Subgroups — C₄×C₃₁×#427 #21477208

Order 7936.

1182 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	2	yes	maximal
4	C₄×C₃₁×#107	C₂	1	4	yes	maximal, complemented
5	C₄×C₃₁×#106	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1735	C₂	1	2	yes	maximal
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	4	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#427	C₄	1	16	yes	direct product factor
12	C₂×C₃₁×#107	C₂²	1	4	yes	—
13	C₂×C₃₁×#106	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₃₁	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
18	C₄×C₂²⋊C₄×C₃₁	—	2	4	no	—
19	C₃₁×#376	—	2	2	no	—
20	C₄³×C₃₁	—	2	2	no	—
21	C₃₁×#340	C₂²	1	8	yes	—
22	C₂³×C₄×C₃₁	D₄	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	—	2	4	no	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#117	—	2	8	no	—
32	C₃₁×#117	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄²×C₃₁	—	2	4	no	—
36	C₂×C₄²×C₃₁	—	2	8	no	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₃₁×#107	C₂×C₄	1	16	yes	complemented
43	C₃₁×#106	C₂×C₄	1	16	yes	complemented
44	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
46	C₃₁×#95	—	2	8	no	—
47	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	8	no	—
49	C₂²×C₄×C₃₁	—	2	8	no	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₃₁	#41	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₃₁	—	2	8	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	C₄²	1	16	yes	complemented
62	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
67	C₄⋊C₄×C₃₁	—	2	8	no	—
68	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
69	C₂²⋊C₄×C₃₁	—	2	16	no	—
70	C₄²×C₃₁	—	2	16	no	—
71	C₄²×C₃₁	—	4	16	no	—
72	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
74	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
75	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
76	C₄×#427	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	2	4	no	—
78	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₃₁	C₄×D₄	1	4	yes	—
80	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	16	no	—
83	C₂×C₄×C₃₁	—	2	16	no	—
84	C₂×C₄×C₃₁	—	4	16	no	—
85	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₃₁	#117	1	4	yes	—
90	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
92	C₂×C₄×C₃₁	#135	1	8	yes	—
93	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	C₂×C₄²	1	4	yes	—
99	C₈×C₃₁	C₂×C₄²	1	4	yes	—
100	C₂×#427	C₂×C₃₁	1	2	yes	—
101	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
102	C₄×#117	C₂×C₃₁	1	2	yes	—
103	C₄×#107	C₂×C₃₁	1	4	yes	complemented
104	C₄×#106	C₂×C₃₁	1	4	yes	complemented
105	#1735	C₂×C₃₁	1	2	yes	—
106	C₂²×C₃₁	—	2	4	no	—
107	C₂²×C₃₁	—	2	8	no	—
108	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
109	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
111	C₂²×C₃₁	C₂×#135	1	4	yes	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	4	16	no	—
114	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
115	C₄×C₃₁	#427	1	8	yes	direct product factor
116	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
117	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
118	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
119	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×#117	C₂²×C₃₁	1	4	yes	—
122	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
123	#427	C₄×C₃₁	1	16	yes	direct product factor
124	C₂×#107	C₂²×C₃₁	1	4	yes	—
125	C₂×#106	C₂²×C₃₁	1	4	yes	—
126	C₂×C₄×C₈	C₄×C₃₁	1	8	yes	complemented
127	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
129	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
130	C₄×C₂²⋊C₄	—	2	4	no	—
131	#376	—	2	2	no	—
132	C₄³	—	2	2	no	—
133	#340	C₂²×C₃₁	1	8	yes	—
134	C₂×C₃₁	—	2	4	no	—
135	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#427	1	2	yes	—
137	C₂×C₃₁	C₄×#135	1	4	yes	—
138	C₂³×C₄	D₄×C₃₁	1	2	yes	—
139	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	—	2	4	no	—
141	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
144	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
145	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
147	#117	—	2	8	no	—
148	#117	C₂×C₄×C₃₁	1	16	yes	—
149	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₄²	—	2	4	no	—
152	C₂×C₄²	—	2	8	no	—
153	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
154	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
155	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
156	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
157	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
158	#107	C₂×C₄×C₃₁	1	16	yes	complemented
159	#106	C₂×C₄×C₃₁	1	16	yes	complemented
160	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
161	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
162	#95	—	2	8	no	—
163	C₃₁	C₄×#427	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
168	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
170	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
174	C₂²×C₄	#41×C₃₁	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄²×C₃₁	1	16	yes	complemented
179	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
180	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
186	C₂²⋊C₄	—	2	16	no	—
187	C₄²	—	2	16	no	—
188	C₄²	—	4	16	no	—
189	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
190	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
191	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
192	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
195	C₂³	C₄×D₄×C₃₁	1	4	yes	—
196	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	4	16	no	—
201	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
205	C₂×C₄	C₃₁×#117	1	4	yes	—
206	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
207	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
208	C₂×C₄	C₃₁×#135	1	8	yes	—
209	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
211	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄²×C₃₁	1	4	yes	—
215	C₈	C₂×C₄²×C₃₁	1	4	yes	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
221	C₂²	C₂×C₃₁×#135	1	4	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	4	16	no	—
224	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
225	C₄	C₃₁×#427	1	8	yes	direct product factor
226	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
227	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
228	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
231	C₂	C₂×C₃₁×#427	1	2	yes	—
232	C₂	C₄×C₃₁×#135	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄×C₃₁×#107	C₂	semidirect
5. C₄×C₃₁×#106	C₂	semidirect
11. C₃₁×#427	C₄	direct (N × H)
14. C₂×C₄×C₈×C₃₁	C₄	semidirect
42. C₃₁×#107	C₂×C₄	semidirect
43. C₃₁×#106	C₂×C₄	semidirect
61. C₂×C₈×C₃₁	C₄²	semidirect
76. C₄×#427	C₃₁	direct (N × H)
103. C₄×#107	C₂×C₃₁	semidirect
104. C₄×#106	C₂×C₃₁	semidirect
115. C₄×C₃₁	#427	direct (N × H)
123. #427	C₄×C₃₁	direct (N × H)
126. C₂×C₄×C₈	C₄×C₃₁	semidirect
158. #107	C₂×C₄×C₃₁	semidirect
159. #106	C₂×C₄×C₃₁	semidirect
163. C₃₁	C₄×#427	direct (N × H)
178. C₂×C₈	C₄²×C₃₁	semidirect
225. C₄	C₃₁×#427	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×#427	2
C₂²×C₄×C₈×C₃₁	2
C₄×C₃₁×#117	2
C₄×C₃₁×#107	2
C₄×C₃₁×#106	2
C₃₁×#1735	2
C₄×#427	31

Subgroups — C4×C31×#427 #21477208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#427 #21477208