Subgroups — C₃₁×#8340 #21465099

Order 7936.

1030 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#332	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₃₁	—	2	1	no	—
6	C₂×C₃₁×C₄⋊Q₈	—	2	8	no	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
12	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×#336	—	2	1	no	—
20	C₃₁×#332	—	2	8	no	—
21	C₂²×Q₈×C₃₁	—	2	8	no	—
22	C₂²×Q₈×C₃₁	—	2	2	no	—
23	C₂²×Q₈×C₃₁	—	2	2	no	—
24	C₂²×Q₈×C₃₁	—	4	4	no	—
25	C₂²×Q₈×C₃₁	—	4	2	no	—
26	C₂²×Q₈×C₃₁	—	4	2	no	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₃₁×C₄⋊Q₈	—	2	8	no	—
34	C₃₁×C₄⋊Q₈	—	2	8	no	—
35	C₃₁×C₄⋊Q₈	—	2	8	no	—
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	D₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₃₁	—	2	4	no	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	4	no	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₂²⋊C₈	—	4	2	no	—
52	C₃₁×C₂²⋊C₈	—	4	2	no	—
53	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₃₁	—	2	8	no	—
55	C₂×Q₈×C₃₁	—	2	8	no	—
56	C₂×Q₈×C₃₁	—	4	4	no	—
57	C₂×Q₈×C₃₁	—	4	16	no	—
58	C₂×Q₈×C₃₁	—	4	16	no	—
59	C₂×Q₈×C₃₁	—	4	4	no	—
60	C₂×Q₈×C₃₁	—	4	2	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	8	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	4	4	no	—
73	C₂²×C₄×C₃₁	—	4	4	no	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	D₈	1	2	yes	—
78	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	—	4	2	no	—
87	C₂×C₈×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	—	2	8	no	—
91	C₄⋊C₄×C₃₁	—	2	8	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	8	no	—
94	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	C₂×D₄	1	4	yes	—
99	#8340	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	1	no	—
101	C₂³×C₃₁	—	2	1	no	—
102	C₂³×C₃₁	—	2	1	no	—
103	C₂³×C₃₁	—	2	1	no	—
104	C₂³×C₃₁	#101	1	2	yes	—
105	C₂³×C₃₁	#102	1	2	yes	—
106	C₂³×C₃₁	#103	1	2	yes	—
107	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	16	no	—
109	Q₈×C₃₁	—	4	8	no	—
110	Q₈×C₃₁	—	4	8	no	—
111	C₂×C₄×C₃₁	—	2	8	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	4	no	—
123	C₂×C₄×C₃₁	—	2	4	no	—
124	C₂×C₄×C₃₁	—	2	8	no	—
125	C₂×C₄×C₃₁	—	4	16	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	#101	1	2	yes	—
129	C₂×C₄×C₃₁	#102	1	2	yes	—
130	C₂×C₄×C₃₁	#103	1	2	yes	—
131	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₃₁	#120	1	2	yes	—
133	C₂×C₄×C₃₁	#120	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₃₁	#137	1	2	yes	—
140	C₈×C₃₁	—	8	2	no	—
141	C₈×C₃₁	—	8	2	no	—
142	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
143	#1797	C₂×C₃₁	1	1	yes	characteristic
144	#1797	C₂×C₃₁	1	1	yes	characteristic
145	C₂×#332	C₂×C₃₁	1	4	yes	complemented
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	2	no	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	2	2	no	—
151	C₂²×C₃₁	—	2	2	no	—
152	C₂²×C₃₁	#332	1	4	yes	complemented
153	C₂²×C₃₁	#379	1	1	yes	characteristic
154	C₂²×C₃₁	C₂×#101	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×#101	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
158	C₂²×C₃₁	#418	1	1	yes	characteristic
159	C₂²×C₃₁	#419	1	1	yes	characteristic
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	4	8	no	—
165	C₄×C₃₁	#447	1	2	yes	—
166	C₄×C₃₁	#448	1	2	yes	—
167	C₄×C₃₁	#450	1	2	yes	—
168	C₄×C₃₁	#451	1	2	yes	—
169	C₂³×Q₈	—	2	1	no	—
170	C₂×C₄⋊Q₈	—	2	8	no	—
171	C₂×C₄⋊Q₈	—	2	1	no	—
172	C₂×C₄⋊Q₈	—	2	1	no	—
