Subgroups — C₃₁×#33688 #21490447

Order 7936.

564 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	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2238	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2230	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2213	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2190	C₂	1	2	yes	maximal, complemented
7	C₃₁×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#575	C₂²	1	1	yes	characteristic
10	C₃₁×#568	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#509	C₂²	1	4	yes	—
13	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#469	—	2	1	no	—
17	C₃₁×#469	C₂²	1	2	yes	—
18	C₃₁×#444	—	2	1	no	—
19	C₃₁×#444	C₂²	1	2	yes	—
20	C₃₁×#443	—	2	1	no	—
21	C₃₁×#443	C₂²	1	2	yes	—
22	C₃₁×#442	—	2	1	no	—
23	C₃₁×#442	C₂²	1	2	yes	complemented
24	C₃₁×#430	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#104	—	2	1	no	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#365	C₂²	1	4	yes	complemented
28	C₃₁×#364	C₂²	1	2	yes	complemented
29	C₃₁×#361	C₂²	1	4	yes	complemented
30	C₃₁×#360	C₂²	1	4	yes	—
31	C₃₁×#349	C₂²	1	2	yes	—
32	C₃₁×#347	C₂²	1	2	yes	complemented
33	C₂×#41×C₃₁	—	2	1	no	—
34	C₂×#41×C₃₁	—	4	1	no	—
35	C₃₁×#137	—	2	2	no	—
36	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
37	C₃₁×#135	—	2	1	no	—
38	C₃₁×#135	C₂³	1	2	yes	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#131	—	2	2	no	—
42	C₃₁×#131	—	2	2	no	—
43	C₃₁×#131	—	4	2	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	—	4	2	no	—
46	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
47	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#124	—	4	1	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	4	1	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	4	1	no	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₃₁×SD₃₂	—	2	4	no	—
57	C₃₁×D₁₆	—	2	4	no	—
58	C₃₁×M₃₂	C₂³	1	4	yes	—
59	C₃₁×#108	—	2	1	no	—
60	C₃₁×#108	C₂³	1	2	yes	—
61	C₃₁×#106	—	2	1	no	—
62	C₃₁×#106	C₂³	1	2	yes	—
63	C₃₁×#104	—	2	2	no	—
64	C₃₁×#104	—	2	2	no	—
65	C₃₁×#102	—	2	2	no	—
66	C₃₁×#102	—	2	2	no	—
67	C₃₁×#97	D₄	1	1	yes	characteristic
68	C₃₁×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	#41×C₃₁	—	2	2	no	—
72	#41×C₃₁	—	2	2	no	—
73	#41×C₃₁	—	4	1	no	—
74	#41×C₃₁	—	4	1	no	—
75	#41×C₃₁	—	8	4	no	—
76	C₂×Q₈×C₃₁	—	2	1	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	2	1	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	—	2	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₃₁	—	2	1	no	—
89	Q₁₆×C₃₁	#41	1	2	yes	complemented
90	SD₁₆×C₃₁	—	2	2	no	—
91	SD₁₆×C₃₁	—	2	2	no	—
92	D₈×C₃₁	—	2	1	no	—
93	D₈×C₃₁	—	2	4	no	—
94	D₈×C₃₁	—	4	2	no	—
95	D₈×C₃₁	#41	1	2	yes	complemented
96	M₁₆×C₃₁	—	2	2	no	—
97	M₁₆×C₃₁	—	2	2	no	—
98	M₁₆×C₃₁	—	2	2	no	—
99	M₁₆×C₃₁	—	4	2	no	—
100	M₁₆×C₃₁	—	4	1	no	—
101	M₁₆×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	4	2	no	—
103	C₂×C₈×C₃₁	—	4	1	no	—
104	C₂×C₈×C₃₁	—	4	2	no	—
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	2	1	no	—
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₃₁	—	2	4	no	—
117	#33688	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	4	1	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈×C₃₁	—	2	2	no	—
122	Q₈×C₃₁	—	4	1	no	—
123	Q₈×C₃₁	—	8	2	no	—
124	D₄×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	4	1	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	4	no	—
143	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	—	4	2	no	—
148	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
149	C₈×C₃₁	C₂×#41	1	2	yes	—
150	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2956	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2238	C₂×C₃₁	1	2	yes	complemented
153	#2230	C₂×C₃₁	1	2	yes	complemented
154	#2213	C₂×C₃₁	1	2	yes	complemented
155	#2190	C₂×C₃₁	1	2	yes	complemented
156	AGL(1,ℤ/16)	C₂×C₃₁	1	4	yes	complemented
157	#2165	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	4	1	no	—
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	16	1	no	—
