Subgroups — C₃₁×#21359 #21478118

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1948	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1853	C₂	1	4	yes	maximal
6	C₄×C₃₁×#106	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#117	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₃₁	—	2	1	no	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#431	C₄	1	8	yes	complemented
12	C₂×C₃₁×#106	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	C₂²	1	2	yes	—
16	C₃₁×#384	C₂²	1	2	yes	—
17	C₃₁×#382	—	2	1	no	—
18	C₃₁×#380	—	2	1	no	—
19	C₃₁×#380	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
21	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₃₁×#95	—	2	1	no	—
27	C₃₁×#340	C₂²	1	8	yes	—
28	C₂³×C₄×C₃₁	—	2	2	no	—
29	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×#117	—	4	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	—	2	2	no	—
44	C₂×C₄²×C₃₁	—	4	4	no	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#106	—	2	16	no	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
53	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
55	C₃₁×#95	—	2	2	no	—
56	C₃₁×#95	—	2	2	no	—
57	C₃₁×#95	—	2	2	no	—
58	C₃₁×#95	—	2	2	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	4	2	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	—	4	4	no	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₃₁	—	2	8	no	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	#41	1	8	yes	—
87	C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₄²×C₃₁	—	4	4	no	—
90	C₄²×C₃₁	—	4	4	no	—
91	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
92	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
93	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
95	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
96	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
97	#21359	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
102	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
103	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	16	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	#106	1	16	yes	complemented
118	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
127	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
130	C₈×C₃₁	—	2	8	no	—
131	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
132	#2285	C₂×C₃₁	1	1	yes	characteristic
133	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
134	#1948	C₂×C₃₁	1	4	yes	complemented
135	#1853	C₂×C₃₁	1	4	yes	—
136	C₄×#106	C₂×C₃₁	1	4	yes	complemented
137	C₂²×C₃₁	—	2	4	no	—
138	C₂²×C₃₁	—	2	4	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	C₂×#106	1	4	yes	—
142	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₃₁	#525	1	2	yes	—
144	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
146	C₂²×C₃₁	#575	1	2	yes	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	4	8	no	—
151	C₄×C₃₁	—	8	4	no	—
152	C₄×C₃₁	C₂×#106	1	4	yes	—
153	C₄×C₃₁	C₂×#106	1	4	yes	—
154	C₄×C₃₁	C₄×#41	1	2	yes	—
155	C₄×C₃₁	#533	1	2	yes	—
156	C₂×#117	—	2	1	no	—
157	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
158	C₂²×C₄²	—	2	1	no	—
159	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
160	#431	C₄×C₃₁	1	8	yes	complemented
161	C₂×#106	C₂²×C₃₁	1	8	yes	—
162	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
165	#384	C₂²×C₃₁	1	2	yes	—
166	#382	—	2	1	no	—
167	#380	—	2	1	no	—
168	#380	C₂²×C₃₁	1	2	yes	—
169	C₄×C₄⋊C₄	—	2	1	no	—
170	C₄×C₄⋊C₄	—	2	1	no	—
171	C₄×C₄⋊C₄	—	2	1	no	—
172	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
175	C₂×#95	—	2	1	no	—
176	#340	C₂²×C₃₁	1	8	yes	—
177	C₂×C₃₁	—	2	4	no	—
178	C₂×C₃₁	#2296	1	1	yes	characteristic
179	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
180	C₂×C₃₁	#2911	1	1	yes	characteristic
181	C₂×C₃₁	#3101	1	2	yes	—
182	C₂×C₃₁	#3103	1	2	yes	—
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
188	#117	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	4	4	no	—
200	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
201	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
204	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
206	#106	—	2	16	no	—
207	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
208	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
209	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	C₃₁	#21359	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
224	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
225	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
226	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
235	C₂×C₈	—	2	8	no	—
236	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	#41×C₃₁	1	8	yes	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
248	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
249	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
250	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
251	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
252	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
256	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
257	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
258	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₃₁×#106	1	16	yes	complemented
273	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
277	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
278	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
279	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
281	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
282	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
285	C₈	—	2	8	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₂×C₃₁×#106	1	4	yes	—
291	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₂²	C₃₁×#525	1	2	yes	—
293	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
294	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
295	C₂²	C₃₁×#575	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	8	no	—
300	C₄	—	8	4	no	—
301	C₄	C₂×C₃₁×#106	1	4	yes	—
302	C₄	C₂×C₃₁×#106	1	4	yes	—
303	C₄	C₄×#41×C₃₁	1	2	yes	—
304	C₄	C₃₁×#533	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₃₁×#2296	1	1	yes	characteristic
307	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
308	C₂	C₃₁×#2911	1	1	yes	characteristic
309	C₂	C₃₁×#3101	1	2	yes	—
310	C₂	C₃₁×#3103	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1948	C₂	semidirect
6. C₄×C₃₁×#106	C₂	semidirect
11. C₃₁×#431	C₄	semidirect
52. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
54. C₄×C₈×C₃₁	C₂×C₄	semidirect
97. #21359	C₃₁	direct (N × H)
117. C₂×C₄×C₃₁	#106	semidirect
134. #1948	C₂×C₃₁	semidirect
136. C₄×#106	C₂×C₃₁	semidirect
160. #431	C₄×C₃₁	semidirect
207. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
209. C₄×C₈	C₂×C₄×C₃₁	semidirect
214. C₃₁	#21359	direct (N × H)
272. C₂×C₄	C₃₁×#106	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
C₃₁×#2285	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1948	2
C₃₁×#1853	2
C₄×C₃₁×#106	2
#21359	31

Subgroups — C31×#21359 #21478118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21359 #21478118