Subgroups — C₃₁×#23834 #21480593

Order 7936.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2961	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2080	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2053	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2050	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2031	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1765	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1757	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#446	—	2	2	no	—
16	C₃₁×#446	—	2	2	no	—
17	C₃₁×#444	—	2	2	no	—
18	C₃₁×#443	—	2	2	no	—
19	C₃₁×#430	C₂²	1	1	yes	characteristic
20	C₃₁×#424	C₂²	1	1	yes	characteristic
21	C₃₁×#424	C₂²	1	1	yes	characteristic
22	C₃₁×#423	C₂²	1	1	yes	characteristic
23	C₃₁×#421	C₂²	1	2	yes	—
24	C₂×C₃₁×#104	C₂²	1	2	yes	—
25	C₃₁×#416	C₂²	1	2	yes	—
26	C₃₁×#413	C₂²	1	2	yes	—
27	C₃₁×#410	C₂²	1	2	yes	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#405	C₂²	1	2	yes	—
30	C₃₁×#405	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₃₁×#344	C₂²	1	2	yes	—
33	C₃₁×#344	C₂²	1	2	yes	—
34	C₃₁×#341	C₂²	1	2	yes	—
35	C₃₁×#339	C₂²	1	2	yes	—
36	C₃₁×#338	C₂²	1	1	yes	characteristic
37	C₃₁×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
38	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	C₂³	1	2	yes	—
40	C₃₁×#137	—	2	2	no	—
41	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
42	C₃₁×#135	—	2	2	no	—
43	C₃₁×#131	—	2	2	no	—
44	C₃₁×#131	—	2	2	no	—
45	C₃₁×#131	—	2	2	no	—
46	C₃₁×#131	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×#125	—	2	1	no	—
57	C₃₁×C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₃₁×#117	C₂³	1	2	yes	—
60	C₃₁×#117	C₂³	1	1	yes	characteristic
61	C₃₁×#117	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#108	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×#104	—	2	4	no	—
68	C₃₁×#104	—	2	2	no	—
69	C₃₁×#104	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₃₁×#101	—	2	2	no	—
73	C₃₁×#100	—	2	2	no	—
74	C₃₁×#99	—	2	4	no	—
75	C₃₁×#97	—	2	1	no	—
76	C₃₁×#97	—	2	1	no	—
77	C₃₁×#97	—	2	1	no	—
78	C₃₁×#97	—	2	2	no	—
79	C₃₁×#97	D₄	1	2	yes	—
80	C₄×C₈×C₃₁	—	2	1	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₄×C₈×C₃₁	D₄	1	2	yes	—
83	#41×C₃₁	—	2	2	no	—
84	#41×C₃₁	—	2	2	no	—
85	#41×C₃₁	—	2	2	no	—
86	#41×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	#41	1	2	yes	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	#41	1	2	yes	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	Q₁₆×C₃₁	—	2	2	no	—
95	SD₁₆×C₃₁	—	2	4	no	—
96	D₈×C₃₁	—	2	2	no	—
97	M₁₆×C₃₁	—	2	2	no	—
98	M₁₆×C₃₁	—	2	2	no	—
99	M₁₆×C₃₁	—	2	2	no	—
100	M₁₆×C₃₁	—	2	2	no	—
101	M₁₆×C₃₁	—	2	2	no	—
102	M₁₆×C₃₁	—	2	2	no	—
103	M₁₆×C₃₁	—	4	2	no	—
104	M₁₆×C₃₁	—	4	2	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	4	2	no	—
111	C₂×C₈×C₃₁	—	4	2	no	—
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	#41	1	2	yes	—
116	C₂×C₈×C₃₁	#41	1	2	yes	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₂²⋊C₄×C₃₁	#41	1	2	yes	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	C₂×D₄	1	2	yes	—
129	C₄²×C₃₁	#41	1	1	yes	characteristic
130	C₄²×C₃₁	#41	1	1	yes	characteristic
131	#23834	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
134	Q₈×C₃₁	—	2	2	no	—
135	Q₈×C₃₁	—	2	2	no	—
136	Q₈×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	2	2	no	—
138	D₄×C₃₁	—	2	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	8	1	no	—
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
152	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
153	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	4	1	no	—
156	C₈×C₃₁	—	4	1	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₈×C₃₁	#124	1	1	yes	characteristic
160	C₈×C₃₁	#124	1	2	yes	—
161	C₈×C₃₁	#124	1	1	yes	characteristic
162	#2961	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#2955	C₂×C₃₁	1	1	yes	characteristic
164	#2103	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#2102	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2080	C₂×C₃₁	1	2	yes	complemented
167	#2053	C₂×C₃₁	1	1	yes	characteristic
168	#2050	C₂×C₃₁	1	1	yes	characteristic
169	#2031	C₂×C₃₁	1	2	yes	complemented
170	#1765	C₂×C₃₁	1	2	yes	complemented
171	#1757	C₂×C₃₁	1	2	yes	complemented
172	#1747	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₂²×C₃₁	—	4	1	no	—
174	C₂²×C₃₁	—	4	2	no	—
175	C₂²×C₃₁	—	8	2	no	—
176	C₂²×C₃₁	#525	1	1	yes	characteristic
177	C₂²×C₃₁	#529	1	2	yes	—