173	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
174	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
175	C₂²×C₄⋊C₄	—	2	1	no	—
176	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
177	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
180	C₂×C₂²⋊C₈	—	2	1	no	—
181	C₂×C₂²⋊C₈	—	2	1	no	—
182	#336	—	2	1	no	—
183	#336	—	2	1	no	—
184	#332	—	2	8	no	—
185	C₂×C₃₁	—	2	4	no	—
186	C₂×C₃₁	C₂×#332	1	1	yes	characteristic
187	C₂×C₃₁	#1546	1	1	yes	characteristic
188	C₂×C₃₁	#1549	1	1	yes	characteristic
189	C₂×C₃₁	#1556	1	1	yes	characteristic
190	C₂×C₃₁	#1784	1	1	yes	characteristic
191	C₂×C₃₁	#1786	1	1	yes	characteristic
192	C₂×C₃₁	#1787	1	1	yes	characteristic
193	C₂²×Q₈	—	2	8	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	4	4	no	—
197	C₂²×Q₈	—	4	2	no	—
198	C₂²×Q₈	—	4	2	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
202	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	8	no	—
206	C₄⋊Q₈	—	2	8	no	—
207	C₄⋊Q₈	—	2	8	no	—
208	C₄⋊Q₈	—	2	4	no	—
209	C₄⋊Q₈	D₄×C₃₁	1	8	yes	—
210	C₂×C₄⋊C₄	—	2	8	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	4	no	—
223	C₂²⋊C₈	—	4	2	no	—
224	C₂²⋊C₈	—	4	2	no	—
225	C₃₁	#8340	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	16	no	—
231	C₂×Q₈	—	4	16	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₃₁	1	2	yes	—
251	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
252	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
253	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂×D₄×C₃₁	1	4	yes	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	C₃₁×#101	1	2	yes	—
277	C₂³	C₃₁×#102	1	2	yes	—
278	C₂³	C₃₁×#103	1	2	yes	—
279	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
280	Q₈	—	4	16	no	—
281	Q₈	—	4	8	no	—
282	Q₈	—	4	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	C₃₁×#101	1	2	yes	—
301	C₂×C₄	C₃₁×#102	1	2	yes	—
302	C₂×C₄	C₃₁×#103	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
304	C₂×C₄	C₃₁×#120	1	2	yes	—
305	C₂×C₄	C₃₁×#120	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
311	C₂×C₄	C₃₁×#137	1	2	yes	—
312	C₈	—	8	2	no	—
313	C₈	—	8	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	C₃₁×#332	1	4	yes	complemented
321	C₂²	C₃₁×#379	1	1	yes	characteristic
322	C₂²	C₂×C₃₁×#101	1	1	yes	characteristic
323	C₂²	C₂×C₃₁×#101	1	1	yes	characteristic
324	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
325	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
326	C₂²	C₃₁×#418	1	1	yes	characteristic
327	C₂²	C₃₁×#419	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	8	no	—
333	C₄	C₃₁×#447	1	2	yes	—
334	C₄	C₃₁×#448	1	2	yes	—
335	C₄	C₃₁×#450	1	2	yes	—
336	C₄	C₃₁×#451	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	C₂×C₃₁×#332	1	1	yes	characteristic
339	C₂	C₃₁×#1546	1	1	yes	characteristic
340	C₂	C₃₁×#1549	1	1	yes	characteristic
341	C₂	C₃₁×#1556	1	1	yes	characteristic
342	C₂	C₃₁×#1784	1	1	yes	characteristic
343	C₂	C₃₁×#1786	1	1	yes	characteristic
344	C₂	C₃₁×#1787	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂×C₃₁×#332	C₂	semidirect
99. #8340	C₃₁	direct (N × H)
145. C₂×#332	C₂×C₃₁	semidirect
152. C₂²×C₃₁	#332	semidirect
225. C₃₁	#8340	direct (N × H)
320. C₂²	C₃₁×#332	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⋊Q₈	2
C₃₁×#1797	2
C₂×C₃₁×#332	2
#8340	31

Subgroups — C31×#8340 #21465099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8340 #21465099