Subgroups — C₃₁×#12848 #21469607

Order 7936.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1379	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#405	—	2	1	no	—
11	C₃₁×#405	—	2	1	no	—
12	C₃₁×#405	C₄	1	2	yes	—
13	C₃₁×#405	C₂²	1	2	yes	—
14	C₂×C₃₁×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	—	2	1	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
19	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	2	4	no	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#117	—	2	1	no	—
33	C₃₁×#117	—	2	1	no	—
34	C₃₁×#117	C₈	1	2	yes	—
35	C₃₁×#117	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₂×C₄²×C₃₁	—	2	1	no	—
40	C₃₁×M₃₂	—	2	4	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₂×C₁₆×C₃₁	—	2	4	no	—
44	C₃₁×#97	—	2	2	no	—
45	C₃₁×#97	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	—	2	2	no	—
48	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₃₁	—	2	4	no	—
60	M₁₆×C₃₁	—	2	2	no	—
61	M₁₆×C₃₁	M₁₆	1	4	yes	—
62	C₂×C₈×C₃₁	—	2	4	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	#41	1	4	yes	—
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
77	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
78	C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	2	2	no	—
81	C₄²×C₃₁	—	2	2	no	—
82	C₁₆×C₃₁	—	4	2	no	—
83	C₁₆×C₃₁	—	4	2	no	—
84	#12848	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	—	4	1	no	—
88	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#117	1	2	yes	—
106	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	4	no	—
111	C₈×C₃₁	#123	1	2	yes	—
112	C₈×C₃₁	#123	1	2	yes	—
113	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
114	#2112	C₂×C₃₁	1	1	yes	characteristic
115	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
116	#1379	C₂×C₃₁	1	4	yes	complemented
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	2	1	no	—
124	C₂²×C₃₁	—	4	4	no	—
125	C₂²×C₃₁	#350	1	2	yes	—
126	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₃₁	#432	1	2	yes	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	#381	1	1	yes	characteristic
133	C₄×C₃₁	#407	1	1	yes	characteristic
134	C₄×C₃₁	#431	1	2	yes	—
135	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
136	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
138	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
139	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
140	#405	—	2	1	no	—
141	#405	—	2	1	no	—
142	#405	C₄×C₃₁	1	2	yes	—
143	#405	C₂²×C₃₁	1	2	yes	—
144	C₂×#97	—	2	1	no	—
145	C₂×C₄×C₈	—	2	1	no	—
146	C₂²⋊C₁₆	—	2	1	no	—
147	C₂²⋊C₁₆	—	2	1	no	—
148	C₂²⋊C₁₆	—	2	1	no	—
149	C₂²⋊C₁₆	—	2	1	no	—
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	—	2	2	no	—
152	C₂×C₃₁	—	4	1	no	—
153	C₂×C₃₁	#1817	1	1	yes	characteristic
154	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
155	C₂×C₃₁	#2117	1	1	yes	characteristic
156	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	#117	—	2	1	no	—
169	#117	—	2	1	no	—
170	#117	C₈×C₃₁	1	2	yes	—
171	#117	C₂×C₄×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	1	no	—
175	C₂×C₄²	—	2	1	no	—
176	M₃₂	—	2	4	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	4	no	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₃₁	#12848	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	M₁₆×C₃₁	1	4	yes	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	#41×C₃₁	1	4	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
214	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₁₆	—	4	2	no	—
220	C₁₆	—	4	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
225	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
226	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#117	1	2	yes	—
242	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	C₃₁×#123	1	2	yes	—
248	C₈	C₃₁×#123	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₃₁×#350	1	2	yes	—
258	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
259	C₂²	C₃₁×#432	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₃₁×#381	1	1	yes	characteristic
265	C₄	C₃₁×#407	1	1	yes	characteristic
266	C₄	C₃₁×#431	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	C₃₁×#1817	1	1	yes	characteristic
271	C₂	C₂×C₃₁×#350	1	1	yes	characteristic
272	C₂	C₃₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₃₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1379	C₂	semidirect
84. #12848	C₃₁	direct (N × H)
116. #1379	C₂×C₃₁	semidirect
184. C₃₁	#12848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#405	2
C₃₁×#2112	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1379	2
#12848	31

Subgroups — C31×#12848 #21469607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12848 #21469607