Subgroups — C₃₁×#33443 #21490202

Order 7936.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3407	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2171	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2170	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2117	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
9	C₂×C₃₁×#131	C₂²	1	2	yes	—
10	C₃₁×#504	—	2	4	no	—
11	C₃₁×#504	C₂²	1	8	yes	complemented
12	C₂×C₃₁×M₃₂	—	2	4	no	—
13	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
16	C₃₁×#443	C₄	1	2	yes	—
17	C₃₁×#443	C₄	1	2	yes	—
18	C₃₁×#443	C₂²	1	2	yes	—
19	C₃₁×#443	C₂²	1	2	yes	—
20	C₂×C₃₁×#108	C₄	1	1	yes	characteristic
21	C₂×C₃₁×#104	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#364	C₂²	1	2	yes	complemented
24	C₃₁×#364	C₂²	1	2	yes	complemented
25	C₃₁×#350	C₂²	1	8	yes	complemented
26	C₃₁×#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
28	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
29	C₃₁×#135	C₈	1	2	yes	—
30	C₃₁×#135	C₈	1	2	yes	—
31	C₃₁×#135	C₂×C₄	1	2	yes	—
32	C₃₁×#135	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₃₁	C₈	1	1	yes	characteristic
34	C₂×SD₁₆×C₃₁	C₈	1	2	yes	—
35	C₂×D₈×C₃₁	C₈	1	1	yes	characteristic
36	C₃₁×#131	—	2	2	no	—
37	C₃₁×#131	—	2	2	no	—
38	C₃₁×#131	C₂×C₄	1	4	yes	—
39	C₃₁×#131	C₂³	1	4	yes	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₃₁×M₃₂	—	2	2	no	—
47	C₃₁×M₃₂	—	2	1	no	—
48	C₃₁×M₃₂	—	2	1	no	—
49	C₃₁×M₃₂	—	2	8	no	—
50	C₃₁×M₃₂	D₄	1	2	yes	—
51	C₃₁×M₃₂	D₄	1	2	yes	—
52	C₃₁×M₃₂	C₂³	1	4	yes	—
53	C₂×C₁₆×C₃₁	—	2	2	no	—
54	C₂×C₁₆×C₃₁	—	2	8	no	—
55	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
57	C₃₁×#108	C₂×C₄	1	2	yes	—
58	C₃₁×#108	C₂×C₄	1	2	yes	—
59	C₃₁×#104	C₂×C₄	1	4	yes	—
60	C₃₁×#104	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	#41×C₃₁	—	2	2	no	—
66	#41×C₃₁	—	2	2	no	—
67	#41×C₃₁	C₂×C₈	1	4	yes	—
68	#41×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	Q₁₆×C₃₁	C₂×C₈	1	2	yes	—
76	Q₁₆×C₃₁	C₂×C₈	1	2	yes	—
77	SD₁₆×C₃₁	C₂×C₈	1	4	yes	—
78	SD₁₆×C₃₁	C₂×C₈	1	4	yes	—
79	D₈×C₃₁	C₂×C₈	1	2	yes	—
80	D₈×C₃₁	C₂×C₈	1	2	yes	—
81	M₁₆×C₃₁	—	2	2	no	—
82	M₁₆×C₃₁	—	2	4	no	—
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
98	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₁₆×C₃₁	—	2	4	no	—
104	C₁₆×C₃₁	—	2	2	no	—
105	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
106	#33443	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	#131	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	2	no	—
110	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
111	D₄×C₃₁	—	2	4	no	—
112	D₄×C₃₁	—	2	4	no	—
113	D₄×C₃₁	—	2	2	no	—
114	D₄×C₃₁	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₃₁	—	2	4	no	—
128	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
129	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
130	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
132	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
133	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
134	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
136	#3407	C₂×C₃₁	1	2	yes	complemented
137	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
138	#2171	C₂×C₃₁	1	4	yes	complemented
139	#2170	C₂×C₃₁	1	4	yes	complemented
140	C₂×#364	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#2117	C₂×C₃₁	1	2	yes	complemented
142	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
148	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
153	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
154	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
156	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
157	C₂×#131	C₂²×C₃₁	1	2	yes	—
158	#504	—	2	4	no	—
159	#504	C₂²×C₃₁	1	8	yes	complemented
160	C₂×M₃₂	—	2	4	no	—
161	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
162	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
164	#443	C₄×C₃₁	1	2	yes	—
165	#443	C₄×C₃₁	1	2	yes	—
166	#443	C₂²×C₃₁	1	2	yes	—
167	#443	C₂²×C₃₁	1	2	yes	—
168	C₂×#108	C₄×C₃₁	1	1	yes	characteristic
