Subgroups — C₃₁×#21349 #21478108

Order 7936.

1030 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1939	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1850	C₂	1	4	yes	maximal
6	C₄×C₃₁×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#431	—	2	4	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	—	2	4	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#376	C₄	1	2	yes	—
24	C₄³×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	—	2	4	no	—
30	C₂²×C₈×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×#117	—	2	4	no	—
34	C₃₁×#117	C₈	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	D₄	1	4	yes	—
45	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	16	no	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₂²⋊C₈	—	2	4	no	—
52	C₄×C₈×C₃₁	—	2	4	no	—
53	C₄×C₈×C₃₁	—	2	4	no	—
54	C₄×C₈×C₃₁	—	2	4	no	—
55	C₃₁×#95	C₂×C₄	1	4	yes	—
56	C₃₁×#95	C₂×C₄	1	4	yes	—
57	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	16	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	8	no	—
79	C₂×C₈×C₃₁	#41	1	8	yes	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	complemented
82	C₂²⋊C₄×C₃₁	—	2	4	no	—
83	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
84	C₄²×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	—	2	8	no	—
86	C₄²×C₃₁	—	2	8	no	—
87	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
88	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
89	C₄²×C₃₁	C₂×C₈	1	8	yes	complemented
90	C₄²×C₃₁	C₂×D₄	1	2	yes	—
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
93	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
94	#21349	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	—	2	2	no	—
96	C₂³×C₃₁	—	2	2	no	—
97	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
99	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
100	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	C₄⋊C₈	1	16	yes	complemented
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
118	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₃₁	#131	1	4	yes	—
123	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
126	C₈×C₃₁	—	2	8	no	—
127	C₈×C₃₁	—	4	4	no	—
128	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
129	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
130	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
131	#1939	C₂×C₃₁	1	4	yes	complemented
132	#1850	C₂×C₃₁	1	4	yes	—
133	C₄×C₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	4	no	—
136	C₂²×C₃₁	—	2	4	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
139	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₃₁	C₄×#41	1	2	yes	—
141	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
142	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
143	C₂²×C₃₁	C₂×#131	1	2	yes	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	2	8	no	—
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	4	4	no	—
148	C₄×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
149	C₄×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
150	C₄×C₃₁	#424	1	4	yes	—
151	C₄×C₃₁	#525	1	2	yes	—
152	C₄×C₃₁	#533	1	2	yes	—
153	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
155	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
157	#431	—	2	4	no	—
158	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
160	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄×C₈	—	2	4	no	—
162	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
165	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
167	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
168	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
169	#376	C₄×C₃₁	1	2	yes	—
170	C₄³	C₂²×C₃₁	1	2	yes	—
171	#336	C₂²×C₃₁	1	8	yes	—
172	C₂×C₃₁	—	2	4	no	—
173	C₂×C₃₁	#2296	1	1	yes	characteristic
174	C₂×C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
175	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
176	C₂×C₃₁	C₈×#41	1	2	yes	—
177	C₂×C₃₁	#2965	1	2	yes	—
178	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	2	8	no	—
183	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
184	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
185	#117	—	2	4	no	—
186	#117	C₈×C₃₁	1	8	yes	complemented
187	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
189	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
196	C₂×C₄²	D₄×C₃₁	1	4	yes	—
197	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
198	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	16	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	#95	C₂×C₄×C₃₁	1	4	yes	—
208	#95	C₂×C₄×C₃₁	1	4	yes	—
209	C₃₁	#21349	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
215	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
216	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
227	C₂×C₈	—	2	16	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	#41×C₃₁	1	8	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	complemented
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	8	no	—
240	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
241	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
242	C₄²	C₂×C₈×C₃₁	1	8	yes	complemented
243	C₄²	C₂×D₄×C₃₁	1	2	yes	—
244	C₄²	C₂×D₄×C₃₁	1	2	yes	—
245	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
246	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
250	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
251	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
252	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	C₃₁×C₄⋊C₈	1	16	yes	complemented
264	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
269	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
270	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
271	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
272	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
273	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
274	C₂×C₄	C₃₁×#131	1	4	yes	—
275	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
278	C₈	—	2	8	no	—
279	C₈	—	4	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
285	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
286	C₂²	C₄×#41×C₃₁	1	2	yes	—
287	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
288	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
289	C₂²	C₂×C₃₁×#131	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	8	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
295	C₄	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
296	C₄	C₃₁×#424	1	4	yes	—
297	C₄	C₃₁×#525	1	2	yes	—
298	C₄	C₃₁×#533	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	C₃₁×#2296	1	1	yes	characteristic
301	C₂	C₂²×C₃₁×C₄⋊C₈	1	1	yes	characteristic
302	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
303	C₂	C₈×#41×C₃₁	1	2	yes	—
304	C₂	C₃₁×#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂³
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1939	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
34. C₃₁×#117	C₈	semidirect
81. C₄⋊C₄×C₃₁	C₂×C₈	semidirect
89. C₄²×C₃₁	C₂×C₈	semidirect
94. #21349	C₃₁	direct (N × H)
111. C₂×C₄×C₃₁	C₄⋊C₈	semidirect
131. #1939	C₂×C₃₁	semidirect
133. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
186. #117	C₈×C₃₁	semidirect
209. C₃₁	#21349	direct (N × H)
234. C₄⋊C₄	C₂×C₈×C₃₁	semidirect
242. C₄²	C₂×C₈×C₃₁	semidirect
263. C₂×C₄	C₃₁×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₂²×C₄×C₈×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1939	2
C₃₁×#1850	2
C₄×C₃₁×C₄⋊C₈	2
#21349	31

Subgroups — C31×#21349 #21478108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21349 #21478108