Subgroups — C₃₁×#21943 #21478702

Order 7936.

1090 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#407	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2280	C₂	1	2	yes	maximal
3	C₃₁×#1854	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1809	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#117	C₂²	1	4	yes	—
8	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
9	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
10	C₃₁×#407	C₄	1	8	yes	complemented
11	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₃₁×#381	—	2	2	no	—
13	C₃₁×#380	C₂²	1	4	yes	—
14	C₃₁×#380	C₂²	1	4	yes	—
15	C₃₁×#379	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
17	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
18	C₃₁×#340	C₂²	1	8	yes	complemented
19	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	—	4	2	no	—
21	C₂³×C₄×C₃₁	—	4	2	no	—
22	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	D₄	1	2	yes	—
25	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	4	2	no	—
29	C₂×M₁₆×C₃₁	—	4	2	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₃₁×#117	—	4	4	no	—
33	C₃₁×#117	—	4	4	no	—
34	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
35	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄²×C₃₁	—	2	4	no	—
39	C₃₁×#107	—	4	4	no	—
40	C₃₁×#106	—	4	4	no	—
41	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	—
42	C₃₁×#95	—	2	4	no	—
43	C₃₁×#95	—	2	4	no	—
44	C₂⁴×C₃₁	—	4	2	no	—
45	C₂⁴×C₃₁	—	4	4	no	—
46	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
47	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
48	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	8	no	—
54	C₂²×C₄×C₃₁	—	4	8	no	—
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	—	4	4	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₃₁	—	4	8	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
72	C₄⋊C₄×C₃₁	—	2	8	no	—
73	C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₂²⋊C₄×C₃₁	—	4	8	no	—
75	C₂²⋊C₄×C₃₁	—	4	8	no	—
76	C₂²⋊C₄×C₃₁	—	4	8	no	—
77	C₄²×C₃₁	—	4	8	no	—
78	#21943	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	—	2	4	no	—
80	C₂³×C₃₁	—	2	2	no	—
81	C₂³×C₃₁	—	2	4	no	—
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	—	2	4	no	—
84	C₂³×C₃₁	—	4	4	no	—
85	C₂³×C₃₁	—	4	8	no	—
86	C₂³×C₃₁	—	4	16	no	—
87	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	8	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	—	4	8	no	—
102	C₂×C₄×C₃₁	—	4	8	no	—
103	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
108	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
109	C₈×C₃₁	—	2	8	no	—
110	C₂×#407	C₂×C₃₁	1	1	yes	characteristic
111	#2280	C₂×C₃₁	1	2	yes	—
112	#1854	C₂×C₃₁	1	8	yes	complemented
113	#1809	C₂×C₃₁	1	4	yes	complemented
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	2	4	no	—
116	C₂²×C₃₁	—	2	4	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	—	2	4	no	—
119	C₂²×C₃₁	—	4	16	no	—
120	C₂²×C₃₁	—	4	8	no	—
121	C₂²×C₃₁	—	4	4	no	—
122	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₃₁	#534	1	2	yes	—
124	C₂²×C₃₁	#575	1	4	yes	—
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	8	8	no	—
128	C₄×C₃₁	#518	1	2	yes	—
129	C₄×C₃₁	#543	1	1	yes	characteristic
130	C₄×C₃₁	#543	1	1	yes	characteristic
131	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
132	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
133	C₂×#117	C₂²×C₃₁	1	4	yes	—
134	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
135	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
136	#407	C₄×C₃₁	1	8	yes	complemented
137	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
138	#381	—	2	2	no	—
139	#380	C₂²×C₃₁	1	4	yes	—
140	#380	C₂²×C₃₁	1	4	yes	—
141	#379	—	2	2	no	—
142	C₄×C₂²⋊C₄	—	2	2	no	—
143	C₄×C₂²⋊C₄	—	2	2	no	—
144	#340	C₂²×C₃₁	1	8	yes	complemented
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	—	4	4	no	—
147	C₂×C₃₁	#2315	1	1	yes	characteristic
148	C₂×C₃₁	#2911	1	2	yes	—
149	C₂×C₃₁	#3197	1	4	yes	—
150	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
151	C₂³×C₄	—	4	2	no	—
152	C₂³×C₄	—	4	2	no	—
153	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
154	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	D₄×C₃₁	1	2	yes	—
156	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
157	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
158	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
159	C₂×M₁₆	—	4	2	no	—
160	C₂×M₁₆	—	4	2	no	—
161	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
162	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
163	#117	—	4	4	no	—
164	#117	—	4	4	no	—
165	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
166	C₂×C₂²⋊C₄	—	2	4	no	—
167	C₂×C₂²⋊C₄	—	2	4	no	—
168	C₂×C₂²⋊C₄	—	2	4	no	—
169	C₂×C₄²	—	2	4	no	—
170	#107	—	4	4	no	—
171	#106	—	4	4	no	—
172	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₃₁	#21943	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	4	2	no	—
177	C₂⁴	—	4	4	no	—
178	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
179	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
180	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
192	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
193	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
198	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	4	8	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₄²	—	4	8	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	8	no	—
217	C₂³	—	4	16	no	—
218	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
219	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
220	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
239	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
240	C₈	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	16	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	4	no	—
249	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂²	C₃₁×#534	1	2	yes	—
251	C₂²	C₃₁×#575	1	4	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	8	no	—
255	C₄	C₃₁×#518	1	2	yes	—
256	C₄	C₃₁×#543	1	1	yes	characteristic
257	C₄	C₃₁×#543	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	4	4	no	—
260	C₂	C₃₁×#2315	1	1	yes	characteristic
261	C₂	C₃₁×#2911	1	2	yes	—
262	C₂	C₃₁×#3197	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1854	C₂	semidirect
4. C₃₁×#1809	C₂	semidirect
8. C₂×C₃₁×#107	C₂²	semidirect
9. C₂×C₃₁×#106	C₂²	semidirect
10. C₃₁×#407	C₄	semidirect
18. C₃₁×#340	C₂²	semidirect
78. #21943	C₃₁	direct (N × H)
112. #1854	C₂×C₃₁	semidirect
113. #1809	C₂×C₃₁	semidirect
134. C₂×#107	C₂²×C₃₁	semidirect
135. C₂×#106	C₂²×C₃₁	semidirect
136. #407	C₄×C₃₁	semidirect
144. #340	C₂²×C₃₁	semidirect
175. C₃₁	#21943	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#407	2
C₃₁×#2280	2
C₃₁×#1854	2
C₃₁×#1809	2
#21943	31

Subgroups — C31×#21943 #21478702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21943 #21478702