Subgroups — C₃₁×#29154 #21485913

Order 7936.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2031	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1976	C₂	1	2	yes	maximal
5	C₃₁×#1833	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1814	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1812	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₃₁×#533	C₂²	1	2	yes	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#499	—	2	1	no	—
14	C₃₁×#499	—	2	1	no	—
15	C₃₁×#498	—	2	1	no	—
16	C₃₁×#443	—	2	1	no	—
17	C₃₁×#443	—	2	1	no	—
18	C₃₁×#443	—	2	1	no	—
19	C₃₁×#443	—	2	1	no	—
20	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
22	C₃₁×#427	C₂²	1	1	yes	characteristic
23	C₃₁×#427	C₂²	1	2	yes	—
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₃₁×#421	C₂²	1	4	yes	—
26	C₂×C₃₁×#104	C₂²	1	2	yes	—
27	C₃₁×#410	C₂²	1	4	yes	—
28	C₃₁×#405	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#344	C₂²	1	4	yes	—
31	C₃₁×#339	C₂²	1	2	yes	—
32	C₃₁×H(3,ℤ/4)	C₂²	1	4	yes	—
33	C₂×#41×C₃₁	C₂³	1	2	yes	—
34	C₃₁×#135	—	2	1	no	—
35	C₃₁×#135	—	2	1	no	—
36	C₃₁×#135	—	2	1	no	—
37	C₃₁×#135	—	2	1	no	—
38	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₃₁	Q₈	1	2	yes	complemented
40	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic, complemented
41	C₃₁×#131	—	2	2	no	—
42	C₃₁×#131	—	2	2	no	—
43	C₃₁×#131	—	2	2	no	—
44	C₃₁×#131	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×#125	—	2	2	no	—
53	C₃₁×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₃₁	—	4	2	no	—
55	C₃₁×#117	C₂³	1	2	yes	—
56	C₃₁×#117	C₂³	1	4	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#108	—	2	1	no	—
59	C₃₁×#108	—	2	1	no	—
60	C₃₁×#108	—	2	1	no	—
61	C₃₁×#108	—	2	1	no	—
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	2	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×#106	—	2	2	no	—
67	C₃₁×#106	—	2	1	no	—
68	C₃₁×#104	—	2	2	no	—
69	C₃₁×#104	—	2	2	no	—
70	C₃₁×#104	—	4	4	no	—
71	C₃₁×#99	—	4	4	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
75	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	#41×C₃₁	—	2	2	no	—
79	#41×C₃₁	—	2	2	no	—
80	#41×C₃₁	—	2	2	no	—
81	#41×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₃₁	—	2	1	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₃₁	—	2	1	no	—
90	Q₁₆×C₃₁	—	2	1	no	—
91	SD₁₆×C₃₁	—	2	2	no	—
92	SD₁₆×C₃₁	—	2	2	no	—
93	D₈×C₃₁	—	2	1	no	—
94	D₈×C₃₁	—	2	1	no	—
95	M₁₆×C₃₁	—	2	2	no	—
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	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	4	2	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	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
114	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
115	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
116	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂²⋊C₄×C₃₁	—	2	4	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	4	no	—
126	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	#29154	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	2	no	—
130	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	Q₈×C₃₁	—	2	2	no	—
132	Q₈×C₃₁	—	2	2	no	—
133	Q₈×C₃₁	—	8	4	no	—
134	D₄×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	2	2	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	4	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
152	C₈×C₃₁	—	4	2	no	—
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
157	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
158	C₈×C₃₁	#124	1	1	yes	characteristic
159	C₈×C₃₁	#124	1	1	yes	characteristic
160	C₈×C₃₁	#124	1	1	yes	characteristic
161	C₈×C₃₁	#124	1	1	yes	characteristic
162	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
163	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
164	#2031	C₂×C₃₁	1	4	yes	complemented
165	#1976	C₂×C₃₁	1	2	yes	—
166	#1833	C₂×C₃₁	1	2	yes	complemented
167	#1814	C₂×C₃₁	1	2	yes	complemented
168	#1812	C₂×C₃₁	1	2	yes	complemented
169	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
170	C₂²×C₃₁	—	4	1	no	—
171	C₂²×C₃₁	—	4	2	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	#535	1	1	yes	characteristic
174	C₂²×C₃₁	#554	1	1	yes	characteristic
175	C₂²×C₃₁	#555	1	1	yes	characteristic
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	4	1	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	4	no	—
