Subgroups — C₃₁×#33261 #21490020

Order 7936.

606 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	2	yes	maximal
2	C₃₁×#1944	C₂	1	2	yes	maximal
3	C₃₁×#1943	C₂	1	2	yes	maximal
4	C₃₁×#1942	C₂	1	2	yes	maximal
5	C₃₁×#1942	C₂	1	2	yes	maximal
6	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1836	C₂	1	2	yes	maximal
9	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#400	—	2	4	no	—
21	C₃₁×#398	—	2	4	no	—
22	C₃₁×#389	C₂²	1	4	yes	—
23	C₃₁×#389	C₂²	1	4	yes	—
24	C₃₁×#384	C₂²	1	2	yes	—
25	C₃₁×#382	C₂²	1	2	yes	—
26	C₃₁×#376	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	—	2	4	no	—
42	C₃₁×#107	—	2	4	no	—
43	C₃₁×#107	—	2	4	no	—
44	C₃₁×#106	—	2	4	no	—
45	C₃₁×#97	—	2	2	no	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	Q₈	1	4	yes	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	—	2	2	no	—
50	C₄×C₈×C₃₁	Q₈	1	4	yes	—
51	C₃₁×#95	—	2	4	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	—	2	4	no	—
54	C₃₁×#95	D₄	1	2	yes	—
55	C₃₁×#95	D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	1	no	—
64	C₂×C₈×C₃₁	—	2	1	no	—
65	C₂×C₈×C₃₁	—	2	1	no	—
66	C₂×C₈×C₃₁	—	2	1	no	—
67	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₃₁	#41	1	2	yes	—
72	C₂×C₈×C₃₁	#41	1	2	yes	—
73	C₂×C₈×C₃₁	#41	1	2	yes	—
74	C₂×C₈×C₃₁	#41	1	2	yes	—
75	C₄⋊C₄×C₃₁	—	2	4	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	8	4	no	—
80	C₄⋊C₄×C₃₁	—	8	4	no	—
81	C₄⋊C₄×C₃₁	—	8	4	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	—	2	1	no	—
84	C₄²×C₃₁	—	2	1	no	—
85	C₄²×C₃₁	—	2	1	no	—
86	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
87	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
88	#33261	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	4	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	#125	1	4	yes	—
115	C₈×C₃₁	#125	1	4	yes	—
116	#2647	C₂×C₃₁	1	2	yes	—
117	#1944	C₂×C₃₁	1	2	yes	—
118	#1943	C₂×C₃₁	1	2	yes	—
119	#1942	C₂×C₃₁	1	2	yes	—
120	#1942	C₂×C₃₁	1	2	yes	—
121	#1846	C₂×C₃₁	1	1	yes	characteristic
122	#1845	C₂×C₃₁	1	1	yes	characteristic
123	#1836	C₂×C₃₁	1	2	yes	—
124	#1726	C₂×C₃₁	1	1	yes	characteristic
125	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₃₁	#532	1	1	yes	characteristic
127	C₂²×C₃₁	#532	1	1	yes	characteristic
128	C₂²×C₃₁	#576	1	2	yes	—
129	C₂²×C₃₁	#578	1	2	yes	—
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	—	4	1	no	—
133	C₄×C₃₁	—	4	1	no	—
134	C₄×C₃₁	—	8	4	no	—
135	C₄×C₃₁	—	8	4	no	—
136	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
137	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
138	C₄×C₃₁	#533	1	1	yes	characteristic
139	C₄×C₃₁	#533	1	1	yes	characteristic
140	C₂×#107	C₂²×C₃₁	1	2	yes	—
141	C₂×#107	C₂²×C₃₁	1	2	yes	—
142	C₂×#107	C₂²×C₃₁	1	2	yes	—
143	C₂×#106	C₂²×C₃₁	1	2	yes	—
144	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	#400	—	2	4	no	—
151	#398	—	2	4	no	—
152	#389	C₂²×C₃₁	1	4	yes	—
153	#389	C₂²×C₃₁	1	4	yes	—
154	#384	C₂²×C₃₁	1	2	yes	—
155	#382	C₂²×C₃₁	1	2	yes	—
156	#376	C₂²×C₃₁	1	1	yes	characteristic
157	#340	C₂²×C₃₁	1	4	yes	—
158	#340	C₂²×C₃₁	1	4	yes	—
159	C₂×C₃₁	#2859	1	1	yes	characteristic
160	C₂×C₃₁	#3164	1	2	yes	—
161	C₂×C₃₁	#3175	1	2	yes	—
162	C₂×C₃₁	#3176	1	1	yes	characteristic
163	C₂×C₃₁	#3177	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
172	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
173	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#106	—	2	4	no	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	#97	Q₈×C₃₁	1	4	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	Q₈×C₃₁	1	4	yes	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	#95	D₄×C₃₁	1	2	yes	—
190	#95	D₄×C₃₁	1	2	yes	—
191	C₃₁	#33261	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
205	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
206	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
207	C₂×C₈	#41×C₃₁	1	2	yes	—
208	C₂×C₈	#41×C₃₁	1	2	yes	—
209	C₂×C₈	#41×C₃₁	1	2	yes	—
210	C₂×C₈	#41×C₃₁	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	8	4	no	—
216	C₄⋊C₄	—	8	4	no	—
217	C₄⋊C₄	—	8	4	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
223	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
224	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
241	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	C₃₁×#125	1	4	yes	—
250	C₈	C₃₁×#125	1	4	yes	—
251	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
252	C₂²	C₃₁×#532	1	1	yes	characteristic
253	C₂²	C₃₁×#532	1	1	yes	characteristic
254	C₂²	C₃₁×#576	1	2	yes	—
255	C₂²	C₃₁×#578	1	2	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	8	4	no	—
261	C₄	—	8	4	no	—
262	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
263	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
264	C₄	C₃₁×#533	1	1	yes	characteristic
265	C₄	C₃₁×#533	1	1	yes	characteristic
266	C₂	C₃₁×#2859	1	1	yes	characteristic
267	C₂	C₃₁×#3164	1	2	yes	—
268	C₂	C₃₁×#3175	1	2	yes	—
269	C₂	C₃₁×#3176	1	1	yes	characteristic
270	C₂	C₃₁×#3177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
88. #33261	C₃₁	direct (N × H)
191. C₃₁	#33261	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×#2647	2
C₃₁×#1944	2
C₃₁×#1943	2
C₃₁×#1942	2
C₃₁×#1846	2
C₃₁×#1845	2
C₃₁×#1836	2
C₃₁×#1726	2
#33261	31

Subgroups — C31×#33261 #21490020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33261 #21490020