Subgroups — C₃₁×#33774 #21490533

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2248	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2207	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2187	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#533	—	2	1	no	—
11	C₂×C₃₁×#127	—	2	1	no	—
12	C₂×C₃₁×SD₃₂	C₂²	1	8	yes	complemented
13	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#493	—	2	1	no	—
16	C₃₁×#493	—	2	1	no	—
17	C₃₁×#493	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	2	yes	—
19	C₃₁×#487	—	2	1	no	—
20	C₃₁×#467	—	2	1	no	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₃₁	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₃₁×#357	C₂²	1	8	yes	complemented
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₂²×D₄×C₃₁	—	2	2	no	—
30	C₂²×D₄×C₃₁	—	4	1	no	—
31	C₂²×D₄×C₃₁	—	4	1	no	—
32	C₂×Q₁₆×C₃₁	—	2	1	no	—
33	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
34	C₂×D₈×C₃₁	—	2	2	no	—
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	2	no	—
37	C₂×D₈×C₃₁	—	4	4	no	—
38	C₂×D₈×C₃₁	—	4	4	no	—
39	C₂×D₈×C₃₁	C₂³	1	4	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×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×#127	—	2	2	no	—
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	—	4	1	no	—
47	C₃₁×#127	—	4	1	no	—
48	C₃₁×#125	—	4	1	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	4	1	no	—
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	4	1	no	—
53	C₃₁×#117	—	4	1	no	—
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×SD₃₂	—	4	8	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
57	C₃₁×#107	—	2	1	no	—
58	C₃₁×#107	C₂³	1	2	yes	—
59	C₃₁×#103	—	2	2	no	—
60	C₃₁×#103	—	2	2	no	—
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂⁴×C₃₁	—	4	2	no	—
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	4	4	no	—
70	C₂×D₄×C₃₁	—	4	4	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂×D₄×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	—	8	4	no	—
77	C₂×D₄×C₃₁	—	8	4	no	—
78	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	2	4	no	—
82	Q₁₆×C₃₁	—	4	2	no	—
83	D₈×C₃₁	—	2	8	no	—
84	D₈×C₃₁	—	4	4	no	—
85	D₈×C₃₁	—	4	4	no	—
86	D₈×C₃₁	—	4	4	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	4	1	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	D₈	1	1	yes	characteristic
102	C₄²×C₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₁₆×C₃₁	—	2	4	no	—
106	#33774	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	4	4	no	—
108	C₂³×C₃₁	—	8	4	no	—
109	C₂³×C₃₁	—	8	4	no	—
110	C₂³×C₃₁	—	8	4	no	—
111	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₃₁	—	4	2	no	—
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	4	no	—
117	D₄×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	8	4	no	—
119	D₄×C₃₁	—	8	4	no	—
120	D₄×C₃₁	—	8	4	no	—
121	D₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	SD₃₂	1	4	yes	complemented
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
137	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
138	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
140	#2248	C₂×C₃₁	1	2	yes	complemented
141	#2240	C₂×C₃₁	1	2	yes	complemented
142	#2207	C₂×C₃₁	1	4	yes	complemented
143	#2187	C₂×C₃₁	1	4	yes	complemented
144	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	—	8	4	no	—
148	C₂²×C₃₁	—	8	4	no	—
149	C₂²×C₃₁	—	8	4	no	—
150	C₂²×C₃₁	—	16	8	no	—
151	C₂²×C₃₁	C₂×SD₃₂	1	2	yes	—
152	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	C₂×SD₃₂	1	2	yes	—
158	C₄×C₃₁	C₂×SD₃₂	1	2	yes	—
159	C₄×C₃₁	#509	1	2	yes	—
160	C₄×C₃₁	#535	1	1	yes	characteristic
161	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×D₈	—	2	1	no	—
163	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
164	#533	—	2	1	no	—
165	C₂×#127	—	2	1	no	—
166	C₂×SD₃₂	C₂²×C₃₁	1	8	yes	complemented
