Subgroups — C₂²×C₃₁×#431 #21516873

Order 7936.

3086 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₁	C₂²	1	4	yes	—
10	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₄	1	12	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₃₁	C₂²	1	12	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#431	C₂²	1	64	yes	direct product factor
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	24	yes	complemented
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	24	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	24	yes	complemented
20	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	8	no	—
22	C₂³×C₄×C₃₁	—	2	8	no	—
23	C₂³×C₄×C₃₁	C₈	1	16	yes	complemented
24	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
28	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
31	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
33	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	—	2	16	no	—
35	C₂²×C₈×C₃₁	C₂³	1	24	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	48	yes	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	16	yes	—
44	C₃₁×C₄⋊C₈	C₂³	1	32	yes	complemented
45	C₃₁×C₂²⋊C₈	C₂³	1	32	yes	—
46	C₄×C₈×C₃₁	C₂³	1	32	yes	complemented
47	C₂⁴×C₃₁	—	2	4	no	—
48	C₂⁴×C₃₁	—	2	4	no	—
49	C₂⁴×C₃₁	C₂×C₈	1	8	yes	—
50	C₂⁴×C₃₁	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	24	no	—
55	C₂²×C₄×C₃₁	—	2	48	no	—
56	C₂²×C₄×C₃₁	—	2	24	no	—
57	C₂²×C₄×C₃₁	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₃₁	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₃₁	C₂×C₈	1	96	yes	complemented
60	C₂²×C₄×C₃₁	M₁₆	1	8	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₃₁	#41	1	8	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
73	C₂×C₈×C₃₁	—	2	48	no	—
74	C₂×C₈×C₃₁	C₂⁴	1	16	yes	—
75	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
76	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
77	C₄²×C₃₁	C₂²×C₄	1	12	yes	—
78	C₄²×C₃₁	C₂²×C₄	1	12	yes	—
79	C₄²×C₃₁	C₂⁴	1	4	yes	—
80	C₄²×C₃₁	C₂⁴	1	4	yes	—
81	C₂²×#431	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	2	12	no	—
83	C₂³×C₃₁	—	2	12	no	—
84	C₂³×C₃₁	—	2	32	no	—
85	C₂³×C₃₁	#117	1	4	yes	—
86	C₂³×C₃₁	C₂²×C₈	1	24	yes	—
87	C₂³×C₃₁	C₂²×C₈	1	4	yes	—
88	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
89	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
90	C₂³×C₃₁	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₃₁	—	2	12	no	—
92	C₂×C₄×C₃₁	—	2	12	no	—
93	C₂×C₄×C₃₁	—	2	24	no	—
94	C₂×C₄×C₃₁	—	2	32	no	—
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	#117	1	12	yes	—
97	C₂×C₄×C₃₁	C₂²×C₈	1	24	yes	—
98	C₂×C₄×C₃₁	C₂²×C₈	1	24	yes	—
99	C₂×C₄×C₃₁	C₂²×C₈	1	64	yes	complemented
100	C₂×C₄×C₃₁	C₂×M₁₆	1	24	yes	—
101	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₃₁	C₂³×C₄	1	3	yes	—
104	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
108	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
109	C₈×C₃₁	—	2	16	no	—
110	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic
111	C₂×#431	C₂×C₃₁	1	24	yes	direct product factor
112	C₂²×C₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
113	C₂²×C₂²⋊C₈	C₂×C₃₁	1	2	yes	—
114	C₂²×C₄×C₈	C₂×C₃₁	1	2	yes	complemented
115	C₂²×C₃₁	—	2	4	no	—
