Subgroups — C₃₁×#33784 #21490543

Order 7936.

584 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2251	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2245	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2209	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2188	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2185	C₂	1	2	yes	maximal
9	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₃₁	—	2	1	no	—
11	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₃₁×#529	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
16	C₃₁×#499	C₂²	1	1	yes	characteristic
17	C₃₁×#495	C₂²	1	1	yes	characteristic
18	C₃₁×#494	—	2	1	no	—
19	C₃₁×#494	—	2	1	no	—
20	C₃₁×#494	C₂²	1	2	yes	—
21	C₃₁×#494	C₂²	1	2	yes	—
22	C₃₁×#488	—	2	1	no	—
23	C₃₁×#467	C₂²	1	1	yes	characteristic
24	C₃₁×#466	C₂²	1	1	yes	characteristic
25	C₃₁×#465	—	2	1	no	—
26	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₃₁	—	2	1	no	—
28	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	4	yes	—
32	C₃₁×#358	C₂²	1	4	yes	—
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×#346	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂×Q₁₆×C₃₁	—	2	2	no	—
37	C₂×Q₁₆×C₃₁	—	2	2	no	—
38	C₂×Q₁₆×C₃₁	—	2	2	no	—
39	C₂×Q₁₆×C₃₁	—	4	4	no	—
40	C₂×Q₁₆×C₃₁	—	4	4	no	—
41	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
43	C₂×SD₁₆×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	—	4	1	no	—
50	C₃₁×C₄⋊Q₈	—	4	1	no	—
51	C₃₁×#126	—	4	1	no	—
52	C₃₁×#125	—	2	1	no	—
53	C₃₁×#124	—	2	1	no	—
54	C₃₁×#123	—	4	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	1	no	—
56	C₃₁×#121	—	2	1	no	—
57	C₄×Q₈×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	4	1	no	—
60	C₃₁×#117	—	4	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
63	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
64	C₃₁×Q₃₂	—	4	4	no	—
65	C₃₁×SD₃₂	—	4	4	no	—
66	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#106	—	2	1	no	—
70	C₃₁×#103	—	2	1	no	—
71	C₃₁×#103	—	2	1	no	—
72	C₃₁×#102	—	2	1	no	—
73	C₃₁×#102	—	2	1	no	—
74	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
75	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₃₁	—	2	1	no	—
79	C₂×Q₈×C₃₁	—	2	4	no	—
80	C₂×Q₈×C₃₁	—	4	4	no	—
81	C₂×Q₈×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₃₁	—	2	4	no	—
91	Q₁₆×C₃₁	—	2	2	no	—
92	Q₁₆×C₃₁	—	2	4	no	—
93	Q₁₆×C₃₁	—	4	4	no	—
94	Q₁₆×C₃₁	—	4	4	no	—
95	Q₁₆×C₃₁	—	4	4	no	—
96	SD₁₆×C₃₁	—	4	2	no	—
97	D₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	1	no	—
113	C₄⋊C₄×C₃₁	—	4	1	no	—
114	C₂²⋊C₄×C₃₁	—	4	1	no	—
115	C₂²⋊C₄×C₃₁	—	4	1	no	—
116	C₂²⋊C₄×C₃₁	—	4	1	no	—
117	C₂²⋊C₄×C₃₁	—	4	1	no	—
118	C₂²⋊C₄×C₃₁	—	4	1	no	—
119	C₄²×C₃₁	—	4	1	no	—
120	C₄²×C₃₁	D₈	1	1	yes	characteristic
121	C₄²×C₃₁	D₈	1	1	yes	characteristic
122	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₃₁	—	2	4	no	—
125	#33784	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₃₁	—	4	1	no	—
127	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₃₁	—	4	4	no	—
129	Q₈×C₃₁	—	4	1	no	—
130	Q₈×C₃₁	—	4	1	no	—
131	Q₈×C₃₁	—	4	4	no	—
132	Q₈×C₃₁	—	8	4	no	—
133	D₄×C₃₁	—	4	1	no	—
134	D₄×C₃₁	—	4	1	no	—
135	D₄×C₃₁	—	8	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	4	no	—
144	C₂×C₄×C₃₁	—	8	4	no	—
145	C₂×C₄×C₃₁	—	8	2	no	—
146	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₃₁	—	2	1	no	—
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
155	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
156	#3168	C₂×C₃₁	1	1	yes	characteristic
157	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2251	C₂×C₃₁	1	2	yes	complemented
159	#2245	C₂×C₃₁	1	2	yes	complemented
160	#2209	C₂×C₃₁	1	2	yes	complemented
161	#2208	C₂×C₃₁	1	2	yes	complemented
162	#2188	C₂×C₃₁	1	2	yes	complemented
163	#2185	C₂×C₃₁	1	2	yes	—
164	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₂²×C₃₁	—	2	1	no	—
166	C₂²×C₃₁	—	2	1	no	—
167	C₂²×C₃₁	—	8	1	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	—	8	1	no	—
170	C₂²×C₃₁	#510	1	2	yes	—
171	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	8	1	no	—
177	C₄×C₃₁	—	8	1	no	—
178	C₄×C₃₁	—	8	4	no	—
179	C₄×C₃₁	—	8	1	no	—