167	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
168	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
169	#493	—	2	1	no	—
170	#493	—	2	1	no	—
171	#493	C₂²×C₃₁	1	2	yes	—
172	#493	C₂²×C₃₁	1	2	yes	—
173	#487	—	2	1	no	—
174	#467	—	2	1	no	—
175	#467	C₂²×C₃₁	1	2	yes	—
176	C₄×Q₁₆	—	2	1	no	—
177	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
178	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
180	#357	C₂²×C₃₁	1	8	yes	complemented
181	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
182	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
183	C₂×C₃₁	—	2	2	no	—
184	C₂×C₃₁	—	16	4	no	—
185	C₂×C₃₁	#3205	1	1	yes	characteristic
186	C₂×C₃₁	C₂²×SD₃₂	1	1	yes	characteristic
187	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	—	4	1	no	—
190	C₂²×D₄	—	4	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂×D₈	C₂³×C₃₁	1	4	yes	—
199	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	2	no	—
204	#127	—	2	2	no	—
205	#127	—	4	1	no	—
206	#127	—	4	1	no	—
207	#125	—	4	1	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	4	1	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	4	1	no	—
212	#117	—	4	1	no	—
213	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
214	SD₃₂	—	4	8	no	—
215	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
216	#107	—	2	1	no	—
217	#107	C₂³×C₃₁	1	2	yes	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
221	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₃₁	#33774	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂×D₄	—	8	4	no	—
238	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	D₈	—	2	8	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	4	no	—
246	D₈	—	4	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	D₈×C₃₁	1	1	yes	characteristic
262	C₄²	D₈×C₃₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₁₆	—	2	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	—	8	4	no	—
270	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	8	2	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₃₁×SD₃₂	1	4	yes	complemented
287	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
296	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×C₃₁×SD₃₂	1	2	yes	—
304	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₃₁×SD₃₂	1	2	yes	—
310	C₄	C₂×C₃₁×SD₃₂	1	2	yes	—
311	C₄	C₃₁×#509	1	2	yes	—
312	C₄	C₃₁×#535	1	1	yes	characteristic
313	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	C₃₁×#3205	1	1	yes	characteristic
317	C₂	C₂²×C₃₁×SD₃₂	1	1	yes	characteristic
318	C₂	C₂×C₃₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	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×#2248	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2207	C₂	semidirect
6. C₃₁×#2187	C₂	semidirect
7. C₃₁×#2162	C₂	semidirect
12. C₂×C₃₁×SD₃₂	C₂²	semidirect
13. C₃₁×#500	C₂²	semidirect
14. C₃₁×#494	C₂²	semidirect
23. C₄×Q₁₆×C₃₁	C₂²	semidirect
25. C₃₁×C₄⋊C₁₆	C₂²	semidirect
26. C₃₁×#357	C₂²	semidirect
28. C₄×C₁₆×C₃₁	C₂²	semidirect
106. #33774	C₃₁	direct (N × H)
127. C₂×C₄×C₃₁	SD₃₂	semidirect
138. #3170	C₂×C₃₁	semidirect
140. #2248	C₂×C₃₁	semidirect
141. #2240	C₂×C₃₁	semidirect
142. #2207	C₂×C₃₁	semidirect
143. #2187	C₂×C₃₁	semidirect
144. #2162	C₂×C₃₁	semidirect
166. C₂×SD₃₂	C₂²×C₃₁	semidirect
167. #500	C₂²×C₃₁	semidirect
168. #494	C₂²×C₃₁	semidirect
177. C₄×Q₁₆	C₂²×C₃₁	semidirect
179. C₄⋊C₁₆	C₂²×C₃₁	semidirect
180. #357	C₂²×C₃₁	semidirect
182. C₄×C₁₆	C₂²×C₃₁	semidirect
224. C₃₁	#33774	direct (N × H)
286. C₂×C₄	C₃₁×SD₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3170	2
C₂×C₃₁×#493	2
C₃₁×#2248	2
C₃₁×#2240	2
C₃₁×#2207	2
C₃₁×#2187	2
C₃₁×#2162	2
#33774	31

Subgroups — C31×#33774 #21490533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33774 #21490533