Subgroups — C₃₁×#33139 #21489898

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2625	C₂	1	2	yes	maximal
4	C₃₁×#1969	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1942	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1935	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1894	C₂	1	2	yes	maximal
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#467	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₂×C₃₁×#103	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×#400	—	2	2	no	—
23	C₃₁×#398	—	2	2	no	—
24	C₃₁×#397	—	2	4	no	—
25	C₃₁×#391	C₂²	1	4	yes	—
26	C₃₁×#389	C₂²	1	4	yes	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	D₄	1	4	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×C₂²⋊Q₈	—	2	4	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₃₁×#107	—	2	2	no	—
56	C₃₁×#107	—	2	4	no	—
57	C₃₁×#103	—	2	4	no	—
58	C₃₁×#103	—	2	8	no	—
59	C₃₁×#97	—	2	4	no	—
60	C₄×C₈×C₃₁	—	2	4	no	—
61	C₃₁×#95	—	2	4	no	—
62	C₃₁×#95	—	2	4	no	—
63	C₃₁×#95	—	2	4	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	#41	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	4	4	no	—
82	M₁₆×C₃₁	—	2	8	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	8	no	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	#41	1	4	yes	—
90	C₂×C₈×C₃₁	#41	1	4	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	8	4	no	—
99	C₂²⋊C₄×C₃₁	—	2	4	no	—
100	C₂²⋊C₄×C₃₁	—	4	4	no	—
101	C₂²⋊C₄×C₃₁	—	8	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	#33139	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	#121	1	2	yes	—
110	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	4	2	no	—
113	Q₈×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	4	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	4	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	#121	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
135	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
136	C₈×C₃₁	—	2	2	no	—
137	C₈×C₃₁	—	2	4	no	—
138	C₈×C₃₁	#123	1	4	yes	—
139	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
140	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
141	#2625	C₂×C₃₁	1	2	yes	—
142	#1969	C₂×C₃₁	1	2	yes	complemented
143	#1942	C₂×C₃₁	1	2	yes	complemented
144	#1935	C₂×C₃₁	1	4	yes	complemented
145	#1894	C₂×C₃₁	1	2	yes	—
146	#1808	C₂×C₃₁	1	1	yes	characteristic
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	2	no	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	4	2	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
153	C₂²×C₃₁	#532	1	2	yes	—
154	C₂²×C₃₁	#576	1	1	yes	characteristic
155	C₂²×C₃₁	#578	1	2	yes	—
156	C₂²×C₃₁	#578	1	1	yes	characteristic
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	—	8	4	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
164	C₄×C₃₁	#525	1	1	yes	characteristic
165	C₄×C₃₁	#529	1	2	yes	—
166	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
170	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
171	#467	—	2	4	no	—
172	C₂×#107	C₂²×C₃₁	1	2	yes	—
173	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#103	C₂²×C₃₁	1	4	yes	—
175	C₂×#103	C₂²×C₃₁	1	2	yes	—
176	#405	—	2	4	no	—
177	C₂×#97	C₂²×C₃₁	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
179	#400	—	2	2	no	—
180	#398	—	2	2	no	—
181	#397	—	2	4	no	—
182	#391	C₂²×C₃₁	1	4	yes	—
183	#389	C₂²×C₃₁	1	4	yes	—
184	#380	C₂²×C₃₁	1	2	yes	—
185	#340	C₂²×C₃₁	1	2	yes	—
186	#340	C₂²×C₃₁	1	4	yes	—
187	C₂×C₃₁	—	4	2	no	—
188	C₂×C₃₁	#2854	1	1	yes	characteristic
189	C₂×C₃₁	#3059	1	1	yes	characteristic
190	C₂×C₃₁	#3061	1	1	yes	characteristic
191	C₂×C₃₁	#3175	1	2	yes	—
192	C₂×C₃₁	#3177	1	2	yes	—
193	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
194	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	D₄×C₃₁	1	4	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²⋊Q₈	—	2	4	no	—
205	C₂²⋊Q₈	—	2	4	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
217	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
218	#107	—	2	2	no	—
219	#107	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	8	no	—
222	#97	—	2	4	no	—
223	C₄×C₈	—	2	4	no	—
224	#95	—	2	2	no	—
225	#95	—	2	4	no	—
226	#95	—	2	4	no	—
227	#95	—	2	4	no	—
228	C₃₁	#33139	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	#41×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
245	Q₁₆	—	4	4	no	—
246	M₁₆	—	2	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
253	C₂×C₈	#41×C₃₁	1	4	yes	—
254	C₂×C₈	#41×C₃₁	1	4	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	8	4	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	8	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	1	no	—
272	C₂³	C₃₁×#121	1	2	yes	—
273	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₃₁×#121	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
297	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
298	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	4	no	—
301	C₈	C₃₁×#123	1	4	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
308	C₂²	C₃₁×#532	1	2	yes	—
309	C₂²	C₃₁×#576	1	1	yes	characteristic
310	C₂²	C₃₁×#578	1	2	yes	—
311	C₂²	C₃₁×#578	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	4	no	—
317	C₄	—	8	2	no	—
318	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
319	C₄	C₃₁×#525	1	1	yes	characteristic
320	C₄	C₃₁×#529	1	2	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₃₁×#2854	1	1	yes	characteristic
323	C₂	C₃₁×#3059	1	1	yes	characteristic
324	C₂	C₃₁×#3061	1	1	yes	characteristic
325	C₂	C₃₁×#3175	1	2	yes	—
326	C₂	C₃₁×#3177	1	2	yes	—

	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
4. C₃₁×#1969	C₂	semidirect
5. C₃₁×#1942	C₂	semidirect
6. C₃₁×#1935	C₂	semidirect
104. #33139	C₃₁	direct (N × H)
142. #1969	C₂×C₃₁	semidirect
143. #1942	C₂×C₃₁	semidirect
144. #1935	C₂×C₃₁	semidirect
228. C₃₁	#33139	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×#467	2
C₂×C₃₁×#405	2
C₃₁×#2625	2
C₃₁×#1969	2
C₃₁×#1942	2
C₃₁×#1935	2
C₃₁×#1894	2
C₃₁×#1808	2
#33139	31

Subgroups — C31×#33139 #21489898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33139 #21489898