Subgroups — C₃₁×#10832 #21467591

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#336	C₂	1	4	yes	maximal
4	C₂³×C₈×C₃₁	C₂²	1	4	yes	—
5	C₂²×C₄²×C₃₁	C₄	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
10	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₄³×C₃₁	—	2	4	no	—
13	C₃₁×#336	—	2	8	no	—
14	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	—	2	4	no	—
19	C₂²×C₈×C₃₁	—	2	4	no	—
20	C₂²×C₈×C₃₁	—	2	8	no	—
21	C₂²×C₈×C₃₁	—	2	8	no	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₃₁	—	2	4	no	—
24	C₂×C₄²×C₃₁	—	2	4	no	—
25	C₂×C₄²×C₃₁	—	2	8	no	—
26	C₂×C₄²×C₃₁	—	2	4	no	—
27	C₂×C₄²×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	—	2	4	no	—
29	C₂×C₄²×C₃₁	C₈	1	8	yes	complemented
30	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₃₁	D₄	1	4	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	4	yes	—
34	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
35	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
36	C₃₁×C₄⋊C₈	—	2	16	no	—
37	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	—	2	4	no	—
39	C₂²×C₄×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	—	2	4	no	—
41	C₂²×C₄×C₃₁	—	2	4	no	—
42	C₂²×C₄×C₃₁	—	2	4	no	—
43	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
44	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×C₈	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₃₁	M₁₆	1	8	yes	—
53	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₃₁	#41	1	4	yes	—
61	C₂×C₈×C₃₁	—	2	8	no	—
62	C₂×C₈×C₃₁	—	2	16	no	—
63	C₂×C₈×C₃₁	—	4	8	no	—
64	C₂×C₈×C₃₁	—	4	4	no	—
65	C₂×C₈×C₃₁	—	4	4	no	—
66	C₄²×C₃₁	—	2	16	no	—
67	C₄²×C₃₁	—	2	16	no	—
68	C₄²×C₃₁	—	2	8	no	—
69	C₄²×C₃₁	—	2	8	no	—
70	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
71	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
72	#10832	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
74	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₃₁	#117	1	2	yes	—
78	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
79	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
80	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
81	C₂×C₄×C₃₁	—	2	8	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	—	2	8	no	—
86	C₂×C₄×C₃₁	—	2	8	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	2	8	no	—
90	C₂×C₄×C₃₁	#95	1	4	yes	—
91	C₂×C₄×C₃₁	#95	1	4	yes	—
92	C₂×C₄×C₃₁	C₄×C₈	1	4	yes	—
93	C₂×C₄×C₃₁	#97	1	4	yes	—
94	C₂×C₄×C₃₁	C₂²⋊C₈	1	8	yes	—
95	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
96	C₂×C₄×C₃₁	C₄⋊C₈	1	16	yes	complemented
97	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
101	C₂×C₄×C₃₁	#121	1	4	yes	—
102	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₃₁	#124	1	2	yes	—
104	C₂×C₄×C₃₁	#125	1	2	yes	—
105	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
108	C₈×C₃₁	—	4	8	no	—
109	C₂²×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
110	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
111	C₂×#336	C₂×C₃₁	1	4	yes	—
112	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
113	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
114	C₂²×C₃₁	#380	1	2	yes	—
115	C₂²×C₃₁	#382	1	1	yes	characteristic
116	C₂²×C₃₁	#384	1	1	yes	characteristic
117	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₃₁	C₂×#97	1	1	yes	characteristic
119	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
120	C₂²×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
121	C₂²×C₃₁	#407	1	2	yes	—
122	C₂²×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
123	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
124	C₂²×C₃₁	#423	1	4	yes	—
125	C₂²×C₃₁	#423	1	2	yes	—
126	C₂²×C₃₁	#431	1	4	yes	—
127	C₂²×C₃₁	#432	1	4	yes	—
128	C₄×C₃₁	—	2	16	no	—
129	C₄×C₃₁	—	2	4	no	—
130	C₄×C₃₁	#336	1	8	yes	complemented
131	C₄×C₃₁	#389	1	4	yes	—
132	C₄×C₃₁	#390	1	2	yes	—
133	C₄×C₃₁	#391	1	2	yes	—
134	C₂³×C₈	C₂²×C₃₁	1	4	yes	—
135	C₂²×C₄²	C₄×C₃₁	1	2	yes	—
136	C₂²×C₄²	C₄×C₃₁	1	2	yes	—
137	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
138	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	2	4	no	—
140	C₂×C₄⋊C₈	—	2	4	no	—
141	C₂×C₄⋊C₈	C₄×C₃₁	1	8	yes	complemented
142	C₄³	—	2	4	no	—
143	#336	—	2	8	no	—
144	C₂×C₃₁	C₄⋊#95	1	1	yes	characteristic
