Subgroups — C₄×C₃₁×#424 #21477185

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1768	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1729	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#424	C₄	1	16	yes	direct product factor
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
15	C₄×M₁₆×C₃₁	—	2	4	no	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	—	2	4	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
23	C₃₁×#376	C₂²	1	2	yes	—
24	C₄³×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₃₁×#117	—	2	4	no	—
37	C₃₁×#117	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	D₄	1	4	yes	—
45	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×C₄⋊C₈	C₂×C₄	1	32	yes	complemented
51	C₃₁×#97	—	2	4	no	—
52	C₄×C₈×C₃₁	—	2	4	no	—
53	C₄×C₈×C₃₁	—	2	4	no	—
54	C₄×C₈×C₃₁	—	2	4	no	—
55	C₃₁×#95	C₂×C₄	1	8	yes	—
56	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	8	no	—
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₃₁	#41	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₃₁	—	2	16	no	—
70	C₂×C₈×C₃₁	—	2	16	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	C₄²	1	8	yes	—
79	C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
81	C₄²×C₃₁	—	2	16	no	—
82	C₄²×C₃₁	—	2	4	no	—
83	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
84	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
85	C₄²×C₃₁	C₂×D₄	1	2	yes	—
86	C₄²×C₃₁	C₂×D₄	1	2	yes	—
87	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
88	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
89	C₄×#424	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	4	no	—
91	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
92	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₃₁	#117	1	2	yes	—
94	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	16	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	8	no	—
100	C₂×C₄×C₃₁	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₃₁	#117	1	2	yes	—
106	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
108	C₂×C₄×C₃₁	#131	1	8	yes	—
109	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
113	C₈×C₃₁	—	2	8	no	—
114	C₈×C₃₁	—	2	8	no	—
115	C₂×#424	C₂×C₃₁	1	2	yes	—
116	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
117	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
118	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
119	#1768	C₂×C₃₁	1	4	yes	complemented
120	C₄×C₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
121	#1729	C₂×C₃₁	1	2	yes	—
122	C₂²×C₃₁	—	2	4	no	—
123	C₂²×C₃₁	—	2	4	no	—
124	C₂²×C₃₁	—	2	4	no	—
125	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
126	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
127	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
129	C₂²×C₃₁	C₂×#131	1	4	yes	—
130	C₄×C₃₁	—	2	8	no	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	4	4	no	—
133	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
134	C₄×C₃₁	#424	1	8	yes	direct product factor
135	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
136	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
137	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
139	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
141	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
142	#424	C₄×C₃₁	1	16	yes	direct product factor
143	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
144	C₄×M₁₆	—	2	4	no	—
145	C₂×#97	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄×C₈	—	2	4	no	—
147	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
150	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
151	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
152	#376	C₂²×C₃₁	1	2	yes	—
153	C₄³	C₂²×C₃₁	1	2	yes	—
154	#336	C₂²×C₃₁	1	8	yes	—
155	C₂×C₃₁	—	2	4	no	—
156	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃₁	C₄×#131	1	2	yes	—
158	C₂×C₃₁	C₂×#424	1	2	yes	—
159	C₂×C₃₁	#2922	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
161	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
168	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
169	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
170	#117	—	2	4	no	—
171	#117	C₂×C₄×C₃₁	1	8	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
177	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
178	C₂×C₄²	D₄×C₃₁	1	4	yes	—
179	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
181	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
182	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
184	C₄⋊C₈	C₂×C₄×C₃₁	1	32	yes	complemented
185	#97	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	2	4	no	—
189	#95	C₂×C₄×C₃₁	1	8	yes	—
190	C₃₁	C₄×#424	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
194	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
202	C₂²×C₄	#41×C₃₁	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
204	M₁₆	—	2	16	no	—
205	C₂×C₈	—	2	16	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
211	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	C₄²×C₃₁	1	8	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
216	C₄²	—	2	16	no	—
217	C₄²	—	2	4	no	—
218	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
219	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
220	C₄²	C₂×D₄×C₃₁	1	2	yes	—
221	C₄²	C₂×D₄×C₃₁	1	2	yes	—
222	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
223	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
226	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
227	C₂³	C₃₁×#117	1	2	yes	—
228	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	C₂×C₄²×C₃₁	1	8	yes	—
235	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
239	C₂×C₄	C₃₁×#117	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
241	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
242	C₂×C₄	C₃₁×#131	1	8	yes	—
243	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
247	C₈	—	2	8	no	—
248	C₈	—	2	8	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
253	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
254	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
256	C₂²	C₂×C₃₁×#131	1	4	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
261	C₄	C₃₁×#424	1	8	yes	direct product factor
262	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
263	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
264	C₂	—	2	4	no	—
265	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂	C₄×C₃₁×#131	1	2	yes	—
267	C₂	C₂×C₃₁×#424	1	2	yes	—
268	C₂	C₃₁×#2922	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1768	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
13. C₃₁×#424	C₄	direct (N × H)
50. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
89. C₄×#424	C₃₁	direct (N × H)
119. #1768	C₂×C₃₁	semidirect
120. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
134. C₄×C₃₁	#424	direct (N × H)
142. #424	C₄×C₃₁	direct (N × H)
184. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
190. C₃₁	C₄×#424	direct (N × H)
261. C₄	C₃₁×#424	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×#424	2
C₂×C₄×M₁₆×C₃₁	2
C₂²×C₄×C₈×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1768	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1729	2
C₄×#424	31

Subgroups — C4×C31×#424 #21477185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#424 #21477185