178	C₄×C₃₁	—	4	1	no	—
179	C₄×C₃₁	—	4	1	no	—
180	C₄×C₃₁	—	4	2	no	—
181	C₄×C₃₁	—	4	2	no	—
182	C₄×C₃₁	—	4	2	no	—
183	C₄×C₃₁	—	8	2	no	—
184	C₄×C₃₁	#525	1	1	yes	characteristic
185	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
186	C₄×C₃₁	#529	1	2	yes	—
187	#575	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×#131	C₂²×C₃₁	1	2	yes	—
189	#533	C₂²×C₃₁	1	1	yes	characteristic
190	#446	—	2	2	no	—
191	#446	—	2	2	no	—
192	#444	—	2	2	no	—
193	#443	—	2	2	no	—
194	#430	C₂²×C₃₁	1	1	yes	characteristic
195	#424	C₂²×C₃₁	1	1	yes	characteristic
196	#424	C₂²×C₃₁	1	1	yes	characteristic
197	#423	C₂²×C₃₁	1	1	yes	characteristic
198	#421	C₂²×C₃₁	1	2	yes	—
199	C₂×#104	C₂²×C₃₁	1	2	yes	—
200	#416	C₂²×C₃₁	1	2	yes	—
201	#413	C₂²×C₃₁	1	2	yes	—
202	#410	C₂²×C₃₁	1	2	yes	—
203	#405	C₂²×C₃₁	1	1	yes	characteristic
204	#405	C₂²×C₃₁	1	2	yes	—
205	#405	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
207	#344	C₂²×C₃₁	1	2	yes	—
208	#344	C₂²×C₃₁	1	2	yes	—
209	#341	C₂²×C₃₁	1	2	yes	—
210	#339	C₂²×C₃₁	1	2	yes	—
211	#338	C₂²×C₃₁	1	1	yes	characteristic
212	H(3,ℤ/4)	C₂²×C₃₁	1	1	yes	characteristic
213	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
214	C₂×C₃₁	—	2	1	no	—
215	C₂×C₃₁	—	2	2	no	—
216	C₂×C₃₁	—	8	2	no	—
217	C₂×C₃₁	#2398	1	1	yes	characteristic
218	C₂×#41	C₂³×C₃₁	1	2	yes	—
219	#137	—	2	2	no	—
220	AGL(1,ℤ/8)	—	2	2	no	—
221	#135	—	2	2	no	—
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	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
230	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
234	C₄⋊Q₈	—	2	1	no	—
235	#125	—	2	1	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	#117	C₂³×C₃₁	1	2	yes	—
239	#117	C₂³×C₃₁	1	1	yes	characteristic
240	#117	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	#108	—	2	2	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	#104	—	2	4	no	—
247	#104	—	2	2	no	—
248	#104	—	2	2	no	—
249	#103	—	2	2	no	—
250	#102	—	2	2	no	—
251	#101	—	2	2	no	—
252	#100	—	2	2	no	—
253	#99	—	2	4	no	—
254	#97	—	2	1	no	—
255	#97	—	2	1	no	—
256	#97	—	2	1	no	—
257	#97	—	2	2	no	—
258	#97	D₄×C₃₁	1	2	yes	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	2	no	—
261	C₄×C₈	D₄×C₃₁	1	2	yes	—
262	C₃₁	#23834	1	1	yes	characteristic, direct product factor, Sylow
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	#41×C₃₁	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	#41×C₃₁	1	2	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	SD₁₆	—	2	4	no	—
276	D₈	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₈	#41×C₃₁	1	2	yes	—
296	C₂×C₈	#41×C₃₁	1	2	yes	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	#41×C₃₁	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	2	2	no	—
308	C₄²	C₂×D₄×C₃₁	1	2	yes	—
309	C₄²	#41×C₃₁	1	1	yes	characteristic
310	C₄²	#41×C₃₁	1	1	yes	characteristic
311	C₂³	—	2	2	no	—
312	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	8	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
331	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
332	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	C₃₁×#124	1	1	yes	characteristic
339	C₈	C₃₁×#124	1	2	yes	—
340	C₈	C₃₁×#124	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₃₁×#525	1	1	yes	characteristic
345	C₂²	C₃₁×#529	1	2	yes	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₃₁×#525	1	1	yes	characteristic
353	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
354	C₄	C₃₁×#529	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₃₁×#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2398
prime radical	C₃₁×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2961	C₂	semidirect
3. C₃₁×#2103	C₂	semidirect
4. C₃₁×#2102	C₂	semidirect
5. C₃₁×#2080	C₂	semidirect
8. C₃₁×#2031	C₂	semidirect
9. C₃₁×#1765	C₂	semidirect
10. C₃₁×#1757	C₂	semidirect
11. C₃₁×#1747	C₂	semidirect
131. #23834	C₃₁	direct (N × H)
162. #2961	C₂×C₃₁	semidirect
164. #2103	C₂×C₃₁	semidirect
165. #2102	C₂×C₃₁	semidirect
166. #2080	C₂×C₃₁	semidirect
169. #2031	C₂×C₃₁	semidirect
170. #1765	C₂×C₃₁	semidirect
171. #1757	C₂×C₃₁	semidirect
172. #1747	C₂×C₃₁	semidirect
262. C₃₁	#23834	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×#2961	2
C₃₁×#2955	2
C₃₁×#2103	2
C₃₁×#2102	2
C₃₁×#2080	2
C₃₁×#2053	2
C₃₁×#2050	2
C₃₁×#2031	2
C₃₁×#1765	2
C₃₁×#1757	2
C₃₁×#1747	2
#23834	31

Subgroups — C31×#23834 #21480593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23834 #21480593