Subgroups — C₃₁×#32859 #21489618

Order 7936.

708 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2860	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2088	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2054	C₂	1	2	yes	maximal
8	C₃₁×#2054	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#126	—	2	2	no	—
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₃₁×#390	C₂²	1	2	yes	—
22	C₃₁×#390	C₂²	1	2	yes	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#388	—	2	2	no	—
28	C₃₁×#388	—	2	2	no	—
29	C₃₁×#388	—	2	2	no	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
36	C₂×D₈×C₃₁	—	4	2	no	—
37	C₂×D₈×C₃₁	—	4	1	no	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#126	—	4	8	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	D₄	1	2	yes	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#107	—	4	1	no	—
60	C₃₁×#106	—	4	2	no	—
61	C₃₁×C₄⋊C₈	—	4	1	no	—
62	C₃₁×C₄⋊C₈	—	4	2	no	—
63	C₃₁×C₄⋊C₈	—	4	2	no	—
64	C₃₁×C₄⋊C₈	—	4	1	no	—
65	C₃₁×#102	—	4	2	no	—
66	C₃₁×#102	—	4	2	no	—
67	C₃₁×#102	—	4	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	—	2	2	no	—
71	C₂⁴×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	D₈×C₃₁	—	4	4	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	4	no	—
101	C₄⋊C₄×C₃₁	—	4	4	no	—
102	C₄⋊C₄×C₃₁	—	4	4	no	—
103	C₂²⋊C₄×C₃₁	—	4	4	no	—
104	C₂²⋊C₄×C₃₁	—	4	4	no	—
105	C₂²⋊C₄×C₃₁	—	4	4	no	—
106	C₂²⋊C₄×C₃₁	—	4	4	no	—
107	C₂²⋊C₄×C₃₁	—	4	4	no	—
108	C₂²⋊C₄×C₃₁	—	4	4	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	4	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	2	1	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	2	1	no	—
115	C₄²×C₃₁	—	4	1	no	—
116	#32859	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	—	4	2	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	4	2	no	—
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	D₄×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	4	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	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	#135	1	2	yes	—
148	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
149	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
152	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	4	1	no	—
155	C₈×C₃₁	—	4	1	no	—
156	#2860	C₂×C₃₁	1	2	yes	complemented
157	#2088	C₂×C₃₁	1	2	yes	complemented
158	#2087	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2067	C₂×C₃₁	1	2	yes	complemented
160	#2065	C₂×C₃₁	1	2	yes	complemented
161	#2065	C₂×C₃₁	1	2	yes	complemented
162	#2054	C₂×C₃₁	1	2	yes	—
163	#2054	C₂×C₃₁	1	1	yes	characteristic
164	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	#537	1	2	yes	—
173	C₂²×C₃₁	#537	1	1	yes	characteristic
174	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
175	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
176	C₂²×C₃₁	#575	1	1	yes	characteristic
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	4	2	no	—
180	C₄×C₃₁	—	4	1	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	—	8	2	no	—
184	C₄×C₃₁	#560	1	1	yes	characteristic
185	C₄×C₃₁	#563	1	2	yes	—
186	C₄×C₃₁	#563	1	1	yes	characteristic
187	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×#126	—	2	2	no	—
189	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×#106	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
195	C₂×#102	C₂²×C₃₁	1	2	yes	—
196	C₂×#102	C₂²×C₃₁	1	2	yes	—
197	C₂×#102	C₂²×C₃₁	1	2	yes	—
198	#390	C₂²×C₃₁	1	2	yes	—
199	#390	C₂²×C₃₁	1	2	yes	—
200	#390	C₂²×C₃₁	1	2	yes	—
201	#389	C₂²×C₃₁	1	2	yes	—
202	#389	C₂²×C₃₁	1	2	yes	—
203	#389	C₂²×C₃₁	1	2	yes	—
204	#388	—	2	2	no	—
205	#388	—	2	2	no	—
206	#388	—	2	2	no	—
207	C₄³	C₂²×C₃₁	1	1	yes	characteristic
208	#340	C₂²×C₃₁	1	2	yes	—
209	#340	C₂²×C₃₁	1	1	yes	characteristic
210	#340	C₂²×C₃₁	1	1	yes	characteristic
211	#340	C₂²×C₃₁	1	2	yes	—
212	C₂×C₃₁	—	16	2	no	—
213	C₂×C₃₁	#2823	1	1	yes	characteristic
214	C₂×C₃₁	#3247	1	2	yes	—
215	C₂×C₃₁	#3250	1	1	yes	characteristic
216	C₂×C₃₁	#3251	1	2	yes	—
217	C₂×C₃₁	#3253	1	1	yes	characteristic
218	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
219	C₂×D₈	—	4	2	no	—
220	C₂×D₈	—	4	1	no	—
221	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	#126	—	4	8	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	D₄×C₃₁	1	2	yes	—
238	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	#107	—	4	1	no	—
243	#106	—	4	2	no	—
244	C₄⋊C₈	—	4	1	no	—
245	C₄⋊C₈	—	4	2	no	—
246	C₄⋊C₈	—	4	2	no	—
247	C₄⋊C₈	—	4	1	no	—
248	#102	—	4	2	no	—
249	#102	—	4	2	no	—
250	#102	—	4	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	C₃₁	#32859	1	1	yes	characteristic, direct product factor, Sylow
255	C₂⁴	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
270	D₈	—	4	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	4	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	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	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	C₃₁×#135	1	2	yes	—
331	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
332	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
333	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
334	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
335	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	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	C₃₁×#537	1	2	yes	—
347	C₂²	C₃₁×#537	1	1	yes	characteristic
348	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
349	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
350	C₂²	C₃₁×#575	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₃₁×#560	1	1	yes	characteristic
359	C₄	C₃₁×#563	1	2	yes	—
360	C₄	C₃₁×#563	1	1	yes	characteristic
361	C₂	—	16	2	no	—
362	C₂	C₃₁×#2823	1	1	yes	characteristic
363	C₂	C₃₁×#3247	1	2	yes	—
364	C₂	C₃₁×#3250	1	1	yes	characteristic
365	C₂	C₃₁×#3251	1	2	yes	—
366	C₂	C₃₁×#3253	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2860	C₂	semidirect
2. C₃₁×#2088	C₂	semidirect
3. C₃₁×#2087	C₂	semidirect
4. C₃₁×#2067	C₂	semidirect
5. C₃₁×#2065	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
9. C₃₁×#1842	C₂	semidirect
116. #32859	C₃₁	direct (N × H)
156. #2860	C₂×C₃₁	semidirect
157. #2088	C₂×C₃₁	semidirect
158. #2087	C₂×C₃₁	semidirect
159. #2067	C₂×C₃₁	semidirect
160. #2065	C₂×C₃₁	semidirect
161. #2065	C₂×C₃₁	semidirect
164. #1842	C₂×C₃₁	semidirect
254. C₃₁	#32859	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×#2860	2
C₃₁×#2088	2
C₃₁×#2087	2
C₃₁×#2067	2
C₃₁×#2065	2
C₃₁×#2054	2
C₃₁×#1842	2
#32859	31

Subgroups — C31×#32859 #21489618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32859 #21489618