Subgroups — C₃₁×#33907 #21490666

Order 7936.

480 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×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2217	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2158	C₂	1	2	yes	maximal, complemented
12	C₃₁×#556	—	2	1	no	—
13	C₃₁×#555	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#485	—	2	2	no	—
19	C₃₁×#482	—	2	2	no	—
20	C₃₁×#476	—	2	1	no	—
21	C₃₁×#475	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#427	C₂²	1	2	yes	—
28	C₃₁×#427	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×#367	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	2	yes	—
33	C₃₁×#366	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×#126	—	4	2	no	—
47	C₃₁×#124	—	4	1	no	—
48	C₃₁×#124	—	4	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	4	1	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	4	1	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×#117	—	2	2	no	—
58	C₃₁×#117	—	4	1	no	—
59	C₃₁×SD₃₂	—	2	2	no	—
60	C₂×C₁₆×C₃₁	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	C₂³	1	1	yes	characteristic
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#107	C₂³	1	2	yes	—
70	C₃₁×#106	—	2	2	no	—
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#103	—	2	2	no	—
75	C₃₁×#102	—	2	1	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	Q₁₆×C₃₁	—	2	1	no	—
88	Q₁₆×C₃₁	—	2	1	no	—
89	D₈×C₃₁	—	2	1	no	—
90	D₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	1	no	—
110	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
111	C₄²×C₃₁	—	4	2	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	1	no	—
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₃₁	—	2	2	no	—
116	C₁₆×C₃₁	—	2	1	no	—
117	C₁₆×C₃₁	#41	1	2	yes	—
118	#33907	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	4	1	no	—
120	C₂³×C₃₁	C₂×D₈	1	2	yes	—
121	Q₈×C₃₁	—	4	1	no	—
122	Q₈×C₃₁	—	4	1	no	—
123	Q₈×C₃₁	—	8	2	no	—
124	Q₈×C₃₁	—	16	1	no	—
125	D₄×C₃₁	—	2	2	no	—
126	D₄×C₃₁	—	4	1	no	—
127	D₄×C₃₁	—	4	1	no	—
128	D₄×C₃₁	—	16	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	4	1	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	8	2	no	—
137	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
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	2	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
145	#3342	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#3341	C₂×C₃₁	1	1	yes	characteristic
147	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2237	C₂×C₃₁	1	2	yes	complemented
149	#2234	C₂×C₃₁	1	2	yes	complemented
150	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2217	C₂×C₃₁	1	2	yes	complemented
153	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
154	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2158	C₂×C₃₁	1	2	yes	complemented
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	8	1	no	—
160	C₂²×C₃₁	—	8	1	no	—
161	C₂²×C₃₁	—	8	1	no	—
162	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	8	1	no	—
166	C₄×C₃₁	—	8	1	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	—	8	1	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	#508	1	2	yes	—
171	C₄×C₃₁	#547	1	1	yes	characteristic
172	C₄×C₃₁	#578	1	1	yes	characteristic
173	#556	—	2	1	no	—
174	#555	—	2	1	no	—
175	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
177	#500	C₂²×C₃₁	1	1	yes	characteristic
178	#500	C₂²×C₃₁	1	1	yes	characteristic
179	#485	—	2	2	no	—
180	#482	—	2	2	no	—
181	#476	—	2	1	no	—
182	#475	—	2	1	no	—
183	#467	C₂²×C₃₁	1	2	yes	—
184	#466	C₂²×C₃₁	1	2	yes	—
185	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
187	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
188	#427	C₂²×C₃₁	1	2	yes	—
189	#427	C₂²×C₃₁	1	2	yes	—
190	#367	C₂²×C₃₁	1	1	yes	characteristic
191	#367	C₂²×C₃₁	1	2	yes	—
192	#367	C₂²×C₃₁	1	2	yes	—
193	#366	C₂²×C₃₁	1	2	yes	—
194	#366	C₂²×C₃₁	1	2	yes	—
195	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
196	#358	C₂²×C₃₁	1	1	yes	characteristic
197	#358	C₂²×C₃₁	1	2	yes	—
198	#357	C₂²×C₃₁	1	1	yes	characteristic
199	#357	C₂²×C₃₁	1	2	yes	—
200	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
201	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×C₃₁	—	4	2	no	—
203	C₂×C₃₁	—	16	1	no	—
204	C₂×C₃₁	#3334	1	1	yes	characteristic
205	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
206	C₂×C₃₁	#3416	1	1	yes	characteristic
207	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	#126	—	4	2	no	—
213	#124	—	4	1	no	—
214	#124	—	4	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	#121	—	2	2	no	—
217	#121	—	4	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
222	#117	—	2	2	no	—
223	#117	—	2	2	no	—
224	#117	—	4	1	no	—
225	SD₃₂	—	2	2	no	—
226	C₂×C₁₆	—	2	4	no	—
227	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
229	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	C₂³×C₃₁	1	1	yes	characteristic
233	#107	C₂³×C₃₁	1	2	yes	—
234	#107	C₂³×C₃₁	1	1	yes	characteristic
235	#107	C₂³×C₃₁	1	2	yes	—
236	#106	—	2	2	no	—
237	#106	—	2	2	no	—
238	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
239	#103	—	2	1	no	—
240	#103	—	2	2	no	—
241	#102	—	2	1	no	—
242	#102	—	2	2	no	—
243	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
244	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₃₁	#33907	1	1	yes	characteristic, direct product factor, Sylow
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	D₈	—	2	1	no	—
257	D₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
262	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	#41×C₃₁	1	2	yes	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂×D₈×C₃₁	1	2	yes	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	16	1	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	16	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
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	2	no	—
308	C₈	—	2	1	no	—
309	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
310	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	C₃₁×#508	1	2	yes	—
326	C₄	C₃₁×#547	1	1	yes	characteristic
327	C₄	C₃₁×#578	1	1	yes	characteristic
328	C₂	—	4	2	no	—
329	C₂	—	16	1	no	—
330	C₂	C₃₁×#3334	1	1	yes	characteristic
331	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
332	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3342	C₂	semidirect
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2234	C₂	semidirect
6. C₃₁×#2227	C₂	semidirect
7. C₃₁×#2224	C₂	semidirect
8. C₃₁×#2217	C₂	semidirect
9. C₄×C₃₁×SD₃₂	C₂	semidirect
10. D₄×C₁₆×C₃₁	C₂	semidirect
11. C₃₁×#2158	C₂	semidirect
118. #33907	C₃₁	direct (N × H)
145. #3342	C₂×C₃₁	semidirect
147. #2254	C₂×C₃₁	semidirect
148. #2237	C₂×C₃₁	semidirect
149. #2234	C₂×C₃₁	semidirect
150. #2227	C₂×C₃₁	semidirect
151. #2224	C₂×C₃₁	semidirect
152. #2217	C₂×C₃₁	semidirect
153. C₄×SD₃₂	C₂×C₃₁	semidirect
154. D₄×C₁₆	C₂×C₃₁	semidirect
155. #2158	C₂×C₃₁	semidirect
245. C₃₁	#33907	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×#3342	2
C₃₁×#3341	2
C₃₁×#2254	2
C₃₁×#2237	2
C₃₁×#2234	2
C₃₁×#2227	2
C₃₁×#2224	2
C₃₁×#2217	2
C₄×C₃₁×SD₃₂	2
D₄×C₁₆×C₃₁	2
C₃₁×#2158	2
#33907	31

Subgroups — C31×#33907 #21490666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33907 #21490666