Subgroups — C₃₁×#11444 #21468203

Order 7936.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	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×#1323	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	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	—	2	1	no	—
10	C₃₁×#424	—	2	1	no	—
11	C₃₁×#424	—	2	1	no	—
12	C₃₁×#424	C₄	1	2	yes	—
13	C₃₁×#424	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
15	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	4	2	no	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	4	2	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	—	4	2	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₂×C₁₆×C₃₁	—	4	2	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	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	—	4	2	no	—
60	M₁₆×C₃₁	—	4	2	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	4	1	no	—
65	C₂×C₈×C₃₁	—	4	1	no	—
66	C₂×C₈×C₃₁	—	4	1	no	—
67	C₂×C₈×C₃₁	—	4	1	no	—
68	C₂×C₈×C₃₁	—	4	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	C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
76	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
77	C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	2	2	no	—
81	C₁₆×C₃₁	—	4	2	no	—
82	C₁₆×C₃₁	—	4	2	no	—
83	#11444	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	4	1	no	—
87	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	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	1	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#99	1	2	yes	—
104	C₂×C₄×C₃₁	#101	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	4	2	no	—
112	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
113	#2112	C₂×C₃₁	1	1	yes	characteristic
114	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
115	#1323	C₂×C₃₁	1	4	yes	complemented
116	C₂²×C₃₁	—	2	1	no	—
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	—	4	4	no	—
124	C₂²×C₃₁	#324	1	2	yes	—
125	C₂²×C₃₁	#350	1	2	yes	—
126	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	#324	1	2	yes	—
132	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
133	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
134	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
137	C₂×M₃₂	—	2	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	#424	—	2	1	no	—
140	#424	—	2	1	no	—
141	#424	C₄×C₃₁	1	2	yes	—
142	#424	C₂²×C₃₁	1	2	yes	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂²⋊C₁₆	—	2	1	no	—
146	C₂²⋊C₁₆	—	2	1	no	—
147	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
148	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
149	C₂×C₃₁	—	2	2	no	—
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	—	4	1	no	—
152	C₂×C₃₁	C₂×#324	1	1	yes	characteristic
153	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
154	C₂×C₃₁	#2117	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	4	2	no	—
158	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	4	2	no	—
161	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₈	D₄×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	C₂³×C₃₁	1	1	yes	characteristic
167	#117	—	2	1	no	—
168	#117	—	2	1	no	—
169	#117	C₈×C₃₁	1	2	yes	—
170	#117	C₂×C₄×C₃₁	1	2	yes	—
171	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	—	2	1	no	—
175	M₃₂	—	4	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	4	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₃₁	#11444	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
212	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₁₆	—	4	2	no	—
218	C₁₆	—	4	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
223	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
224	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	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	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
237	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₄	C₃₁×#99	1	2	yes	—
239	C₂×C₄	C₃₁×#101	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
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	—	4	4	no	—
255	C₂²	C₃₁×#324	1	2	yes	—
256	C₂²	C₃₁×#350	1	2	yes	—
257	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	C₃₁×#324	1	2	yes	—
263	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
264	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	2	2	no	—
267	C₂	—	4	1	no	—
268	C₂	C₂×C₃₁×#324	1	1	yes	characteristic
269	C₂	C₂×C₃₁×#350	1	1	yes	characteristic
270	C₂	C₃₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1323	C₂	semidirect
83. #11444	C₃₁	direct (N × H)
115. #1323	C₂×C₃₁	semidirect
183. C₃₁	#11444	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×#424	2
C₃₁×#2112	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1323	2
#11444	31

Subgroups — C31×#11444 #21468203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11444 #21468203