165	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	4	1	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	#525	1	1	yes	characteristic
172	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	#575	C₂²×C₃₁	1	1	yes	characteristic
174	#568	—	2	1	no	—
175	#532	—	2	1	no	—
176	#509	C₂²×C₃₁	1	4	yes	—
177	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
178	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
179	#486	—	2	1	no	—
180	#469	—	2	1	no	—
181	#469	C₂²×C₃₁	1	2	yes	—
182	#444	—	2	1	no	—
183	#444	C₂²×C₃₁	1	2	yes	—
184	#443	—	2	1	no	—
185	#443	C₂²×C₃₁	1	2	yes	—
186	#442	—	2	1	no	—
187	#442	C₂²×C₃₁	1	2	yes	complemented
188	#430	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×#104	—	2	1	no	—
190	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	#365	C₂²×C₃₁	1	4	yes	complemented
192	#364	C₂²×C₃₁	1	2	yes	complemented
193	#361	C₂²×C₃₁	1	4	yes	complemented
194	#360	C₂²×C₃₁	1	4	yes	—
195	#349	C₂²×C₃₁	1	2	yes	—
196	#347	C₂²×C₃₁	1	2	yes	complemented
197	C₂×C₃₁	—	2	1	no	—
198	C₂×C₃₁	—	4	1	no	—
199	C₂×C₃₁	—	8	2	no	—
200	C₂×C₃₁	—	16	2	no	—
201	C₂×C₃₁	#3109	1	1	yes	characteristic
202	C₂×#41	—	2	1	no	—
203	C₂×#41	—	4	1	no	—
204	#137	—	2	2	no	—
205	AGL(1,ℤ/8)	—	2	2	no	—
206	#135	—	2	1	no	—
207	#135	C₂³×C₃₁	1	2	yes	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	C₂³×C₃₁	1	2	yes	—
210	#131	—	2	2	no	—
211	#131	—	2	2	no	—
212	#131	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
216	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	#127	—	2	1	no	—
219	#124	—	4	1	no	—
220	#121	—	2	2	no	—
221	#121	—	4	1	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	4	1	no	—
224	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
225	SD₃₂	—	2	4	no	—
226	D₁₆	—	2	4	no	—
227	M₃₂	C₂³×C₃₁	1	4	yes	—
228	#108	—	2	1	no	—
229	#108	C₂³×C₃₁	1	2	yes	—
230	#106	—	2	1	no	—
231	#106	C₂³×C₃₁	1	2	yes	—
232	#104	—	2	2	no	—
233	#104	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	#97	D₄×C₃₁	1	1	yes	characteristic
237	#97	C₂³×C₃₁	1	1	yes	characteristic
238	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₃₁	#33688	1	1	yes	characteristic, direct product factor, Sylow
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	#41×C₃₁	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	D₈	#41×C₃₁	1	2	yes	complemented
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₁₆	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	2	no	—
317	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	C₂×#41×C₃₁	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	16	1	no	—
326	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₃₁×#525	1	1	yes	characteristic
333	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	4	1	no	—
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₃₁×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3184	C₂	semidirect
2. C₃₁×#2956	C₂	semidirect
3. C₃₁×#2238	C₂	semidirect
4. C₃₁×#2230	C₂	semidirect
5. C₃₁×#2213	C₂	semidirect
6. C₃₁×#2190	C₂	semidirect
7. C₃₁×AGL(1,ℤ/16)	C₂	semidirect
8. C₃₁×#2165	C₂	semidirect
13. C₃₁×#498	C₂²	semidirect
14. C₃₁×#493	C₂²	semidirect
23. C₃₁×#442	C₂²	semidirect
27. C₃₁×#365	C₂²	semidirect
28. C₃₁×#364	C₂²	semidirect
29. C₃₁×#361	C₂²	semidirect
32. C₃₁×#347	C₂²	semidirect
89. Q₁₆×C₃₁	#41	semidirect
95. D₈×C₃₁	#41	semidirect
117. #33688	C₃₁	direct (N × H)
150. #3184	C₂×C₃₁	semidirect
151. #2956	C₂×C₃₁	semidirect
152. #2238	C₂×C₃₁	semidirect
153. #2230	C₂×C₃₁	semidirect
154. #2213	C₂×C₃₁	semidirect
155. #2190	C₂×C₃₁	semidirect
156. AGL(1,ℤ/16)	C₂×C₃₁	semidirect
157. #2165	C₂×C₃₁	semidirect
177. #498	C₂²×C₃₁	semidirect
178. #493	C₂²×C₃₁	semidirect
187. #442	C₂²×C₃₁	semidirect
191. #365	C₂²×C₃₁	semidirect
192. #364	C₂²×C₃₁	semidirect
193. #361	C₂²×C₃₁	semidirect
196. #347	C₂²×C₃₁	semidirect
240. C₃₁	#33688	direct (N × H)
259. Q₁₆	#41×C₃₁	semidirect
265. D₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3184	2
C₃₁×#2956	2
C₃₁×#2238	2
C₃₁×#2230	2
C₃₁×#2213	2
C₃₁×#2190	2
C₃₁×AGL(1,ℤ/16)	2
C₃₁×#2165	2
#33688	31

Subgroups — C31×#33688 #21490447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33688 #21490447