₁×#525	1	2	yes	—
349	C₂²	C₃₁×#534	1	1	yes	characteristic
350	C₂²	C₃₁×#576	1	1	yes	characteristic
351	C₂²	C₃₁×#578	1	2	yes	—
352	C₂²	C₃₁×#578	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	C₃₁×#551	1	1	yes	characteristic
361	C₄	C₃₁×#553	1	2	yes	—
362	C₄	C₃₁×#559	1	1	yes	characteristic
363	C₂	—	8	1	no	—
364	C₂	C₃₁×#2600	1	1	yes	characteristic
365	C₂	C₃₁×#3124	1	2	yes	—
366	C₂	C₃₁×#3127	1	2	yes	—
367	C₂	C₃₁×#3202	1	1	yes	characteristic
368	C₂	C₃₁×#3203	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×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×#1928	C₂	semidirect
6. C₃₁×#1918	C₂	semidirect
7. C₃₁×#1878	C₂	semidirect
8. C₃₁×#1863	C₂	semidirect
119. #29427	C₃₁	direct (N × H)
163. #1928	C₂×C₃₁	semidirect
164. #1918	C₂×C₃₁	semidirect
165. #1878	C₂×C₃₁	semidirect
166. #1863	C₂×C₃₁	semidirect
258. C₃₁	#29427	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×#2323	2
C₃₁×#2075	2
C₃₁×#2036	2
C₃₁×#2001	2
C₃₁×#1928	2
C₃₁×#1918	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1818	2
#29427	31

Subgroups — C31×#29427 #21486186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29427 #21486186