Subgroups — C₃₁×#33518 #21490277

Order 7936.

1124 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#510	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2188	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2185	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2141	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₃₁	—	2	1	no	—
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₃₁	—	2	8	no	—
11	C₂²×Q₁₆×C₃₁	—	2	1	no	—
12	C₂²×Q₁₆×C₃₁	—	2	2	no	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#525	—	2	1	no	—
17	C₃₁×#510	—	2	8	no	—
18	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	complemented
19	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	complemented
20	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#465	—	2	2	no	—
24	C₃₁×#427	C₂²	1	1	yes	characteristic
25	C₃₁×#416	—	2	1	no	—
26	C₃₁×#358	C₂²	1	8	yes	complemented
27	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂×#41×C₃₁	—	2	1	no	—
29	C₂²×Q₈×C₃₁	—	2	8	no	—
30	C₂²×Q₈×C₃₁	—	2	4	no	—
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂²×Q₈×C₃₁	—	4	4	no	—
33	C₂²×Q₈×C₃₁	—	4	2	no	—
34	C₂²×Q₈×C₃₁	—	4	1	no	—
35	C₂³×C₄×C₃₁	—	4	1	no	—
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×#135	—	2	4	no	—
38	C₂×Q₁₆×C₃₁	—	2	8	no	—
39	C₂×Q₁₆×C₃₁	—	2	16	no	—
40	C₂×Q₁₆×C₃₁	—	2	16	no	—
41	C₂×Q₁₆×C₃₁	—	2	4	no	—
42	C₂×Q₁₆×C₃₁	—	2	2	no	—
43	C₂×Q₁₆×C₃₁	—	2	4	no	—
44	C₂×Q₁₆×C₃₁	D₄	1	8	yes	complemented
45	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
47	C₂×SD₁₆×C₃₁	—	2	1	no	—
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	—	2	4	no	—
50	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	D₄	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#123	—	4	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#120	—	4	1	no	—
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₃₁×#117	—	2	1	no	—
60	C₃₁×Q₃₂	—	2	8	no	—
61	C₃₁×SD₃₂	—	2	8	no	—
62	C₃₁×M₃₂	—	2	4	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#103	—	2	2	no	—
67	C₃₁×#102	—	2	2	no	—
68	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
69	#41×C₃₁	—	4	2	no	—
70	#41×C₃₁	—	4	2	no	—
71	C₂×Q₈×C₃₁	—	2	16	no	—
72	C₂×Q₈×C₃₁	—	2	8	no	—
73	C₂×Q₈×C₃₁	—	2	4	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	4	8	no	—
76	C₂×Q₈×C₃₁	—	4	8	no	—
77	C₂×Q₈×C₃₁	—	4	16	no	—
78	C₂×Q₈×C₃₁	—	4	4	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×Q₈×C₃₁	—	4	4	no	—
81	C₂×D₄×C₃₁	—	2	1	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	4	8	no	—
87	C₂²×C₄×C₃₁	—	4	4	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	D₈	1	2	yes	—
91	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₃₁	—	2	16	no	—
95	Q₁₆×C₃₁	—	2	8	no	—
96	Q₁₆×C₃₁	—	2	4	no	—
97	Q₁₆×C₃₁	—	2	2	no	—
98	Q₁₆×C₃₁	C₂×D₄	1	8	yes	—
99	SD₁₆×C₃₁	—	4	2	no	—
100	D₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	1	no	—
114	C₁₆×C₃₁	—	2	4	no	—
115	#33518	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	—	2	2	no	—
117	C₂³×C₃₁	—	2	4	no	—
118	C₂³×C₃₁	—	4	1	no	—
119	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
120	C₂³×C₃₁	C₂×D₈	1	2	yes	—
121	Q₈×C₃₁	—	2	8	no	—
122	Q₈×C₃₁	—	4	16	no	—
123	Q₈×C₃₁	—	4	4	no	—
124	Q₈×C₃₁	—	4	8	no	—
125	Q₈×C₃₁	—	4	1	no	—
126	Q₈×C₃₁	—	4	1	no	—
127	D₄×C₃₁	—	4	1	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	8	2	no	—
130	D₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	4	no	—
135	C₂×C₄×C₃₁	—	4	16	no	—
136	C₂×C₄×C₃₁	—	4	4	no	—
137	C₂×C₄×C₃₁	—	4	8	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	8	2	no	—
141	C₂×C₄×C₃₁	#120	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₃₁	—	2	2	no	—
147	C₈×C₃₁	#120	1	2	yes	—
148	C₈×C₃₁	C₂²×D₄	1	2	yes	—
149	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂×#510	C₂×C₃₁	1	2	yes	complemented
151	#3050	C₂×C₃₁	1	1	yes	characteristic
152	#2188	C₂×C₃₁	1	4	yes	complemented
153	#2185	C₂×C₃₁	1	4	yes	complemented
154	#2141	C₂×C₃₁	1	2	yes	complemented
155	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂²×C₃₁	—	2	8	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	2	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	1	no	—
162	C₂²×C₃₁	#447	1	2	yes	—
163	C₂²×C₃₁	#510	1	4	yes	complemented
164	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	4	8	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	—	8	1	no	—
169	C₄×C₃₁	#447	1	2	yes	—
170	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
171	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂³×Q₈	—	2	1	no	—
173	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
174	C₂²×Q₁₆	—	2	8	no	—
175	C₂²×Q₁₆	—	2	1	no	—
