Subgroups — C₃₁×#33792 #21490551

Order 7936.

538 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2252	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2232	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2224	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#532	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₃₁×#500	—	2	1	no	—
11	C₃₁×#500	—	2	1	no	—
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	1	yes	characteristic
16	C₃₁×#486	—	2	1	no	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#466	C₂²	1	2	yes	—
19	C₄×D₈×C₃₁	—	2	1	no	—
20	C₄×D₈×C₃₁	C₂²	1	2	yes	—
21	C₂×C₃₁×#107	—	2	1	no	—
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#366	C₂²	1	8	yes	—
25	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₃₁×#357	C₂²	1	8	yes	—
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
29	C₂×#41×C₃₁	—	4	1	no	—
30	C₂²×Q₈×C₃₁	—	4	1	no	—
31	C₂×D₈×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	4	1	no	—
39	C₃₁×C₄⋊Q₈	—	4	1	no	—
40	C₃₁×#127	—	2	1	no	—
41	C₃₁×#124	—	4	1	no	—
42	C₃₁×#121	—	2	2	no	—
43	C₃₁×#121	—	4	1	no	—
44	C₄×D₄×C₃₁	—	2	2	no	—
45	C₄×D₄×C₃₁	—	4	1	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
48	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
50	C₃₁×#107	—	2	2	no	—
51	C₃₁×#107	—	2	2	no	—
52	C₃₁×#107	—	2	2	no	—
53	C₃₁×#107	—	2	1	no	—
54	C₃₁×#107	C₂³	1	4	yes	—
55	C₃₁×#107	C₂³	1	2	yes	—
56	C₃₁×#102	—	2	2	no	—
57	C₃₁×#102	—	2	2	no	—
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	#41×C₃₁	—	8	4	no	—
63	C₂×Q₈×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	4	2	no	—
65	C₂×Q₈×C₃₁	—	4	2	no	—
66	C₂×Q₈×C₃₁	—	8	4	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	D₈×C₃₁	—	2	4	no	—
77	D₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₄⋊C₄×C₃₁	—	4	2	no	—
89	C₄⋊C₄×C₃₁	—	4	1	no	—
90	C₂²⋊C₄×C₃₁	—	4	2	no	—
91	C₂²⋊C₄×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	D₈	1	1	yes	characteristic
93	C₄²×C₃₁	D₈	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₃₁	—	2	4	no	—
97	#33792	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
100	Q₈×C₃₁	—	8	4	no	—
101	Q₈×C₃₁	—	8	4	no	—
102	Q₈×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	8	4	no	—
106	D₄×C₃₁	—	8	2	no	—
107	D₄×C₃₁	—	8	2	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	8	4	no	—
114	C₂×C₄×C₃₁	—	8	4	no	—
115	C₂×C₄×C₃₁	D₁₆	1	4	yes	—
116	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	#137	1	2	yes	—
125	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
126	#3169	C₂×C₃₁	1	1	yes	characteristic
127	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#2252	C₂×C₃₁	1	2	yes	complemented
129	#2240	C₂×C₃₁	1	2	yes	complemented
130	#2232	C₂×C₃₁	1	4	yes	complemented
131	#2224	C₂×C₃₁	1	4	yes	complemented
132	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	—	8	2	no	—
137	C₂²×C₃₁	—	8	2	no	—
138	C₂²×C₃₁	C₂×D₁₆	1	2	yes	—
139	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	8	2	no	—
143	C₄×C₃₁	—	8	4	no	—
144	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
145	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
146	C₄×C₃₁	#510	1	2	yes	—
147	C₄×C₃₁	#536	1	1	yes	characteristic
148	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
149	#532	—	2	1	no	—
150	C₂×C₄⋊Q₈	—	2	1	no	—
151	#500	—	2	1	no	—
152	#500	—	2	1	no	—
153	#500	C₂²×C₃₁	1	2	yes	—
154	#500	C₂²×C₃₁	1	2	yes	—
155	#500	C₂²×C₃₁	1	1	yes	characteristic
156	#493	C₂²×C₃₁	1	1	yes	characteristic
157	#486	—	2	1	no	—
158	#466	—	2	1	no	—
159	#466	C₂²×C₃₁	1	2	yes	—
160	C₄×D₈	—	2	1	no	—
161	C₄×D₈	C₂²×C₃₁	1	2	yes	—
162	C₂×#107	—	2	1	no	—
163	C₂×#107	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#366	C₂²×C₃₁	1	8	yes	—
166	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
167	#357	C₂²×C₃₁	1	8	yes	—
168	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
169	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
170	C₂×C₃₁	—	2	2	no	—
171	C₂×C₃₁	—	16	2	no	—
172	C₂×C₃₁	#3226	1	1	yes	characteristic
173	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
174	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
175	C₂×#41	—	4	1	no	—
176	C₂²×Q₈	—	4	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	C₂³×C₃₁	1	2	yes	—
179	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	4	1	no	—
185	C₄⋊Q₈	—	4	1	no	—
186	#127	—	2	1	no	—
187	#124	—	4	1	no	—
188	#121	—	2	2	no	—
189	#121	—	4	1	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	4	1	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	4	1	no	—
194	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
196	#107	—	2	2	no	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	2	1	no	—
200	#107	C₂³×C₃₁	1	4	yes	—
201	#107	C₂³×C₃₁	1	2	yes	—
202	#102	—	2	2	no	—
203	#102	—	2	2	no	—
204	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
205	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₃₁	#33792	1	1	yes	characteristic, direct product factor, Sylow
209	#41	—	8	4	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	8	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	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	D₈	—	2	4	no	—
224	D₈	—	4	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	D₈×C₃₁	1	1	yes	characteristic
240	C₄²	D₈×C₃₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₁₆	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
246	Q₈	—	8	4	no	—
247	Q₈	—	8	4	no	—
248	Q₈	—	8	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₃₁×D₁₆	1	4	yes	—
262	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	C₃₁×#137	1	2	yes	—
271	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂×C₃₁×D₁₆	1	2	yes	—
278	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
284	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
285	C₄	C₃₁×#510	1	2	yes	—
286	C₄	C₃₁×#536	1	1	yes	characteristic
287	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₃₁×#3226	1	1	yes	characteristic
291	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
292	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×C₃₁×#500	C₂	semidirect
3. C₃₁×#2252	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2232	C₂	semidirect
6. C₃₁×#2224	C₂	semidirect
7. C₃₁×#2162	C₂	semidirect
97. #33792	C₃₁	direct (N × H)
127. C₂×#500	C₂×C₃₁	semidirect
128. #2252	C₂×C₃₁	semidirect
129. #2240	C₂×C₃₁	semidirect
130. #2232	C₂×C₃₁	semidirect
131. #2224	C₂×C₃₁	semidirect
132. #2162	C₂×C₃₁	semidirect
208. C₃₁	#33792	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×#3169	2
C₂×C₃₁×#500	2
C₃₁×#2252	2
C₃₁×#2240	2
C₃₁×#2232	2
C₃₁×#2224	2
C₃₁×#2162	2
#33792	31

Subgroups — C31×#33792 #21490551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33792 #21490551