Subgroups — C₂×C₃₁×#1534 #21474416

Order 7936.

966 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	2	yes	maximal
3	C₃₁×#1534	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#330	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#125	—	2	2	no	—
6	C₂×C₃₁×#125	—	2	4	no	—
7	C₂×C₃₁×#125	C₄	1	2	yes	—
8	C₂×C₃₁×#125	C₄	1	4	yes	—
9	C₂×C₃₁×#125	C₂²	1	2	yes	—
10	C₂×C₃₁×#125	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#432	—	2	4	no	—
15	C₃₁×#432	C₂²	1	8	yes	—
16	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₃₁×#97	C₂²	1	4	yes	—
19	C₃₁×#330	C₂²	1	16	yes	complemented
20	C₂³×C₄×C₃₁	—	2	2	no	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	2	yes	—
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₃₁×#125	—	2	8	no	—
25	C₃₁×#125	—	2	4	no	—
26	C₃₁×#125	—	2	4	no	—
27	C₃₁×#125	—	2	8	no	—
28	C₃₁×#125	C₂×C₄	1	4	yes	—
29	C₃₁×#125	C₂×C₄	1	8	yes	—
30	C₃₁×#125	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	2	yes	—
42	C₂×C₄²×C₃₁	D₄	1	4	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₄⋊C₈	—	2	8	no	—
46	C₃₁×C₂²⋊C₈	—	2	8	no	—
47	C₃₁×#97	—	2	4	no	—
48	C₃₁×#97	C₂³	1	8	yes	—
49	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	8	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	4	4	no	—
54	C₂²×C₄×C₃₁	—	4	8	no	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₃₁	—	2	8	no	—
64	C₂×C₈×C₃₁	—	4	8	no	—
65	C₄⋊C₄×C₃₁	—	2	8	no	—
66	C₄⋊C₄×C₃₁	—	2	16	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	8	no	—
69	C₄⋊C₄×C₃₁	—	2	8	no	—
70	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
71	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
72	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	C₄²×C₃₁	C₂×D₄	1	4	yes	—
77	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂×#1534	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	—	2	4	no	—
80	C₂³×C₃₁	#101	1	4	yes	—
81	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	8	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	16	no	—
93	C₂×C₄×C₃₁	—	4	8	no	—
94	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
100	C₈×C₃₁	—	4	8	no	—
101	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic
102	C₂×#432	C₂×C₃₁	1	2	yes	—
103	#1534	C₂×C₃₁	1	8	yes	direct product factor
104	C₂×#330	C₂×C₃₁	1	4	yes	complemented
105	C₂²×C₃₁	—	2	4	no	—
106	C₂²×C₃₁	—	2	8	no	—
107	C₂²×C₃₁	C₂×#101	1	4	yes	—
108	C₂²×C₃₁	C₂×#101	1	2	yes	—
109	C₂²×C₃₁	#421	1	4	yes	—
110	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₃₁	—	2	1	no	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₂×#125	—	2	2	no	—
119	C₂×#125	—	2	4	no	—
120	C₂×#125	C₄×C₃₁	1	2	yes	—
121	C₂×#125	C₄×C₃₁	1	4	yes	—
122	C₂×#125	C₂²×C₃₁	1	2	yes	—
123	C₂×#125	C₂²×C₃₁	1	4	yes	—
124	C₂²×C₄⋊C₄	—	2	2	no	—
125	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
126	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
127	#432	—	2	4	no	—
128	#432	C₂²×C₃₁	1	8	yes	—
129	C₂×C₄⋊C₈	—	2	2	no	—
130	C₂×C₂²⋊C₈	—	2	2	no	—
131	C₂×#97	C₂²×C₃₁	1	4	yes	—
132	#330	C₂²×C₃₁	1	16	yes	complemented
133	C₂×C₃₁	—	2	4	no	—
134	C₂×C₃₁	#1534	1	4	yes	direct product factor
135	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#421	1	2	yes	—
137	C₂³×C₄	—	2	2	no	—
138	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂³×C₄	D₄×C₃₁	1	2	yes	—
140	C₂²×C₈	—	2	4	no	—
141	#125	—	2	8	no	—
142	#125	—	2	4	no	—
143	#125	—	2	4	no	—
144	#125	—	2	8	no	—
145	#125	C₂×C₄×C₃₁	1	4	yes	—
146	#125	C₂×C₄×C₃₁	1	8	yes	—
147	#125	C₂³×C₃₁	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	8	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
156	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
158	C₂×C₄²	D₄×C₃₁	1	2	yes	—
159	C₂×C₄²	D₄×C₃₁	1	4	yes	—
160	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
162	C₄⋊C₈	—	2	8	no	—
163	C₂²⋊C₈	—	2	8	no	—
164	#97	—	2	4	no	—
165	#97	C₂³×C₃₁	1	8	yes	—
166	C₃₁	C₂×#1534	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	4	8	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	2	16	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
189	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
190	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
191	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
193	C₄²	C₂×D₄×C₃₁	1	2	yes	—
194	C₄²	C₂×D₄×C₃₁	1	4	yes	—
195	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
196	C₂³	—	2	4	no	—
197	C₂³	C₃₁×#101	1	4	yes	—
198	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
199	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	16	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
215	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
217	C₈	—	4	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	8	no	—
220	C₂²	C₂×C₃₁×#101	1	4	yes	—
221	C₂²	C₂×C₃₁×#101	1	2	yes	—
222	C₂²	C₃₁×#421	1	4	yes	—
223	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	4	8	no	—
231	C₂	—	2	4	no	—
232	C₂	C₃₁×#1534	1	4	yes	direct product factor
233	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
234	C₂	C₂×C₃₁×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1534	C₂	direct (N × H)
4. C₂×C₃₁×#330	C₂	semidirect
19. C₃₁×#330	C₂²	semidirect
78. C₂×#1534	C₃₁	direct (N × H)
103. #1534	C₂×C₃₁	direct (N × H)
104. C₂×#330	C₂×C₃₁	semidirect
132. #330	C₂²×C₃₁	semidirect
134. C₂×C₃₁	#1534	direct (N × H)
166. C₃₁	C₂×#1534	direct (N × H)
232. C₂	C₃₁×#1534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#125	2
C₂×C₃₁×#432	2
C₃₁×#1534	2
C₂×C₃₁×#330	2
C₂×#1534	31

Subgroups — C2×C31×#1534 #21474416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1534 #21474416