Subgroups — C₃₁×#33771 #21490530

Order 7936.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 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	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2246	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2206	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2184	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#532	—	2	1	no	—
11	C₂×C₃₁×#127	—	2	1	no	—
12	C₂×C₃₁×D₁₆	C₂²	1	8	yes	complemented
13	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	—	2	1	no	—
15	C₃₁×#493	—	2	1	no	—
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#493	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#486	—	2	1	no	—
20	C₃₁×#466	—	2	1	no	—
21	C₃₁×#466	C₂²	1	2	yes	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₃₁×#357	C₂²	1	8	yes	complemented
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₃₁	—	4	1	no	—
30	C₂²×D₄×C₃₁	—	2	2	no	—
31	C₂²×D₄×C₃₁	—	4	1	no	—
32	C₂²×D₄×C₃₁	—	4	1	no	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	2	no	—
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	2	no	—
37	C₂×D₈×C₃₁	—	4	4	no	—
38	C₂×D₈×C₃₁	—	4	4	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂×D₈×C₃₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×#127	—	2	1	no	—
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×#127	—	4	1	no	—
48	C₃₁×#127	—	4	1	no	—
49	C₃₁×#124	—	4	1	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	4	1	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	4	1	no	—
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×D₁₆	—	4	8	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
57	C₃₁×#107	—	2	1	no	—
58	C₃₁×#107	C₂³	1	2	yes	—
59	C₃₁×#102	—	2	2	no	—
60	C₃₁×#102	—	2	2	no	—
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂⁴×C₃₁	—	4	2	no	—
66	#41×C₃₁	—	8	4	no	—
67	C₂×Q₈×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	4	no	—
70	C₂×D₄×C₃₁	—	4	4	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	8	4	no	—
81	C₂×D₄×C₃₁	—	8	4	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	D₈×C₃₁	—	2	4	no	—
87	D₈×C₃₁	—	2	8	no	—
88	D₈×C₃₁	—	4	4	no	—
89	D₈×C₃₁	—	4	4	no	—
90	D₈×C₃₁	—	4	2	no	—
91	D₈×C₃₁	—	4	4	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₃₁	D₈	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₃₁	—	2	4	no	—
107	#33771	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	4	4	no	—
110	C₂³×C₃₁	—	8	4	no	—
111	C₂³×C₃₁	—	8	4	no	—
112	C₂³×C₃₁	—	8	4	no	—
113	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	8	4	no	—
120	D₄×C₃₁	—	8	4	no	—
121	D₄×C₃₁	—	8	4	no	—
122	D₄×C₃₁	—	8	4	no	—
123	D₄×C₃₁	—	8	4	no	—
124	D₄×C₃₁	—	8	2	no	—
125	D₄×C₃₁	—	8	2	no	—
126	D₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	8	4	no	—
132	C₂×C₄×C₃₁	D₁₆	1	4	yes	complemented
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₃₁	—	2	1	no	—
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
142	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
143	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2246	C₂×C₃₁	1	2	yes	complemented
146	#2240	C₂×C₃₁	1	2	yes	complemented
147	#2206	C₂×C₃₁	1	4	yes	complemented
148	#2184	C₂×C₃₁	1	4	yes	complemented
149	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	4	no	—
154	C₂²×C₃₁	—	8	4	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	4	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	16	8	no	—
159	C₂²×C₃₁	C₂×D₁₆	1	2	yes	—
160	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
165	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
166	C₄×C₃₁	#509	1	2	yes	—
167	C₄×C₃₁	#535	1	1	yes	characteristic
168	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×D₈	—	2	1	no	—
170	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
171	#532	—	2	1	no	—
172	C₂×#127	—	2	1	no	—
173	C₂×D₁₆	C₂²×C₃₁	1	8	yes	complemented
174	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
175	#493	—	2	1	no	—
176	#493	—	2	1	no	—
177	#493	C₂²×C₃₁	1	2	yes	—
