Subgroups — C₃₁×#2748 #16842681

Order 3968.

464 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#399	C₂	1	2	yes	maximal, complemented
2	C₃₁×#398	C₂	1	2	yes	maximal, complemented
3	C₃₁×#396	C₂	1	2	yes	maximal, complemented
4	C₃₁×#390	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#387	C₂	1	2	yes	maximal, complemented
6	C₃₁×#386	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#386	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#385	C₂	1	2	yes	maximal, complemented
9	C₃₁×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#95	C₂²	1	2	yes	—
27	C₃₁×#95	C₂²	1	2	yes	—
28	C₃₁×#95	C₂²	1	2	yes	—
29	C₃₁×#95	C₂²	1	2	yes	—
30	C₃₁×#95	C₂²	1	2	yes	—
31	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
32	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₃₁	—	4	1	no	—
34	C₂×D₄×C₃₁	—	4	1	no	—
35	C₂×D₄×C₃₁	Q₈	1	4	yes	—
36	C₂²×C₄×C₃₁	—	2	4	no	—
37	C₂²×C₄×C₃₁	—	4	1	no	—
38	C₂²×C₄×C₃₁	—	4	2	no	—
39	C₂²×C₄×C₃₁	—	4	1	no	—
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₃₁	—	2	4	no	—
48	C₄⋊C₄×C₃₁	—	4	2	no	—
49	C₄⋊C₄×C₃₁	—	4	1	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₄⋊C₄×C₃₁	—	4	2	no	—
53	C₂²⋊C₄×C₃₁	—	2	4	no	—
54	C₂²⋊C₄×C₃₁	—	4	2	no	—
55	C₂²⋊C₄×C₃₁	—	4	2	no	—
56	C₂²⋊C₄×C₃₁	—	4	1	no	—
57	C₂²⋊C₄×C₃₁	—	4	1	no	—
58	C₂²⋊C₄×C₃₁	—	4	2	no	—
59	C₄²×C₃₁	—	4	1	no	—
60	C₂³×C₃₁	—	2	2	no	—
61	C₂³×C₃₁	—	2	2	no	—
62	C₂³×C₃₁	—	2	2	no	—
63	C₂³×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	—	2	1	no	—
65	C₂³×C₃₁	—	2	1	no	—
66	C₂³×C₃₁	—	2	1	no	—
67	C₂³×C₃₁	—	2	1	no	—
68	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
71	D₄×C₃₁	—	4	4	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	1	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₃₁	#41	1	4	yes	—
95	#2748	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₃₁	—	2	2	no	—
97	C₂²×C₃₁	—	2	2	no	—
98	C₂²×C₃₁	—	2	4	no	—
99	C₂²×C₃₁	—	2	4	no	—
100	C₂²×C₃₁	—	4	2	no	—
101	C₂²×C₃₁	—	4	2	no	—
102	C₂²×C₃₁	—	4	1	no	—
103	C₂²×C₃₁	—	4	1	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₃₁	C₂×#41	1	2	yes	—
108	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	4	2	no	—
114	C₄×C₃₁	—	4	2	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	—	4	2	no	—
118	#399	C₂×C₃₁	1	2	yes	complemented
119	#398	C₂×C₃₁	1	2	yes	complemented
120	#396	C₂×C₃₁	1	2	yes	complemented
121	#390	C₂×C₃₁	1	1	yes	characteristic
122	#387	C₂×C₃₁	1	2	yes	complemented
123	#386	C₂×C₃₁	1	1	yes	characteristic
124	#386	C₂×C₃₁	1	1	yes	characteristic
125	#385	C₂×C₃₁	1	2	yes	complemented
126	#380	C₂×C₃₁	1	2	yes	complemented
127	C₂×C₃₁	—	4	2	no	—
128	C₂×C₃₁	—	4	2	no	—
129	C₂×C₃₁	#540	1	2	yes	—
130	C₂×C₃₁	#543	1	1	yes	characteristic
131	C₂×C₃₁	#551	1	1	yes	characteristic
132	C₂×C₃₁	#554	1	2	yes	—
133	C₂×C₃₁	#560	1	1	yes	characteristic
134	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	#95	C₂²×C₃₁	1	2	yes	—
151	#95	C₂²×C₃₁	1	2	yes	—
152	#95	C₂²×C₃₁	1	2	yes	—
153	#95	C₂²×C₃₁	1	2	yes	—
154	#95	C₂²×C₃₁	1	2	yes	—
155	C₃₁	#2748	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
157	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
158	C₂×D₄	—	4	1	no	—
159	C₂×D₄	—	4	1	no	—
160	C₂×D₄	Q₈×C₃₁	1	4	yes	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
167	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
170	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
194	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
195	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
196	D₄	—	4	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
219	C₂×C₄	#41×C₃₁	1	4	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
231	C₂²	C₂×#41×C₃₁	1	2	yes	—
232	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
234	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₂	—	4	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₃₁×#540	1	2	yes	—
245	C₂	C₃₁×#543	1	1	yes	characteristic
246	C₂	C₃₁×#551	1	1	yes	characteristic
247	C₂	C₃₁×#554	1	2	yes	—
248	C₂	C₃₁×#560	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#399	C₂	semidirect
2. C₃₁×#398	C₂	semidirect
3. C₃₁×#396	C₂	semidirect
5. C₃₁×#387	C₂	semidirect
8. C₃₁×#385	C₂	semidirect
9. C₃₁×#380	C₂	semidirect
95. #2748	C₃₁	direct (N × H)
118. #399	C₂×C₃₁	semidirect
119. #398	C₂×C₃₁	semidirect
120. #396	C₂×C₃₁	semidirect
122. #387	C₂×C₃₁	semidirect
125. #385	C₂×C₃₁	semidirect
126. #380	C₂×C₃₁	semidirect
155. C₃₁	#2748	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×#399	2
C₃₁×#398	2
C₃₁×#396	2
C₃₁×#390	2
C₃₁×#387	2
C₃₁×#386	2
C₃₁×#385	2
C₃₁×#380	2
#2748	31

Subgroups — C31×#2748 #16842681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2748 #16842681