Subgroups — C₃₁×#33672 #21490431

Order 7936.

586 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2232	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2207	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2185	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2175	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#532	—	2	1	no	—
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#493	C₂²	1	1	yes	characteristic
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#466	—	2	1	no	—
17	C₃₁×#466	C₂²	1	2	yes	—
18	C₄×Q₁₆×C₃₁	—	2	1	no	—
19	C₄×Q₁₆×C₃₁	—	2	1	no	—
20	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₄×D₈×C₃₁	C₂²	1	2	yes	—
24	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#103	—	2	1	no	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#366	C₂²	1	4	yes	—
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×#358	C₂²	1	4	yes	—
30	C₃₁×#357	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#346	C₂²	1	2	yes	—
33	C₂×#41×C₃₁	—	4	1	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂×Q₁₆×C₃₁	—	2	4	no	—
36	C₂×Q₁₆×C₃₁	—	2	1	no	—
37	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#124	—	4	1	no	—
45	C₃₁×#121	—	2	2	no	—
46	C₃₁×#121	—	4	1	no	—
47	C₄×Q₈×C₃₁	—	2	2	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	4	1	no	—
50	C₄×Q₈×C₃₁	—	4	1	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₄×D₄×C₃₁	—	4	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
55	C₂×C₄²×C₃₁	—	4	1	no	—
56	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×SD₃₂	—	2	8	no	—
58	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#107	—	2	1	no	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#107	C₂³	1	2	yes	—
63	C₃₁×#103	—	2	2	no	—
64	C₃₁×#103	—	2	2	no	—
65	C₃₁×#102	—	2	2	no	—
66	C₃₁×#102	—	2	2	no	—
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	#41×C₃₁	—	8	4	no	—
72	C₂×Q₈×C₃₁	—	2	4	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	4	1	no	—
76	C₂×Q₈×C₃₁	—	4	1	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	1	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₃₁	—	2	4	no	—
88	Q₁₆×C₃₁	—	2	2	no	—
89	Q₁₆×C₃₁	#41	1	4	yes	complemented
90	D₈×C₃₁	—	2	4	no	—
91	D₈×C₃₁	—	4	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	4	1	no	—
94	C₂×C₈×C₃₁	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₄²×C₃₁	—	4	2	no	—
107	C₄²×C₃₁	—	4	2	no	—
108	C₄²×C₃₁	D₈	1	1	yes	characteristic
109	C₄²×C₃₁	D₈	1	1	yes	characteristic
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	4	no	—
113	#33672	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₃₁	—	4	2	no	—
115	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₃₁	—	2	4	no	—
117	Q₈×C₃₁	—	4	4	no	—
118	Q₈×C₃₁	—	4	2	no	—
119	Q₈×C₃₁	—	8	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	8	2	no	—
123	D₄×C₃₁	—	8	4	no	—
124	D₄×C₃₁	—	8	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	8	4	no	—
134	C₂×C₄×C₃₁	—	8	2	no	—
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	C₂×#41	1	2	yes	—
145	#3169	C₂×C₃₁	1	1	yes	characteristic
146	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2232	C₂×C₃₁	1	2	yes	complemented
148	#2227	C₂×C₃₁	1	2	yes	complemented
149	#2207	C₂×C₃₁	1	2	yes	complemented
150	#2185	C₂×C₃₁	1	2	yes	complemented
151	#2175	C₂×C₃₁	1	4	yes	complemented
152	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	#509	1	2	yes	—
159	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	4	4	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	#510	1	2	yes	—
168	C₄×C₃₁	#525	1	1	yes	characteristic
169	C₄×C₃₁	#575	1	1	yes	characteristic
170	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
171	#532	—	2	1	no	—
172	C₂×C₄×Q₈	—	2	1	no	—
173	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#493	C₂²×C₃₁	1	1	yes	characteristic
176	#486	—	2	1	no	—
177	#466	—	2	1	no	—
178	#466	C₂²×C₃₁	1	2	yes	—
179	C₄×Q₁₆	—	2	1	no	—
180	C₄×Q₁₆	—	2	1	no	—
181	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
182	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
183	C₄×D₈	—	2	1	no	—
184	C₄×D₈	C₂²×C₃₁	1	2	yes	—
185	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×#103	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
188	#366	C₂²×C₃₁	1	4	yes	—
189	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
190	#358	C₂²×C₃₁	1	4	yes	—
191	#357	C₂²×C₃₁	1	4	yes	—
192	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
193	#346	C₂²×C₃₁	1	2	yes	—
194	C₂×C₃₁	—	2	2	no	—
195	C₂×C₃₁	—	16	2	no	—
196	C₂×C₃₁	#3100	1	1	yes	characteristic
197	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
198	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
199	C₂×#41	—	4	1	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂³×C₃₁	1	2	yes	—
206	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	#127	—	2	1	no	—
210	#124	—	4	1	no	—
211	#121	—	2	2	no	—
212	#121	—	4	1	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	4	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	4	1	no	—
221	C₂×C₄²	—	4	1	no	—
222	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
223	SD₃₂	—	2	8	no	—
224	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	C₂³×C₃₁	1	2	yes	—
228	#107	C₂³×C₃₁	1	2	yes	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₃₁	#33672	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	8	4	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	#41×C₃₁	1	4	yes	complemented
257	D₈	—	2	4	no	—
258	D₈	—	4	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	D₈×C₃₁	1	1	yes	characteristic
276	C₄²	D₈×C₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₂×#41×C₃₁	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₃₁×#509	1	2	yes	—
317	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₃₁×#510	1	2	yes	—
326	C₄	C₃₁×#525	1	1	yes	characteristic
327	C₄	C₃₁×#575	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	2	no	—
330	C₂	C₃₁×#3100	1	1	yes	characteristic
331	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
332	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2232	C₂	semidirect
4. C₃₁×#2227	C₂	semidirect
5. C₃₁×#2207	C₂	semidirect
6. C₃₁×#2185	C₂	semidirect
7. C₃₁×#2175	C₂	semidirect
8. C₃₁×#2163	C₂	semidirect
89. Q₁₆×C₃₁	#41	semidirect
113. #33672	C₃₁	direct (N × H)
146. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
147. #2232	C₂×C₃₁	semidirect
148. #2227	C₂×C₃₁	semidirect
149. #2207	C₂×C₃₁	semidirect
150. #2185	C₂×C₃₁	semidirect
151. #2175	C₂×C₃₁	semidirect
152. #2163	C₂×C₃₁	semidirect
237. C₃₁	#33672	direct (N × H)
256. Q₁₆	#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×#3169	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2232	2
C₃₁×#2227	2
C₃₁×#2207	2
C₃₁×#2185	2
C₃₁×#2175	2
C₃₁×#2163	2
#33672	31

Subgroups — C31×#33672 #21490431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33672 #21490431