180	C₄×C₃₁	#510	1	2	yes	—
181	C₄×C₃₁	#535	1	1	yes	characteristic
182	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×Q₁₆	—	2	1	no	—
184	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
185	C₂×C₄⋊Q₈	—	2	1	no	—
186	#529	—	2	1	no	—
187	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
188	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
189	#499	C₂²×C₃₁	1	1	yes	characteristic
190	#495	C₂²×C₃₁	1	1	yes	characteristic
191	#494	—	2	1	no	—
192	#494	—	2	1	no	—
193	#494	C₂²×C₃₁	1	2	yes	—
194	#494	C₂²×C₃₁	1	2	yes	—
195	#488	—	2	1	no	—
196	#467	C₂²×C₃₁	1	1	yes	characteristic
197	#466	C₂²×C₃₁	1	1	yes	characteristic
198	#465	—	2	1	no	—
199	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
200	C₄×SD₁₆	—	2	1	no	—
201	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
204	#358	C₂²×C₃₁	1	4	yes	—
205	#358	C₂²×C₃₁	1	4	yes	—
206	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
207	#346	C₂²×C₃₁	1	2	yes	—
208	C₂×C₃₁	—	2	2	no	—
209	C₂×C₃₁	—	16	1	no	—
210	C₂×C₃₁	#3205	1	1	yes	characteristic
211	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
212	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
213	C₂²×Q₈	—	2	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	4	4	no	—
218	C₂×Q₁₆	—	4	4	no	—
219	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
220	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×SD₁₆	—	2	1	no	—
222	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
223	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	4	1	no	—
228	C₄⋊Q₈	—	4	1	no	—
229	#126	—	4	1	no	—
230	#125	—	2	1	no	—
231	#124	—	2	1	no	—
232	#123	—	4	1	no	—
233	C₂²⋊Q₈	—	2	1	no	—
234	#121	—	2	1	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	4	1	no	—
238	#117	—	4	1	no	—
239	C₂×C₄⋊C₄	—	4	1	no	—
240	C₂×C₄⋊C₄	—	4	1	no	—
241	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
242	Q₃₂	—	4	4	no	—
243	SD₃₂	—	4	4	no	—
244	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
245	#107	C₂³×C₃₁	1	1	yes	characteristic
246	#107	C₂³×C₃₁	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#103	—	2	1	no	—
249	#103	—	2	1	no	—
250	#102	—	2	1	no	—
251	#102	—	2	1	no	—
252	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
253	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	C₃₁	#33784	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	—	4	4	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	Q₁₆	—	2	4	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	4	no	—
273	Q₁₆	—	4	4	no	—
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	4	2	no	—
276	D₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₄²	—	4	1	no	—
299	C₄²	D₈×C₃₁	1	1	yes	characteristic
300	C₄²	D₈×C₃₁	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₁₆	—	2	4	no	—
304	C₂³	—	4	1	no	—
305	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
306	Q₈	—	4	4	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	8	4	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
333	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₃₁×#510	1	2	yes	—
340	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	1	no	—
349	C₄	C₃₁×#510	1	2	yes	—
350	C₄	C₃₁×#535	1	1	yes	characteristic
351	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₃₁×#3205	1	1	yes	characteristic
355	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
356	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#494	C₂	semidirect
3. C₃₁×#2251	C₂	semidirect
4. C₃₁×#2245	C₂	semidirect
5. C₃₁×#2209	C₂	semidirect
6. C₃₁×#2208	C₂	semidirect
7. C₃₁×#2188	C₂	semidirect
9. C₃₁×#2163	C₂	semidirect
125. #33784	C₃₁	direct (N × H)
157. C₂×#494	C₂×C₃₁	semidirect
158. #2251	C₂×C₃₁	semidirect
159. #2245	C₂×C₃₁	semidirect
160. #2209	C₂×C₃₁	semidirect
161. #2208	C₂×C₃₁	semidirect
162. #2188	C₂×C₃₁	semidirect
164. #2163	C₂×C₃₁	semidirect
256. C₃₁	#33784	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×#3168	2
C₂×C₃₁×#494	2
C₃₁×#2251	2
C₃₁×#2245	2
C₃₁×#2209	2
C₃₁×#2208	2
C₃₁×#2188	2
C₃₁×#2185	2
C₃₁×#2163	2
#33784	31

Subgroups — C31×#33784 #21490543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33784 #21490543