116	C₂²×C₃₁	—	2	48	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	#431	1	16	yes	direct product factor
119	C₂²×C₃₁	C₂×#117	1	6	yes	—
120	C₂²×C₃₁	C₂³×C₈	1	8	yes	—
121	C₂²×C₃₁	C₂³×C₈	1	6	yes	—
122	C₂²×C₃₁	C₂²×M₁₆	1	6	yes	—
123	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	2	4	no	—
126	C₄×C₃₁	C₂×#117	1	6	yes	—
127	C₄×C₃₁	C₂³×C₈	1	8	yes	—
128	C₄×C₃₁	C₂³×C₈	1	8	yes	—
129	C₄×C₃₁	C₂²×M₁₆	1	8	yes	—
130	C₄×C₃₁	C₂²×#41	1	2	yes	—
131	C₂⁴×C₄	C₄×C₃₁	1	1	yes	characteristic
132	C₂⁴×C₄	C₄×C₃₁	1	1	yes	characteristic
133	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₃₁	1	4	yes	—
135	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
136	C₂²×C₄²	C₄×C₃₁	1	12	yes	—
137	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
140	C₂²×C₄²	C₂²×C₃₁	1	12	yes	—
141	#431	C₂²×C₃₁	1	64	yes	direct product factor
142	C₂×C₄⋊C₈	C₂²×C₃₁	1	24	yes	complemented
143	C₂×C₂²⋊C₈	C₂²×C₃₁	1	24	yes	—
144	C₂×C₄×C₈	C₂²×C₃₁	1	24	yes	complemented
145	C₂×C₃₁	—	2	8	no	—
146	C₂×C₃₁	C₂×#431	1	12	yes	direct product factor
147	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
148	C₂×C₃₁	C₂⁴×C₈	1	1	yes	characteristic
149	C₂×C₃₁	C₂³×M₁₆	1	1	yes	characteristic
150	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	C₈×C₃₁	1	16	yes	complemented
154	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
156	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₃₁	1	6	yes	—
158	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
161	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
163	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	—	2	16	no	—
165	C₂²×C₈	C₂³×C₃₁	1	24	yes	—
166	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
167	C₂×C₄²	C₂×C₄×C₃₁	1	48	yes	—
168	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
169	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
170	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
171	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
172	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
173	C₂×C₄²	C₂³×C₃₁	1	16	yes	—
174	C₄⋊C₈	C₂³×C₃₁	1	32	yes	complemented
175	C₂²⋊C₈	C₂³×C₃₁	1	32	yes	—
176	C₄×C₈	C₂³×C₃₁	1	32	yes	complemented
177	C₃₁	C₂²×#431	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	4	no	—
180	C₂⁴	C₂×C₈×C₃₁	1	8	yes	—
181	C₂⁴	C₂²×C₄×C₃₁	1	6	yes	—
182	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	48	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	C₂×C₈×C₃₁	1	8	yes	—
189	C₂²×C₄	C₂×C₈×C₃₁	1	8	yes	—
190	C₂²×C₄	C₂×C₈×C₃₁	1	96	yes	complemented
191	C₂²×C₄	M₁₆×C₃₁	1	8	yes	—
192	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
193	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
194	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
196	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
201	C₂²×C₄	#41×C₃₁	1	8	yes	—
202	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
204	C₂×C₈	—	2	48	no	—
205	C₂×C₈	C₂⁴×C₃₁	1	16	yes	—
206	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
207	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
208	C₄²	C₂²×C₄×C₃₁	1	12	yes	—
209	C₄²	C₂²×C₄×C₃₁	1	12	yes	—
210	C₄²	C₂⁴×C₃₁	1	4	yes	—
211	C₄²	C₂⁴×C₃₁	1	4	yes	—
212	C₂³	—	2	32	no	—
