Subgroups — C₂²×C₃₁×#442 #21516913

Order 7936.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#442	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃₁×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₃₁	—	2	2	no	—
9	C₂²×D₈×C₃₁	C₄	1	16	yes	complemented
10	C₂²×D₈×C₃₁	C₂²	1	12	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	—	2	12	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#442	C₂²	1	64	yes	direct product factor
19	C₂×C₃₁×#106	C₂²	1	12	yes	complemented
20	C₂×C₃₁×#102	C₂²	1	24	yes	complemented
21	C₂×C₃₁×#97	C₂²	1	12	yes	complemented
22	C₂⁵×C₃₁	—	2	2	no	—
23	C₂²×D₄×C₃₁	—	2	6	no	—
24	C₂²×D₄×C₃₁	—	2	8	no	—
25	C₂²×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
26	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	—	2	24	no	—
29	C₂³×C₄×C₃₁	—	4	16	no	—
30	C₂³×C₄×C₃₁	D₄	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×D₈×C₃₁	C₂×C₄	1	96	yes	complemented
33	C₂×D₈×C₃₁	C₂³	1	16	yes	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
36	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
38	C₄×D₄×C₃₁	—	2	48	no	—
39	C₄×D₄×C₃₁	C₂³	1	32	yes	complemented
40	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	24	no	—
43	C₂×C₄²×C₃₁	D₄	1	6	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
45	C₃₁×#106	C₂³	1	16	yes	complemented
46	C₃₁×#102	C₂³	1	32	yes	complemented
47	C₃₁×#97	C₂³	1	16	yes	complemented
48	C₂⁴×C₃₁	—	2	16	no	—
49	C₂⁴×C₃₁	—	2	12	no	—
50	C₂⁴×C₃₁	—	4	8	no	—
51	C₂⁴×C₃₁	—	4	8	no	—
52	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×D₄×C₃₁	—	2	48	no	—
54	C₂×D₄×C₃₁	—	2	12	no	—
55	C₂×D₄×C₃₁	—	2	24	no	—
56	C₂×D₄×C₃₁	C₂²×C₄	1	48	yes	complemented
57	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₃₁	—	2	8	no	—
59	C₂²×C₄×C₃₁	—	2	12	no	—
60	C₂²×C₄×C₃₁	—	2	32	no	—
61	C₂²×C₄×C₃₁	—	4	96	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₃₁	#41	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
68	D₈×C₃₁	C₂²×C₄	1	64	yes	complemented
69	C₂×C₈×C₃₁	—	2	12	no	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	24	yes	complemented
71	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
72	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
73	C₄⋊C₄×C₃₁	—	2	12	no	—
74	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
75	C₂²⋊C₄×C₃₁	—	2	32	no	—
76	C₄²×C₃₁	C₂×D₄	1	12	yes	—
77	C₄²×C₃₁	C₂⁴	1	4	yes	—
78	C₂²×#442	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	—	2	48	no	—
80	C₂³×C₃₁	—	2	8	no	—
81	C₂³×C₃₁	—	4	24	no	—
82	C₂³×C₃₁	—	4	24	no	—
83	C₂³×C₃₁	—	4	64	no	—
84	C₂³×C₃₁	C₄×D₄	1	4	yes	—
85	C₂³×C₃₁	AGL(1,ℤ/8)	1	8	yes	—
86	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
87	D₄×C₃₁	—	2	48	no	—
88	D₄×C₃₁	—	2	8	no	—
89	D₄×C₃₁	C₂³×C₄	1	16	yes	complemented
90	C₂×C₄×C₃₁	—	2	24	no	—
91	C₂×C₄×C₃₁	—	2	8	no	—
92	C₂×C₄×C₃₁	—	4	64	no	—
93	C₂×C₄×C₃₁	—	4	24	no	—
94	C₂×C₄×C₃₁	C₄×D₄	1	12	yes	—
95	C₂×C₄×C₃₁	C₂³×C₄	1	6	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
98	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
99	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
100	C₈×C₃₁	—	2	4	no	—
101	C₈×C₃₁	C₂³×C₄	1	8	yes	complemented
102	C₂³×D₈	C₂×C₃₁	1	1	yes	characteristic
103	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
104	C₂×#442	C₂×C₃₁	1	24	yes	direct product factor
105	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
106	C₂²×#102	C₂×C₃₁	1	2	yes	complemented