181	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
183	C₄×C₃₁	#554	1	1	yes	characteristic
184	C₄×C₃₁	#555	1	1	yes	characteristic
185	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×#131	C₂²×C₃₁	1	2	yes	—
187	#533	C₂²×C₃₁	1	2	yes	—
188	#500	—	2	1	no	—
189	#499	—	2	1	no	—
190	#499	—	2	1	no	—
191	#498	—	2	1	no	—
192	#443	—	2	1	no	—
193	#443	—	2	1	no	—
194	#443	—	2	1	no	—
195	#443	—	2	1	no	—
196	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
198	#427	C₂²×C₃₁	1	1	yes	characteristic
199	#427	C₂²×C₃₁	1	2	yes	—
200	#427	C₂²×C₃₁	1	2	yes	—
201	#421	C₂²×C₃₁	1	4	yes	—
202	C₂×#104	C₂²×C₃₁	1	2	yes	—
203	#410	C₂²×C₃₁	1	4	yes	—
204	#405	C₂²×C₃₁	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
206	#344	C₂²×C₃₁	1	4	yes	—
207	#339	C₂²×C₃₁	1	2	yes	—
208	H(3,ℤ/4)	C₂²×C₃₁	1	4	yes	—
209	C₂×C₃₁	—	2	1	no	—
210	C₂×C₃₁	—	2	1	no	—
211	C₂×C₃₁	—	2	1	no	—
212	C₂×C₃₁	—	8	2	no	—
213	C₂×C₃₁	#2563	1	1	yes	characteristic
214	C₂×#41	C₂³×C₃₁	1	2	yes	—
215	#135	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic, complemented
220	C₂×SD₁₆	Q₈×C₃₁	1	2	yes	complemented
221	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic, complemented
222	#131	—	2	2	no	—
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
228	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₄⋊Q₈	—	2	2	no	—
233	#125	—	2	2	no	—
234	C₂²⋊Q₈	—	4	2	no	—
235	C₄×Q₈	—	4	2	no	—
236	#117	C₂³×C₃₁	1	4	yes	—
237	#117	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
239	#108	—	2	1	no	—
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	2	1	no	—
243	#107	—	2	2	no	—
244	#107	—	2	2	no	—
245	#107	—	2	1	no	—
246	#106	—	2	2	no	—
247	#106	—	2	2	no	—
248	#106	—	2	1	no	—
249	#104	—	2	2	no	—
250	#104	—	2	2	no	—
251	#104	—	4	4	no	—
252	#99	—	4	4	no	—
253	#97	—	2	2	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
256	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
257	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
259	C₃₁	#29154	1	1	yes	characteristic, direct product factor, Sylow
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
292	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
293	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
294	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
295	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
296	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
297	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
298	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	4	no	—
308	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
336	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
337	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
338	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
339	C₈	C₃₁×#124	1	1	yes	characteristic
340	C₈	C₃₁×#124	1	1	yes	characteristic
341	C₈	C₃₁×#124	1	1	yes	characteristic
342	C₈	C₃₁×#124	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₃₁×#535	1	1	yes	characteristic
347	C₂²	C₃₁×#554	1	1	yes	characteristic
348	C₂²	C₃₁×#555	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	4	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
355	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
356	C₄	C₃₁×#554	1	1	yes	characteristic
357	C₄	C₃₁×#555	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₃₁×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2563
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3160	C₂	semidirect
3. C₃₁×#2031	C₂	semidirect
5. C₃₁×#1833	C₂	semidirect
6. C₃₁×#1814	C₂	semidirect
7. C₃₁×#1812	C₂	semidirect
8. C₃₁×#1772	C₂	semidirect
38. C₂×Q₁₆×C₃₁	Q₈	semidirect
39. C₂×SD₁₆×C₃₁	Q₈	semidirect
40. C₂×D₈×C₃₁	Q₈	semidirect
128. #29154	C₃₁	direct (N × H)
162. #3160	C₂×C₃₁	semidirect
164. #2031	C₂×C₃₁	semidirect
166. #1833	C₂×C₃₁	semidirect
167. #1814	C₂×C₃₁	semidirect
168. #1812	C₂×C₃₁	semidirect
169. #1772	C₂×C₃₁	semidirect
219. C₂×Q₁₆	Q₈×C₃₁	semidirect
220. C₂×SD₁₆	Q₈×C₃₁	semidirect
221. C₂×D₈	Q₈×C₃₁	semidirect
259. C₃₁	#29154	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×#3160	2
C₂×C₃₁×#443	2
C₃₁×#2031	2
C₃₁×#1976	2
C₃₁×#1833	2
C₃₁×#1814	2
C₃₁×#1812	2
C₃₁×#1772	2
#29154	31

Subgroups — C31×#29154 #21485913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29154 #21485913