213	C₂³	—	2	12	no	—
214	C₂³	—	2	12	no	—
215	C₂³	C₃₁×#117	1	4	yes	—
216	C₂³	C₂²×C₈×C₃₁	1	24	yes	—
217	C₂³	C₂²×C₈×C₃₁	1	4	yes	—
218	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
219	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
220	C₂³	C₂³×C₄×C₃₁	1	3	yes	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	12	no	—
223	C₂×C₄	—	2	24	no	—
224	C₂×C₄	—	2	32	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	C₃₁×#117	1	12	yes	—
227	C₂×C₄	C₂²×C₈×C₃₁	1	24	yes	—
228	C₂×C₄	C₂²×C₈×C₃₁	1	24	yes	—
229	C₂×C₄	C₂²×C₈×C₃₁	1	64	yes	complemented
230	C₂×C₄	C₂×M₁₆×C₃₁	1	24	yes	—
231	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
232	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
233	C₂×C₄	C₂³×C₄×C₃₁	1	3	yes	—
234	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
235	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
236	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
237	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
238	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
239	C₈	—	2	16	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	48	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₃₁×#431	1	16	yes	direct product factor
244	C₂²	C₂×C₃₁×#117	1	6	yes	—
245	C₂²	C₂³×C₈×C₃₁	1	8	yes	—
246	C₂²	C₂³×C₈×C₃₁	1	6	yes	—
247	C₂²	C₂²×M₁₆×C₃₁	1	6	yes	—
248	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	C₂×C₃₁×#117	1	6	yes	—
252	C₄	C₂³×C₈×C₃₁	1	8	yes	—
253	C₄	C₂³×C₈×C₃₁	1	8	yes	—
254	C₄	C₂²×M₁₆×C₃₁	1	8	yes	—
255	C₄	C₂²×#41×C₃₁	1	2	yes	—
256	C₂	—	2	8	no	—
257	C₂	C₂×C₃₁×#431	1	12	yes	direct product factor
258	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
259	C₂	C₂⁴×C₈×C₃₁	1	1	yes	characteristic
260	C₂	C₂³×M₁₆×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²×C₃₁	C₂²
commutator	C₂	C₂⁴×C₈×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂⁴×C₃₁	C₂²×C₄
prime radical	C₂⁵×C₃₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#431	C₂	direct (N × H)
3. C₂²×C₃₁×C₄⋊C₈	C₂	semidirect
5. C₂²×C₄×C₈×C₃₁	C₂	semidirect
16. C₃₁×#431	C₂²	direct (N × H)
17. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
19. C₂×C₄×C₈×C₃₁	C₂²	semidirect
23. C₂³×C₄×C₃₁	C₈	semidirect
44. C₃₁×C₄⋊C₈	C₂³	semidirect
46. C₄×C₈×C₃₁	C₂³	semidirect
59. C₂²×C₄×C₃₁	C₂×C₈	semidirect
81. C₂²×#431	C₃₁	direct (N × H)
99. C₂×C₄×C₃₁	C₂²×C₈	semidirect
111. C₂×#431	C₂×C₃₁	direct (N × H)
112. C₂²×C₄⋊C₈	C₂×C₃₁	semidirect
114. C₂²×C₄×C₈	C₂×C₃₁	semidirect
118. C₂²×C₃₁	#431	direct (N × H)
141. #431	C₂²×C₃₁	direct (N × H)
142. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
144. C₂×C₄×C₈	C₂²×C₃₁	semidirect
146. C₂×C₃₁	C₂×#431	direct (N × H)
153. C₂³×C₄	C₈×C₃₁	semidirect
174. C₄⋊C₈	C₂³×C₃₁	semidirect
176. C₄×C₈	C₂³×C₃₁	semidirect
177. C₃₁	C₂²×#431	direct (N × H)
190. C₂²×C₄	C₂×C₈×C₃₁	semidirect
229. C₂×C₄	C₂²×C₈×C₃₁	semidirect
243. C₂²	C₃₁×#431	direct (N × H)
257. C₂	C₂×C₃₁×#431	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×C₄²×C₃₁	2
C₂×C₃₁×#431	2
C₂²×C₃₁×C₄⋊C₈	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂²×C₄×C₈×C₃₁	2
C₂²×#431	31

Subgroups — C22×C31×#431 #21516873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#431 #21516873