Subgroups — C₃₁×#9594 #21466353

Order 7936.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1939	C₂	1	2	yes	maximal
2	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#340	C₄	1	2	yes	—
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×#335	—	2	2	no	—
17	C₈²×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#107	—	2	2	no	—
29	C₃₁×#107	C₈	1	4	yes	complemented
30	C₃₁×C₄⋊C₈	—	2	2	no	—
31	C₃₁×C₄⋊C₈	—	2	2	no	—
32	C₃₁×C₄⋊C₈	—	4	4	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	—	2	1	no	—
35	C₄×C₈×C₃₁	—	2	1	no	—
36	C₄×C₈×C₃₁	D₄	1	2	yes	—
37	C₄×C₈×C₃₁	Q₈	1	2	yes	—
38	C₄×C₈×C₃₁	Q₈	1	2	yes	—
39	C₃₁×#95	—	2	2	no	—
40	C₂²×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	1	no	—
48	C₂×C₈×C₃₁	—	2	1	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	2	1	no	—
51	C₂×C₈×C₃₁	—	2	1	no	—
52	C₂×C₈×C₃₁	—	2	1	no	—
53	C₂×C₈×C₃₁	—	2	1	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
61	C₂×C₈×C₃₁	D₈	1	2	yes	—
62	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
67	C₄²×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	—	4	2	no	—
69	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₃₁	#41	1	1	yes	characteristic
73	#9594	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	—	2	1	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
96	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	—	2	2	no	—
99	C₈×C₃₁	—	2	2	no	—
100	C₈×C₃₁	—	2	2	no	—
101	C₈×C₃₁	—	8	2	no	—
102	C₈×C₃₁	C₄⋊C₈	1	4	yes	complemented
103	#1939	C₂×C₃₁	1	2	yes	—
104	C₄×#107	C₂×C₃₁	1	1	yes	characteristic
105	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
106	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
107	#1279	C₂×C₃₁	1	2	yes	—
108	C₂²×C₃₁	#389	1	1	yes	characteristic
109	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
110	C₂²×C₃₁	#436	1	1	yes	characteristic
111	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
112	C₂²×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
113	C₂²×C₃₁	#443	1	1	yes	characteristic
114	C₂²×C₃₁	#443	1	1	yes	characteristic
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	4	4	no	—
118	C₄×C₃₁	—	8	2	no	—
119	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₃₁	#424	1	1	yes	characteristic
121	C₄×C₃₁	C₈×Q₈	1	1	yes	characteristic
122	C₄×C₃₁	#446	1	1	yes	characteristic
123	C₄×C₃₁	#466	1	1	yes	characteristic
124	C₄×C₃₁	#467	1	1	yes	characteristic
125	C₄×C₃₁	#499	1	1	yes	characteristic
126	C₄×C₃₁	#500	1	1	yes	characteristic
127	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
128	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	2	2	no	—
130	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
135	#340	C₄×C₃₁	1	2	yes	—
136	#336	—	2	2	no	—
137	#335	—	2	2	no	—
138	C₈²	—	2	2	no	—
139	C₂×C₃₁	C₈×D₈	1	1	yes	characteristic
140	C₂×C₃₁	C₈×Q₁₆	1	1	yes	characteristic
141	C₂×C₃₁	#1589	1	1	yes	characteristic
142	C₂×C₃₁	#1591	1	1	yes	characteristic
143	C₂×C₃₁	#1939	1	1	yes	characteristic
144	C₂×C₃₁	#1942	1	1	yes	characteristic
145	C₂×C₃₁	#1945	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	#107	C₈×C₃₁	1	4	yes	complemented
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	4	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₄×C₃₁	1	2	yes	—
165	C₄×C₈	Q₈×C₃₁	1	2	yes	—
166	C₄×C₈	Q₈×C₃₁	1	2	yes	—
167	#95	—	2	2	no	—
168	C₃₁	#9594	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
187	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
188	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
189	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
190	C₂×C₈	D₈×C₃₁	1	2	yes	—
191	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
200	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
201	C₄²	#41×C₃₁	1	1	yes	characteristic
202	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
214	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
216	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
220	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
221	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
224	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
225	C₈	—	2	4	no	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	8	2	no	—
230	C₈	C₃₁×C₄⋊C₈	1	4	yes	complemented
231	C₂²	C₃₁×#389	1	1	yes	characteristic
232	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
233	C₂²	C₃₁×#436	1	1	yes	characteristic
234	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
235	C₂²	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
236	C₂²	C₃₁×#443	1	1	yes	characteristic
237	C₂²	C₃₁×#443	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
243	C₄	C₃₁×#424	1	1	yes	characteristic
244	C₄	C₈×Q₈×C₃₁	1	1	yes	characteristic
245	C₄	C₃₁×#446	1	1	yes	characteristic
246	C₄	C₃₁×#466	1	1	yes	characteristic
247	C₄	C₃₁×#467	1	1	yes	characteristic
248	C₄	C₃₁×#499	1	1	yes	characteristic
249	C₄	C₃₁×#500	1	1	yes	characteristic
250	C₂	C₈×D₈×C₃₁	1	1	yes	characteristic
251	C₂	C₈×Q₁₆×C₃₁	1	1	yes	characteristic
252	C₂	C₃₁×#1589	1	1	yes	characteristic
253	C₂	C₃₁×#1591	1	1	yes	characteristic
254	C₂	C₃₁×#1939	1	1	yes	characteristic
255	C₂	C₃₁×#1942	1	1	yes	characteristic
256	C₂	C₃₁×#1945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₃₁×#107	C₈	semidirect
73. #9594	C₃₁	direct (N × H)
102. C₈×C₃₁	C₄⋊C₈	semidirect
157. #107	C₈×C₃₁	semidirect
168. C₃₁	#9594	direct (N × H)
230. C₈	C₃₁×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#1939	2
C₄×C₃₁×#107	2
C₂×C₃₁×#335	2
C₂×C₈²×C₃₁	2
C₃₁×#1279	2
#9594	31

Subgroups — C31×#9594 #21466353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9594 #21466353