145	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
146	C₂×C₃₁	#1728	1	1	yes	characteristic
147	C₂×C₃₁	C₄×C₄⋊C₈	1	2	yes	—
148	C₂×C₃₁	#1767	1	2	yes	—
149	C₂×C₃₁	#1797	1	4	yes	—
150	C₂×C₃₁	#1831	1	2	yes	—
151	C₂×C₃₁	#1842	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
154	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	—	2	4	no	—
163	C₂×C₄²	—	2	8	no	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	—	2	4	no	—
166	C₂×C₄²	—	2	4	no	—
167	C₂×C₄²	C₈×C₃₁	1	8	yes	complemented
168	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
169	C₂×C₄²	D₄×C₃₁	1	4	yes	—
170	C₂×C₄²	D₄×C₃₁	1	2	yes	—
171	C₂×C₄²	D₄×C₃₁	1	4	yes	—
172	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
173	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
174	C₄⋊C₈	—	2	16	no	—
175	C₃₁	#10832	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
183	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
188	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂×C₈×C₃₁	1	8	yes	—
190	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
191	C₂²×C₄	M₁₆×C₃₁	1	8	yes	—
192	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
199	C₂²×C₄	#41×C₃₁	1	4	yes	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	16	no	—
202	C₂×C₈	—	4	8	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	4	no	—
205	C₄²	—	2	16	no	—
206	C₄²	—	2	16	no	—
207	C₄²	—	2	8	no	—
208	C₄²	—	2	8	no	—
209	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
210	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
211	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
212	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
215	C₂³	C₃₁×#117	1	2	yes	—
216	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
217	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
218	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	C₃₁×#95	1	4	yes	—
229	C₂×C₄	C₃₁×#95	1	4	yes	—
230	C₂×C₄	C₄×C₈×C₃₁	1	4	yes	—
231	C₂×C₄	C₃₁×#97	1	4	yes	—
232	C₂×C₄	C₃₁×C₂²⋊C₈	1	8	yes	—
233	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
234	C₂×C₄	C₃₁×C₄⋊C₈	1	16	yes	complemented
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
238	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
239	C₂×C₄	C₃₁×#121	1	4	yes	—
240	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
241	C₂×C₄	C₃₁×#124	1	2	yes	—
242	C₂×C₄	C₃₁×#125	1	2	yes	—
243	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
244	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
245	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
246	C₈	—	4	8	no	—
247	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
248	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
249	C₂²	C₃₁×#380	1	2	yes	—
250	C₂²	C₃₁×#382	1	1	yes	characteristic
251	C₂²	C₃₁×#384	1	1	yes	characteristic
252	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
253	C₂²	C₂×C₃₁×#97	1	1	yes	characteristic
254	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
255	C₂²	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
256	C₂²	C₃₁×#407	1	2	yes	—
257	C₂²	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
258	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
259	C₂²	C₃₁×#423	1	4	yes	—
260	C₂²	C₃₁×#423	1	2	yes	—
261	C₂²	C₃₁×#431	1	4	yes	—
262	C₂²	C₃₁×#432	1	4	yes	—
263	C₄	—	2	16	no	—
264	C₄	—	2	4	no	—
265	C₄	C₃₁×#336	1	8	yes	complemented
266	C₄	C₃₁×#389	1	4	yes	—
267	C₄	C₃₁×#390	1	2	yes	—
268	C₄	C₃₁×#391	1	2	yes	—
269	C₂	C₃₁×C₄⋊#95	1	1	yes	characteristic
270	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
271	C₂	C₃₁×#1728	1	1	yes	characteristic
272	C₂	C₄×C₃₁×C₄⋊C₈	1	2	yes	—
273	C₂	C₃₁×#1767	1	2	yes	—
274	C₂	C₃₁×#1797	1	4	yes	—
275	C₂	C₃₁×#1831	1	2	yes	—
276	C₂	C₃₁×#1842	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
29. C₂×C₄²×C₃₁	C₈	semidirect
72. #10832	C₃₁	direct (N × H)
96. C₂×C₄×C₃₁	C₄⋊C₈	semidirect
130. C₄×C₃₁	#336	semidirect
141. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
167. C₂×C₄²	C₈×C₃₁	semidirect
175. C₃₁	#10832	direct (N × H)
234. C₂×C₄	C₃₁×C₄⋊C₈	semidirect
265. C₄	C₃₁×#336	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊C₈	2
C₂×C₄³×C₃₁	2
C₂×C₃₁×#336	2
#10832	31

Subgroups — C31×#10832 #21467591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10832 #21467591