176	C₂²×Q₁₆	—	2	2	no	—
177	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
178	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
179	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	#525	—	2	1	no	—
181	#510	—	2	8	no	—
182	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	complemented
183	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	complemented
184	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
185	#467	C₂²×C₃₁	1	2	yes	—
186	#466	C₂²×C₃₁	1	2	yes	—
187	#465	—	2	2	no	—
188	#427	C₂²×C₃₁	1	1	yes	characteristic
189	#416	—	2	1	no	—
190	#358	C₂²×C₃₁	1	8	yes	complemented
191	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
192	C₂×C₃₁	—	2	2	no	—
193	C₂×C₃₁	—	2	4	no	—
194	C₂×C₃₁	—	16	1	no	—
195	C₂×C₃₁	C₂×#447	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#510	1	2	yes	—
197	C₂×#41	—	2	1	no	—
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	2	4	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	4	4	no	—
202	C₂²×Q₈	—	4	2	no	—
203	C₂²×Q₈	—	4	1	no	—
204	C₂³×C₄	—	4	1	no	—
205	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
206	#135	—	2	4	no	—
207	C₂×Q₁₆	—	2	8	no	—
208	C₂×Q₁₆	—	2	16	no	—
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	—	2	4	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×Q₁₆	D₄×C₃₁	1	8	yes	complemented
214	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
218	C₂²×C₈	—	2	4	no	—
219	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₃₁	1	2	yes	—
221	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
222	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
223	#123	—	4	1	no	—
224	C₂²⋊Q₈	—	2	2	no	—
225	#121	—	2	2	no	—
226	#120	—	4	1	no	—
227	C₄×D₄	—	4	2	no	—
228	#117	—	2	1	no	—
229	Q₃₂	—	2	8	no	—
230	SD₃₂	—	2	8	no	—
231	M₃₂	—	2	4	no	—
232	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
233	#107	C₂³×C₃₁	1	2	yes	—
234	#106	—	2	1	no	—
235	#103	—	2	2	no	—
236	#102	—	2	2	no	—
237	C₃₁	#33518	1	1	yes	characteristic, direct product factor, Sylow
238	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
239	#41	—	4	2	no	—
240	#41	—	4	2	no	—
241	C₂×Q₈	—	2	16	no	—
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	8	no	—
246	C₂×Q₈	—	4	16	no	—
247	C₂×Q₈	—	4	8	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	8	no	—
257	C₂²×C₄	—	4	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	D₈×C₃₁	1	2	yes	—
261	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	Q₁₆	—	2	16	no	—
265	Q₁₆	—	2	8	no	—
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	C₂×D₄×C₃₁	1	8	yes	—
269	SD₁₆	—	4	2	no	—
270	D₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
277	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	1	no	—
284	C₁₆	—	2	4	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
289	C₂³	C₂×D₈×C₃₁	1	2	yes	—
290	Q₈	—	2	8	no	—
291	Q₈	—	4	16	no	—
292	Q₈	—	4	8	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₃₁×#120	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
312	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	C₃₁×#120	1	2	yes	—
317	C₈	C₂²×D₄×C₃₁	1	2	yes	—
318	C₂²	—	2	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	1	no	—
324	C₂²	C₃₁×#447	1	2	yes	—
325	C₂²	C₃₁×#510	1	4	yes	complemented
326	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	8	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₃₁×#447	1	2	yes	—
332	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
333	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	4	no	—
336	C₂	—	16	1	no	—
337	C₂	C₂×C₃₁×#447	1	1	yes	characteristic
338	C₂	C₂×C₃₁×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₃₁	C₂	semidirect
2. C₂×C₃₁×#510	C₂	semidirect
4. C₃₁×#2188	C₂	semidirect
5. C₃₁×#2185	C₂	semidirect
6. C₃₁×#2141	C₂	semidirect
7. C₃₁×#2112	C₂	semidirect
18. C₂×C₃₁×Q₃₂	C₂²	semidirect
19. C₂×C₃₁×SD₃₂	C₂²	semidirect
26. C₃₁×#358	C₂²	semidirect
44. C₂×Q₁₆×C₃₁	D₄	semidirect
115. #33518	C₃₁	direct (N × H)
149. C₂³×Q₁₆	C₂×C₃₁	semidirect
150. C₂×#510	C₂×C₃₁	semidirect
152. #2188	C₂×C₃₁	semidirect
153. #2185	C₂×C₃₁	semidirect
154. #2141	C₂×C₃₁	semidirect
155. #2112	C₂×C₃₁	semidirect
163. C₂²×C₃₁	#510	semidirect
182. C₂×Q₃₂	C₂²×C₃₁	semidirect
183. C₂×SD₃₂	C₂²×C₃₁	semidirect
190. #358	C₂²×C₃₁	semidirect
213. C₂×Q₁₆	D₄×C₃₁	semidirect
237. C₃₁	#33518	direct (N × H)
325. C₂²	C₃₁×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#510	2
C₃₁×#3050	2
C₃₁×#2188	2
C₃₁×#2185	2
C₃₁×#2141	2
C₃₁×#2112	2
#33518	31

Subgroups — C31×#33518 #21490277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33518 #21490277