Subgroups — C₃₁×#33820 #21490579

Order 7936.

528 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2224	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2220	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2207	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#533	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#494	C₂²	1	1	yes	characteristic
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#487	—	2	1	no	—
18	C₃₁×#486	—	2	1	no	—
19	C₃₁×#467	—	2	1	no	—
20	C₃₁×#467	C₂²	1	2	yes	—
21	C₃₁×#466	—	2	1	no	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₄×Q₁₆×C₃₁	—	2	1	no	—
24	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#366	C₂²	1	4	yes	—
29	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	4	yes	—
32	C₃₁×#357	C₂²	1	4	yes	—
33	C₂×#41×C₃₁	—	4	1	no	—
34	C₂×Q₁₆×C₃₁	—	2	1	no	—
35	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#125	—	4	1	no	—
45	C₃₁×#124	—	4	1	no	—
46	C₃₁×C₂²⋊Q₈	—	2	2	no	—
47	C₃₁×C₂²⋊Q₈	—	4	1	no	—
48	C₃₁×#121	—	2	2	no	—
49	C₃₁×#121	—	4	1	no	—
50	C₄×Q₈×C₃₁	—	2	2	no	—
51	C₄×Q₈×C₃₁	—	4	1	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	4	1	no	—
54	C₃₁×#117	—	4	1	no	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₃₁×SD₃₂	—	4	4	no	—
57	C₃₁×M₃₂	—	4	2	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	2	no	—
73	C₂×Q₈×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂²×C₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₃₁	—	2	2	no	—
84	Q₁₆×C₃₁	—	2	2	no	—
85	Q₁₆×C₃₁	—	4	2	no	—
86	D₈×C₃₁	—	2	2	no	—
87	D₈×C₃₁	—	2	2	no	—
88	D₈×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	1	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	C₂²⋊C₄×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	#33820	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	—	4	2	no	—
117	Q₈×C₃₁	—	4	2	no	—
118	Q₈×C₃₁	—	8	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	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₃₁	—	2	1	no	—
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#3169	C₂×C₃₁	1	1	yes	characteristic
146	#2227	C₂×C₃₁	1	2	yes	complemented
147	#2224	C₂×C₃₁	1	2	yes	complemented
148	#2220	C₂×C₃₁	1	4	yes	complemented
149	#2208	C₂×C₃₁	1	2	yes	complemented
150	#2207	C₂×C₃₁	1	2	yes	complemented
151	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	2	1	no	—
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	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	#509	1	2	yes	—
166	C₄×C₃₁	#510	1	2	yes	—
167	C₄×C₃₁	#537	1	1	yes	characteristic
168	C₄×C₃₁	#575	1	1	yes	characteristic
169	#533	—	2	1	no	—
170	#532	—	2	1	no	—
171	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
172	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
173	#500	C₂²×C₃₁	1	1	yes	characteristic
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#494	C₂²×C₃₁	1	1	yes	characteristic
176	#493	C₂²×C₃₁	1	1	yes	characteristic
177	#487	—	2	1	no	—
178	#486	—	2	1	no	—
179	#467	—	2	1	no	—
180	#467	C₂²×C₃₁	1	2	yes	—
181	#466	—	2	1	no	—
182	#466	C₂²×C₃₁	1	2	yes	—
183	C₄×Q₁₆	—	2	1	no	—
184	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
185	C₄×D₈	—	2	1	no	—
186	C₄×D₈	C₂²×C₃₁	1	2	yes	—
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	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
191	#358	C₂²×C₃₁	1	4	yes	—
192	#357	C₂²×C₃₁	1	4	yes	—
193	C₂×C₃₁	—	4	1	no	—
194	C₂×C₃₁	—	16	2	no	—
195	C₂×C₃₁	#3245	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
197	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
198	C₂×#41	—	4	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	#127	—	2	1	no	—
209	#125	—	4	1	no	—
210	#124	—	4	1	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	#121	—	2	2	no	—
214	#121	—	4	1	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	4	1	no	—
219	#117	—	4	1	no	—
220	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
221	SD₃₂	—	4	4	no	—
222	M₃₂	—	4	2	no	—
223	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	C₂³×C₃₁	1	2	yes	—
227	#107	C₂³×C₃₁	1	2	yes	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
236	C₃₁	#33820	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	8	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	D₈×C₃₁	1	1	yes	characteristic
275	C₄²	D₈×C₃₁	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₁₆	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
308	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₃₁×#509	1	2	yes	—
323	C₄	C₃₁×#510	1	2	yes	—
324	C₄	C₃₁×#537	1	1	yes	characteristic
325	C₄	C₃₁×#575	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₃₁×#3245	1	1	yes	characteristic
329	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
330	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
1. C₃₁×#3170	C₂	semidirect
3. C₃₁×#2227	C₂	semidirect
4. C₃₁×#2224	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2208	C₂	semidirect
7. C₃₁×#2207	C₂	semidirect
8. C₃₁×#2150	C₂	semidirect
113. #33820	C₃₁	direct (N × H)
144. #3170	C₂×C₃₁	semidirect
146. #2227	C₂×C₃₁	semidirect
147. #2224	C₂×C₃₁	semidirect
148. #2220	C₂×C₃₁	semidirect
149. #2208	C₂×C₃₁	semidirect
150. #2207	C₂×C₃₁	semidirect
151. #2150	C₂×C₃₁	semidirect
236. C₃₁	#33820	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×#3170	2
C₃₁×#3169	2
C₃₁×#2227	2
C₃₁×#2224	2
C₃₁×#2220	2
C₃₁×#2208	2
C₃₁×#2207	2
C₃₁×#2150	2
#33820	31

Subgroups — C31×#33820 #21490579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33820 #21490579