107	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂²×C₃₁	—	2	16	no	—
109	C₂²×C₃₁	—	4	96	no	—
110	C₂²×C₃₁	—	4	8	no	—
111	C₂²×C₃₁	—	4	8	no	—
112	C₂²×C₃₁	#442	1	16	yes	direct product factor
113	C₂²×C₃₁	C₂×C₄×D₄	1	6	yes	—
114	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	12	yes	—
115	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
116	C₄×C₃₁	—	2	8	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₄×C₃₁	C₂×C₄×D₄	1	6	yes	—
119	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
121	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
122	C₂⁴×C₄	—	2	2	no	—
123	C₂²×D₈	C₄×C₃₁	1	16	yes	complemented
124	C₂²×D₈	C₂²×C₃₁	1	12	yes	—
125	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
127	C₂×C₄×D₄	—	2	12	no	—
128	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	complemented
129	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
130	C₂²×C₂²⋊C₄	—	2	2	no	—
131	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
132	#442	C₂²×C₃₁	1	64	yes	direct product factor
133	C₂×#106	C₂²×C₃₁	1	12	yes	complemented
134	C₂×#102	C₂²×C₃₁	1	24	yes	complemented
135	C₂×#97	C₂²×C₃₁	1	12	yes	complemented
136	C₂×C₃₁	—	4	16	no	—
137	C₂×C₃₁	C₂×#442	1	12	yes	direct product factor
138	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
139	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
140	C₂⁵	—	2	2	no	—
141	C₂²×D₄	—	2	6	no	—
142	C₂²×D₄	—	2	8	no	—
143	C₂²×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
144	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	—	2	24	no	—
147	C₂³×C₄	—	4	16	no	—
148	C₂³×C₄	D₄×C₃₁	1	2	yes	—
149	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
150	C₂×D₈	C₂×C₄×C₃₁	1	96	yes	complemented
151	C₂×D₈	C₂³×C₃₁	1	16	yes	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
154	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
155	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
156	C₄×D₄	—	2	48	no	—
157	C₄×D₄	C₂³×C₃₁	1	32	yes	complemented
158	C₂×C₄⋊C₄	—	2	6	no	—
159	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
160	C₂×C₂²⋊C₄	—	2	24	no	—
161	C₂×C₄²	D₄×C₃₁	1	6	yes	—
162	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
163	#106	C₂³×C₃₁	1	16	yes	complemented
164	#102	C₂³×C₃₁	1	32	yes	complemented
165	#97	C₂³×C₃₁	1	16	yes	complemented
166	C₃₁	C₂²×#442	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	16	no	—
168	C₂⁴	—	2	12	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	—	4	8	no	—
171	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×D₄	—	2	48	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	2	24	no	—
175	C₂×D₄	C₂²×C₄×C₃₁	1	48	yes	complemented
176	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	2	32	no	—
180	C₂²×C₄	—	4	96	no	—
181	C₂²×C₄	—	4	8	no	—
182	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
183	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
184	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
185	C₂²×C₄	#41×C₃₁	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
187	D₈	C₂²×C₄×C₃₁	1	64	yes	complemented
188	C₂×C₈	—	2	12	no	—
189	C₂×C₈	C₂²×C₄×C₃₁	1	24	yes	complemented
190	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
191	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
192	C₄⋊C₄	—	2	12	no	—
193	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
194	C₂²⋊C₄	—	2	32	no	—
195	C₄²	C₂×D₄×C₃₁	1	12	yes	—
