Subgroups — C₃₁×#33775 #21490534

Order 7936.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 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
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2247	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2241	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2209	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2188	C₂	1	4	yes	maximal
7	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₁	—	2	1	no	—
9	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#533	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₃₁×Q₃₂	C₂²	1	8	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#494	—	2	1	no	—
15	C₃₁×#494	—	2	1	no	—
16	C₃₁×#494	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
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	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₃₁×#358	C₂²	1	8	yes	—
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₁	—	2	2	no	—
30	C₂×Q₁₆×C₃₁	—	2	1	no	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	—	4	4	no	—
35	C₂×Q₁₆×C₃₁	—	4	4	no	—
36	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₃₁	C₂³	1	4	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	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	4	1	no	—
45	C₃₁×C₄⋊Q₈	—	4	1	no	—
46	C₃₁×#125	—	4	1	no	—
47	C₃₁×C₂²⋊Q₈	—	2	2	no	—
48	C₃₁×C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	4	1	no	—
51	C₃₁×#117	—	4	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×Q₃₂	—	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	C₂×Q₈×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	2	4	no	—
67	C₂×Q₈×C₃₁	—	4	4	no	—
68	C₂×Q₈×C₃₁	—	4	4	no	—
69	C₂×Q₈×C₃₁	—	4	1	no	—
70	C₂×Q₈×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₃₁	—	2	4	no	—
76	Q₁₆×C₃₁	—	2	8	no	—
77	Q₁₆×C₃₁	—	4	4	no	—
78	Q₁₆×C₃₁	—	4	4	no	—
79	Q₁₆×C₃₁	—	4	2	no	—
80	Q₁₆×C₃₁	—	4	4	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	4	2	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₂²⋊C₄×C₃₁	—	4	2	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	D₈	1	1	yes	characteristic
100	C₄²×C₃₁	D₈	1	1	yes	characteristic
101	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₁₆×C₃₁	—	2	4	no	—
104	#33775	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₃₁	—	4	4	no	—
107	Q₈×C₃₁	—	4	2	no	—
108	Q₈×C₃₁	—	4	4	no	—
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	8	4	no	—
111	Q₈×C₃₁	—	8	2	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	8	4	no	—
119	C₂×C₄×C₃₁	Q₃₂	1	4	yes	complemented
120	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₃₁	—	2	1	no	—
127	C₈×C₃₁	—	2	1	no	—
128	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
129	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
130	#3170	C₂×C₃₁	1	1	yes	characteristic
131	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
132	#2247	C₂×C₃₁	1	2	yes	complemented
133	#2241	C₂×C₃₁	1	2	yes	complemented
134	#2209	C₂×C₃₁	1	4	yes	complemented
135	#2188	C₂×C₃₁	1	4	yes	—
136	#2162	C₂×C₃₁	1	1	yes	characteristic
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	C₂×Q₃₂	1	2	yes	—
140	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	8	2	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
147	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
148	C₄×C₃₁	#510	1	2	yes	—
149	C₄×C₃₁	#535	1	1	yes	characteristic
150	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂²×Q₁₆	—	2	1	no	—
152	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
153	#533	—	2	1	no	—
154	C₂×C₄⋊Q₈	—	2	1	no	—
155	C₂×Q₃₂	C₂²×C₃₁	1	8	yes	—
156	#500	C₂²×C₃₁	1	1	yes	characteristic
157	#494	—	2	1	no	—
158	#494	—	2	1	no	—
159	#494	C₂²×C₃₁	1	2	yes	—
160	#494	C₂²×C₃₁	1	2	yes	—
161	#494	C₂²×C₃₁	1	1	yes	characteristic
162	#487	—	2	1	no	—
163	#467	—	2	1	no	—
164	#467	C₂²×C₃₁	1	2	yes	—
165	C₄×Q₁₆	—	2	1	no	—
166	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
168	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
169	#358	C₂²×C₃₁	1	8	yes	—
170	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
171	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	#3205	1	1	yes	characteristic
174	C₂×C₃₁	C₂²×Q₃₂	1	1	yes	characteristic
175	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
176	C₂²×Q₈	—	2	2	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	1	no	—
181	C₂×Q₁₆	—	4	4	no	—
182	C₂×Q₁₆	—	4	4	no	—
183	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
184	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
185	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	4	1	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	#125	—	4	1	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	C₂²⋊Q₈	—	4	1	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	4	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	4	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
202	Q₃₂	—	4	8	no	—
203	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
204	#107	—	2	1	no	—
205	#107	C₂³×C₃₁	1	2	yes	—
206	#103	—	2	2	no	—
207	#103	—	2	2	no	—
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
212	C₃₁	#33775	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	Q₁₆	—	2	4	no	—
224	Q₁₆	—	2	8	no	—
225	Q₁₆	—	4	4	no	—
226	Q₁₆	—	4	4	no	—
227	Q₁₆	—	4	2	no	—
228	Q₁₆	—	4	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	D₈×C₃₁	1	1	yes	characteristic
248	C₄²	D₈×C₃₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₁₆	—	2	4	no	—
252	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	C₃₁×Q₃₂	1	4	yes	complemented
267	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
276	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×C₃₁×Q₃₂	1	2	yes	—
280	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
287	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
288	C₄	C₃₁×#510	1	2	yes	—
289	C₄	C₃₁×#535	1	1	yes	characteristic
290	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	C₃₁×#3205	1	1	yes	characteristic
293	C₂	C₂²×C₃₁×Q₃₂	1	1	yes	characteristic
294	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	C₂³×C₃₁	C₂×D₈
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
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2209	C₂	semidirect
104. #33775	C₃₁	direct (N × H)
119. C₂×C₄×C₃₁	Q₃₂	semidirect
132. #2247	C₂×C₃₁	semidirect
133. #2241	C₂×C₃₁	semidirect
134. #2209	C₂×C₃₁	semidirect
212. C₃₁	#33775	direct (N × H)
266. C₂×C₄	C₃₁×Q₃₂	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×#494	2
C₃₁×#2247	2
C₃₁×#2241	2
C₃₁×#2209	2
C₃₁×#2188	2
C₃₁×#2162	2
#33775	31

Subgroups — C31×#33775 #21490534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33775 #21490534