Subgroups — C₃₁×#32851 #21489610

Order 7936.

532 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2864	C₂	1	2	yes	maximal
2	C₃₁×#2101	C₂	1	2	yes	maximal
3	C₃₁×#2101	C₂	1	2	yes	maximal
4	C₃₁×#2088	C₂	1	2	yes	maximal
5	C₃₁×#2087	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2086	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2085	C₂	1	2	yes	maximal
8	C₃₁×#2085	C₂	1	2	yes	maximal
9	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#106	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃₁×#389	C₂²	1	2	yes	—
16	C₃₁×#389	C₂²	1	2	yes	—
17	C₃₁×#389	C₂²	1	2	yes	—
18	C₃₁×#389	C₂²	1	2	yes	—
19	C₃₁×#387	—	2	2	no	—
20	C₃₁×#387	—	2	2	no	—
21	C₃₁×#387	—	2	2	no	—
22	C₃₁×#387	—	2	2	no	—
23	C₃₁×#382	C₂²	1	2	yes	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#340	C₂²	1	2	yes	—
27	C₃₁×#340	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	2	yes	—
30	C₃₁×#340	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	2	yes	—
48	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#107	—	4	1	no	—
52	C₃₁×#107	—	4	1	no	—
53	C₃₁×#106	—	4	2	no	—
54	C₃₁×C₄⋊C₈	—	4	1	no	—
55	C₃₁×C₄⋊C₈	—	4	1	no	—
56	C₃₁×#95	—	2	2	no	—
57	C₃₁×#95	—	2	2	no	—
58	C₃₁×#95	—	2	2	no	—
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	1	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	4	4	no	—
88	C₄⋊C₄×C₃₁	—	4	4	no	—
89	C₄⋊C₄×C₃₁	—	4	4	no	—
90	C₄⋊C₄×C₃₁	—	4	4	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	2	1	no	—
93	C₄²×C₃₁	—	2	1	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	4	1	no	—
97	C₄²×C₃₁	—	4	2	no	—
98	#32851	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	#135	1	2	yes	—
123	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
125	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
126	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
127	C₈×C₃₁	—	4	2	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	4	1	no	—
130	#2864	C₂×C₃₁	1	2	yes	—
131	#2101	C₂×C₃₁	1	2	yes	—
132	#2101	C₂×C₃₁	1	2	yes	—
133	#2088	C₂×C₃₁	1	2	yes	—
134	#2087	C₂×C₃₁	1	1	yes	characteristic
135	#2086	C₂×C₃₁	1	1	yes	characteristic
136	#2085	C₂×C₃₁	1	2	yes	—
137	#2085	C₂×C₃₁	1	2	yes	—
138	#1831	C₂×C₃₁	1	1	yes	characteristic
139	C₂²×C₃₁	#536	1	2	yes	—
140	C₂²×C₃₁	#537	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
142	C₂²×C₃₁	#575	1	2	yes	—
143	C₂²×C₃₁	#575	1	1	yes	characteristic
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	—	4	2	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	8	2	no	—
149	C₄×C₃₁	—	8	2	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	#563	1	1	yes	characteristic
153	C₄×C₃₁	#563	1	1	yes	characteristic
154	C₄×C₃₁	#564	1	2	yes	—
155	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#106	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#389	C₂²×C₃₁	1	2	yes	—
161	#389	C₂²×C₃₁	1	2	yes	—
162	#389	C₂²×C₃₁	1	2	yes	—
163	#389	C₂²×C₃₁	1	2	yes	—
164	#387	—	2	2	no	—
165	#387	—	2	2	no	—
166	#387	—	2	2	no	—
167	#387	—	2	2	no	—
168	#382	C₂²×C₃₁	1	2	yes	—
169	#382	C₂²×C₃₁	1	2	yes	—
170	C₄³	C₂²×C₃₁	1	1	yes	characteristic
171	#340	C₂²×C₃₁	1	2	yes	—
172	#340	C₂²×C₃₁	1	2	yes	—
173	#340	C₂²×C₃₁	1	2	yes	—
174	#340	C₂²×C₃₁	1	2	yes	—
175	#340	C₂²×C₃₁	1	1	yes	characteristic
176	#340	C₂²×C₃₁	1	1	yes	characteristic
177	#340	C₂²×C₃₁	1	2	yes	—
178	#336	C₂²×C₃₁	1	2	yes	—
179	#336	C₂²×C₃₁	1	2	yes	—
180	C₂×C₃₁	#2822	1	1	yes	characteristic
181	C₂×C₃₁	#3233	1	2	yes	—
182	C₂×C₃₁	#3240	1	2	yes	—
183	C₂×C₃₁	#3250	1	1	yes	characteristic
184	C₂×C₃₁	#3256	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
196	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₃₁	1	2	yes	—
198	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	#107	—	4	1	no	—
202	#107	—	4	1	no	—
203	#106	—	4	2	no	—
204	C₄⋊C₈	—	4	1	no	—
205	C₄⋊C₈	—	4	1	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	C₃₁	#32851	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₃₁×#135	1	2	yes	—
273	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
275	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
276	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
277	C₈	—	4	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₂²	C₃₁×#536	1	2	yes	—
281	C₂²	C₃₁×#537	1	1	yes	characteristic
282	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
283	C₂²	C₃₁×#575	1	2	yes	—
284	C₂²	C₃₁×#575	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₃₁×#563	1	1	yes	characteristic
294	C₄	C₃₁×#563	1	1	yes	characteristic
295	C₄	C₃₁×#564	1	2	yes	—
296	C₂	C₃₁×#2822	1	1	yes	characteristic
297	C₂	C₃₁×#3233	1	2	yes	—
298	C₂	C₃₁×#3240	1	2	yes	—
299	C₂	C₃₁×#3250	1	1	yes	characteristic
300	C₂	C₃₁×#3256	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
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
98. #32851	C₃₁	direct (N × H)
214. C₃₁	#32851	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×#2864	2
C₃₁×#2101	2
C₃₁×#2088	2
C₃₁×#2087	2
C₃₁×#2086	2
C₃₁×#2085	2
C₃₁×#1831	2
#32851	31

Subgroups — C31×#32851 #21489610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32851 #21489610