196	C₄²	C₂⁴×C₃₁	1	4	yes	—
197	C₂³	—	2	48	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	64	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	4	24	no	—
202	C₂³	C₄×D₄×C₃₁	1	4	yes	—
203	C₂³	C₃₁×AGL(1,ℤ/8)	1	8	yes	—
204	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
205	D₄	—	2	48	no	—
206	D₄	—	2	8	no	—
207	D₄	C₂³×C₄×C₃₁	1	16	yes	complemented
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	4	64	no	—
211	C₂×C₄	—	4	24	no	—
212	C₂×C₄	C₄×D₄×C₃₁	1	12	yes	—
213	C₂×C₄	C₂³×C₄×C₃₁	1	6	yes	—
214	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
215	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
216	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
217	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	C₂³×C₄×C₃₁	1	8	yes	complemented
220	C₂²	—	2	16	no	—
221	C₂²	—	4	96	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₃₁×#442	1	16	yes	direct product factor
225	C₂²	C₂×C₄×D₄×C₃₁	1	6	yes	—
226	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	12	yes	—
227	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	C₂×C₄×D₄×C₃₁	1	6	yes	—
231	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
232	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
233	C₂	—	4	16	no	—
234	C₂	C₂×C₃₁×#442	1	12	yes	direct product factor
235	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
236	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂³×D₈×C₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₃₁	C₂	semidirect
3. C₂×C₃₁×#442	C₂	direct (N × H)
4. C₂²×C₃₁×#106	C₂	semidirect
5. C₂²×C₃₁×#102	C₂	semidirect
6. C₂²×C₃₁×#97	C₂	semidirect
9. C₂²×D₈×C₃₁	C₄	semidirect
14. C₂×C₄×D₄×C₃₁	C₂²	semidirect
18. C₃₁×#442	C₂²	direct (N × H)
19. C₂×C₃₁×#106	C₂²	semidirect
20. C₂×C₃₁×#102	C₂²	semidirect
21. C₂×C₃₁×#97	C₂²	semidirect
25. C₂²×D₄×C₃₁	C₂×C₄	semidirect
32. C₂×D₈×C₃₁	C₂×C₄	semidirect
35. C₂²×C₈×C₃₁	C₂×C₄	semidirect
39. C₄×D₄×C₃₁	C₂³	semidirect
45. C₃₁×#106	C₂³	semidirect
46. C₃₁×#102	C₂³	semidirect
47. C₃₁×#97	C₂³	semidirect
56. C₂×D₄×C₃₁	C₂²×C₄	semidirect
68. D₈×C₃₁	C₂²×C₄	semidirect
70. C₂×C₈×C₃₁	C₂²×C₄	semidirect
78. C₂²×#442	C₃₁	direct (N × H)
89. D₄×C₃₁	C₂³×C₄	semidirect
101. C₈×C₃₁	C₂³×C₄	semidirect
103. C₂²×C₄×D₄	C₂×C₃₁	semidirect
104. C₂×#442	C₂×C₃₁	direct (N × H)
105. C₂²×#106	C₂×C₃₁	semidirect
106. C₂²×#102	C₂×C₃₁	semidirect
107. C₂²×#97	C₂×C₃₁	semidirect
112. C₂²×C₃₁	#442	direct (N × H)
123. C₂²×D₈	C₄×C₃₁	semidirect
128. C₂×C₄×D₄	C₂²×C₃₁	semidirect
132. #442	C₂²×C₃₁	direct (N × H)
133. C₂×#106	C₂²×C₃₁	semidirect
134. C₂×#102	C₂²×C₃₁	semidirect
135. C₂×#97	C₂²×C₃₁	semidirect
137. C₂×C₃₁	C₂×#442	direct (N × H)
143. C₂²×D₄	C₂×C₄×C₃₁	semidirect
150. C₂×D₈	C₂×C₄×C₃₁	semidirect
153. C₂²×C₈	C₂×C₄×C₃₁	semidirect
157. C₄×D₄	C₂³×C₃₁	semidirect
163. #106	C₂³×C₃₁	semidirect
164. #102	C₂³×C₃₁	semidirect
165. #97	C₂³×C₃₁	semidirect
166. C₃₁	C₂²×#442	direct (N × H)
175. C₂×D₄	C₂²×C₄×C₃₁	semidirect
187. D₈	C₂²×C₄×C₃₁	semidirect
189. C₂×C₈	C₂²×C₄×C₃₁	semidirect
207. D₄	C₂³×C₄×C₃₁	semidirect
219. C₈	C₂³×C₄×C₃₁	semidirect
224. C₂²	C₃₁×#442	direct (N × H)
234. C₂	C₂×C₃₁×#442	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₂×C₃₁×#442	2
C₂²×C₃₁×#106	2
C₂²×C₃₁×#102	2
C₂²×C₃₁×#97	2
C₂²×#442	31

Subgroups — C22×C31×#442 #21516913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#442 #21516913