169	C₂×#104	C₄×C₃₁	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
171	#364	C₂²×C₃₁	1	2	yes	complemented
172	#364	C₂²×C₃₁	1	2	yes	complemented
173	#350	C₂²×C₃₁	1	8	yes	complemented
174	#346	C₂²×C₃₁	1	2	yes	complemented
175	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	2	1	no	—
179	C₂×C₃₁	—	4	4	no	—
180	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
181	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
182	#135	C₈×C₃₁	1	2	yes	—
183	#135	C₈×C₃₁	1	2	yes	—
184	#135	C₂×C₄×C₃₁	1	2	yes	—
185	#135	C₂×C₄×C₃₁	1	2	yes	—
186	C₂×Q₁₆	C₈×C₃₁	1	1	yes	characteristic
187	C₂×SD₁₆	C₈×C₃₁	1	2	yes	—
188	C₂×D₈	C₈×C₃₁	1	1	yes	characteristic
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	#131	C₂×C₄×C₃₁	1	4	yes	—
192	#131	C₂³×C₃₁	1	4	yes	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
199	M₃₂	—	2	2	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	8	no	—
203	M₃₂	D₄×C₃₁	1	2	yes	—
204	M₃₂	D₄×C₃₁	1	2	yes	—
205	M₃₂	C₂³×C₃₁	1	4	yes	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	8	no	—
208	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
209	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
210	#108	C₂×C₄×C₃₁	1	2	yes	—
211	#108	C₂×C₄×C₃₁	1	2	yes	—
212	#104	C₂×C₄×C₃₁	1	4	yes	—
213	#104	C₂×C₄×C₃₁	1	4	yes	—
214	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
218	C₃₁	#33443	1	1	yes	characteristic, direct product factor, Sylow
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	C₂×C₈×C₃₁	1	4	yes	—
222	#41	C₂²×C₄×C₃₁	1	4	yes	—
223	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
229	Q₁₆	C₂×C₈×C₃₁	1	2	yes	—
230	Q₁₆	C₂×C₈×C₃₁	1	2	yes	—
231	SD₁₆	C₂×C₈×C₃₁	1	4	yes	—
232	SD₁₆	C₂×C₈×C₃₁	1	4	yes	—
233	D₈	C₂×C₈×C₃₁	1	2	yes	—
234	D₈	C₂×C₈×C₃₁	1	2	yes	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
252	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
256	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
257	C₁₆	—	2	4	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₃₁×#131	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×C₈×C₃₁	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
279	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
282	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
283	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
284	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
285	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
286	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
287	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
293	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
294	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
299	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
300	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
301	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×C₈×D₄
prime radical	C₂×C₃₁×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ C₂×C₃₁×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3407	C₂	semidirect
3. C₃₁×#2171	C₂	semidirect
4. C₃₁×#2170	C₂	semidirect
5. C₂×C₃₁×#364	C₂	semidirect
6. C₃₁×#2117	C₂	semidirect
7. C₄×C₃₁×M₃₂	C₂	semidirect
11. C₃₁×#504	C₂²	semidirect
23. C₃₁×#364	C₂²	semidirect
24. C₃₁×#364	C₂²	semidirect
25. C₃₁×#350	C₂²	semidirect
26. C₃₁×#346	C₂²	semidirect
27. C₄×C₁₆×C₃₁	C₂²	semidirect
106. #33443	C₃₁	direct (N × H)
136. #3407	C₂×C₃₁	semidirect
138. #2171	C₂×C₃₁	semidirect
139. #2170	C₂×C₃₁	semidirect
140. C₂×#364	C₂×C₃₁	semidirect
141. #2117	C₂×C₃₁	semidirect
142. C₄×M₃₂	C₂×C₃₁	semidirect
159. #504	C₂²×C₃₁	semidirect
171. #364	C₂²×C₃₁	semidirect
172. #364	C₂²×C₃₁	semidirect
173. #350	C₂²×C₃₁	semidirect
174. #346	C₂²×C₃₁	semidirect
175. C₄×C₁₆	C₂²×C₃₁	semidirect
218. C₃₁	#33443	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3407	2
C₂×C₃₁×#443	2
C₃₁×#2171	2
C₃₁×#2170	2
C₂×C₃₁×#364	2
C₃₁×#2117	2
C₄×C₃₁×M₃₂	2
#33443	31

Subgroups — C31×#33443 #21490202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33443 #21490202