178	#493	C₂²×C₃₁	1	2	yes	—
179	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
180	#486	—	2	1	no	—
181	#466	—	2	1	no	—
182	#466	C₂²×C₃₁	1	2	yes	—
183	C₄×D₈	—	2	1	no	—
184	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
185	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
186	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
187	#357	C₂²×C₃₁	1	8	yes	complemented
188	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
189	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
190	C₂×C₃₁	—	2	2	no	—
191	C₂×C₃₁	—	16	2	no	—
192	C₂×C₃₁	—	16	4	no	—
193	C₂×C₃₁	#3205	1	1	yes	characteristic
194	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
195	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
196	C₂×#41	—	4	1	no	—
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	—	4	1	no	—
199	C₂²×D₄	—	4	1	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	4	4	no	—
205	C₂×D₈	—	4	4	no	—
206	C₂×D₈	C₂³×C₃₁	1	4	yes	—
207	C₂×D₈	C₂³×C₃₁	1	2	yes	—
208	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	#127	—	2	2	no	—
212	#127	—	2	2	no	—
213	#127	—	2	1	no	—
214	#127	—	4	1	no	—
215	#127	—	4	1	no	—
216	#124	—	4	1	no	—
217	#121	—	2	2	no	—
218	#121	—	4	1	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	4	1	no	—
221	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
222	D₁₆	—	4	8	no	—
223	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
224	#107	—	2	1	no	—
225	#107	C₂³×C₃₁	1	2	yes	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₃₁	#33771	1	1	yes	characteristic, direct product factor, Sylow
233	C₂⁴	—	4	2	no	—
234	#41	—	8	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	8	4	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	D₈	—	2	4	no	—
255	D₈	—	2	8	no	—
256	D₈	—	4	4	no	—
257	D₈	—	4	4	no	—
258	D₈	—	4	2	no	—
259	D₈	—	4	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	D₈×C₃₁	1	1	yes	characteristic
271	C₄²	D₈×C₃₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	8	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
281	Q₈	—	8	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	2	no	—
292	D₄	—	8	2	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₃₁×D₁₆	1	4	yes	complemented
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
309	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	16	8	no	—
319	C₂²	C₂×C₃₁×D₁₆	1	2	yes	—
320	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
325	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
326	C₄	C₃₁×#509	1	2	yes	—
327	C₄	C₃₁×#535	1	1	yes	characteristic
328	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	16	4	no	—
331	C₂	—	16	2	no	—
332	C₂	C₃₁×#3205	1	1	yes	characteristic
333	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
334	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. C₂×C₃₁×#493	C₂	semidirect
3. C₃₁×#2246	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2206	C₂	semidirect
6. C₃₁×#2184	C₂	semidirect
7. C₃₁×#2162	C₂	semidirect
12. C₂×C₃₁×D₁₆	C₂²	semidirect
13. C₃₁×#500	C₂²	semidirect
18. C₃₁×#493	C₂²	semidirect
23. C₄×D₈×C₃₁	C₂²	semidirect
25. C₃₁×C₄⋊C₁₆	C₂²	semidirect
26. C₃₁×#357	C₂²	semidirect
28. C₄×C₁₆×C₃₁	C₂²	semidirect
107. #33771	C₃₁	direct (N × H)
132. C₂×C₄×C₃₁	D₁₆	semidirect
143. #3169	C₂×C₃₁	semidirect
144. C₂×#493	C₂×C₃₁	semidirect
145. #2246	C₂×C₃₁	semidirect
146. #2240	C₂×C₃₁	semidirect
147. #2206	C₂×C₃₁	semidirect
148. #2184	C₂×C₃₁	semidirect
149. #2162	C₂×C₃₁	semidirect
173. C₂×D₁₆	C₂²×C₃₁	semidirect
174. #500	C₂²×C₃₁	semidirect
179. #493	C₂²×C₃₁	semidirect
184. C₄×D₈	C₂²×C₃₁	semidirect
186. C₄⋊C₁₆	C₂²×C₃₁	semidirect
187. #357	C₂²×C₃₁	semidirect
189. C₄×C₁₆	C₂²×C₃₁	semidirect
232. C₃₁	#33771	direct (N × H)
299. C₂×C₄	C₃₁×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₂×C₃₁×#493	2
C₃₁×#2246	2
C₃₁×#2240	2
C₃₁×#2206	2
C₃₁×#2184	2
C₃₁×#2162	2
#33771	31

Subgroups — C31×#33771 #21490530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33771 #21490530