Subgroups — C₃₁×#33773 #21490532

Order 7936.

636 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2210	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2186	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₂×C₃₁×#135	C₂²	1	2	yes	—
12	C₃₁×#532	—	2	1	no	—
13	C₃₁×#532	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#493	C₂²	1	1	yes	characteristic
22	C₃₁×#492	—	2	1	no	—
23	C₃₁×#486	—	2	1	no	—
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#466	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×#358	C₂²	1	4	yes	—
31	C₃₁×#357	C₂²	1	4	yes	—
32	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×#41×C₃₁	—	2	2	no	—
36	C₂×#41×C₃₁	—	4	1	no	—
37	C₃₁×#135	—	4	4	no	—
38	C₃₁×#135	—	4	4	no	—
39	C₂×Q₁₆×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₂×D₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×C₄⋊Q₈	—	2	1	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	4	1	no	—
57	C₃₁×#121	—	4	1	no	—
58	C₃₁×#121	—	4	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	—	4	1	no	—
62	C₄×D₄×C₃₁	—	4	1	no	—
63	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
64	C₃₁×SD₃₂	—	4	4	no	—
65	C₃₁×D₁₆	—	4	4	no	—
66	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#107	—	2	1	no	—
69	C₃₁×#107	C₂³	1	2	yes	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	#41×C₃₁	—	4	4	no	—
77	#41×C₃₁	—	4	4	no	—
78	#41×C₃₁	—	8	4	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₃₁	—	2	4	no	—
95	Q₁₆×C₃₁	—	4	2	no	—
96	SD₁₆×C₃₁	—	4	4	no	—
97	SD₁₆×C₃₁	—	4	4	no	—
98	D₈×C₃₁	—	2	4	no	—
99	D₈×C₃₁	—	2	4	no	—
100	D₈×C₃₁	—	4	2	no	—
101	D₈×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	D₈	1	1	yes	characteristic
117	C₄²×C₃₁	D₈	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₃₁	—	2	2	no	—
121	C₁₆×C₃₁	—	2	2	no	—
122	#33773	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	—	4	2	no	—
125	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	8	4	no	—
134	D₄×C₃₁	—	8	4	no	—
135	D₄×C₃₁	—	8	2	no	—
136	D₄×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	8	2	no	—
138	D₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	8	4	no	—
146	C₂×C₄×C₃₁	—	8	4	no	—
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	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
156	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
157	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2248	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#2210	C₂×C₃₁	1	4	yes	complemented
164	#2186	C₂×C₃₁	1	4	yes	complemented
165	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₂²×C₃₁	—	2	1	no	—
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	—	8	2	no	—
174	C₂²×C₃₁	—	8	2	no	—
175	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	2	no	—
181	C₄×C₃₁	#508	1	2	yes	—
182	C₄×C₃₁	#509	1	2	yes	—
183	C₄×C₃₁	#535	1	1	yes	characteristic
184	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×#135	—	2	1	no	—
186	C₂×#135	C₂²×C₃₁	1	2	yes	—
187	#532	—	2	1	no	—
188	#532	—	2	1	no	—
189	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
190	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
191	#500	C₂²×C₃₁	1	1	yes	characteristic
192	#495	—	2	1	no	—
193	#495	C₂²×C₃₁	1	2	yes	—
194	#494	C₂²×C₃₁	1	1	yes	characteristic
195	#493	C₂²×C₃₁	1	1	yes	characteristic
196	#493	C₂²×C₃₁	1	1	yes	characteristic
197	#492	—	2	1	no	—
198	#486	—	2	1	no	—
199	#466	—	2	1	no	—
200	#466	C₂²×C₃₁	1	2	yes	—
201	C₄×D₈	—	2	1	no	—
202	C₄×D₈	C₂²×C₃₁	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
205	#358	C₂²×C₃₁	1	4	yes	—
206	#357	C₂²×C₃₁	1	4	yes	—
207	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
208	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
209	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₃₁	—	4	1	no	—
211	C₂×C₃₁	—	16	2	no	—
212	C₂×C₃₁	—	16	2	no	—
213	C₂×C₃₁	#3205	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
215	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
216	C₂×#41	—	2	2	no	—
217	C₂×#41	—	4	1	no	—
218	#135	—	4	4	no	—
219	#135	—	4	4	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	C₂³×C₃₁	1	2	yes	—
227	C₂×D₈	C₂³×C₃₁	1	2	yes	—
228	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	—	2	1	no	—
234	#124	—	2	2	no	—
235	#124	—	4	1	no	—
236	#121	—	2	2	no	—
237	#121	—	4	1	no	—
238	#121	—	4	1	no	—
239	#121	—	4	1	no	—
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	—	4	1	no	—
243	C₄×D₄	—	4	1	no	—
244	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
245	SD₃₂	—	4	4	no	—
246	D₁₆	—	4	4	no	—
247	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
248	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
249	#107	—	2	1	no	—
250	#107	C₂³×C₃₁	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	2	no	—
253	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
254	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₃₁	#33773	1	1	yes	characteristic, direct product factor, Sylow
258	#41	—	4	4	no	—
259	#41	—	4	4	no	—
260	#41	—	8	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	Q₁₆	—	2	4	no	—
277	Q₁₆	—	4	2	no	—
278	SD₁₆	—	4	4	no	—
279	SD₁₆	—	4	4	no	—
280	D₈	—	2	4	no	—
281	D₈	—	2	4	no	—
282	D₈	—	4	2	no	—
283	D₈	—	4	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	D₈×C₃₁	1	1	yes	characteristic
299	C₄²	D₈×C₃₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	2	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	4	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
337	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₃₁×#508	1	2	yes	—
354	C₄	C₃₁×#509	1	2	yes	—
355	C₄	C₃₁×#535	1	1	yes	characteristic
356	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₂	—	4	1	no	—
358	C₂	—	16	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₃₁×#3205	1	1	yes	characteristic
361	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
362	C₂	C₂×C₃₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
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
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2248	C₂	semidirect
4. C₃₁×#2246	C₂	semidirect
5. C₃₁×#2241	C₂	semidirect
6. C₃₁×#2240	C₂	semidirect
7. C₃₁×#2210	C₂	semidirect
8. C₃₁×#2186	C₂	semidirect
9. C₃₁×#2162	C₂	semidirect
122. #33773	C₃₁	direct (N × H)
157. #3169	C₂×C₃₁	semidirect
158. #3147	C₂×C₃₁	semidirect
159. #2248	C₂×C₃₁	semidirect
160. #2246	C₂×C₃₁	semidirect
161. #2241	C₂×C₃₁	semidirect
162. #2240	C₂×C₃₁	semidirect
163. #2210	C₂×C₃₁	semidirect
164. #2186	C₂×C₃₁	semidirect
165. #2162	C₂×C₃₁	semidirect
257. C₃₁	#33773	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×#3169	2
C₃₁×#3147	2
C₃₁×#2248	2
C₃₁×#2246	2
C₃₁×#2241	2
C₃₁×#2240	2
C₃₁×#2210	2
C₃₁×#2186	2
C₃₁×#2162	2
#33773	31

Subgroups — C31×#33773 #21490532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33773 #21490532