Subgroups — C₂²×C₄×M₁₆×C₃₁ #21516811

Order 7936.

3950 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄²×C₃₁	C₂²
commutator	C₂	C₂³×C₄²×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
3. C₂×C₄×M₁₆×C₃₁	C₂	direct (N × H)
4. C₂²×C₃₁×#97	C₂	semidirect
5. C₂²×C₄×C₈×C₃₁	C₂	semidirect
8. C₂²×M₁₆×C₃₁	C₄	direct (N × H)
16. C₄×M₁₆×C₃₁	C₂²	direct (N × H)
17. C₂×C₃₁×#97	C₂²	semidirect
18. C₂×C₄×C₈×C₃₁	C₂²	semidirect
29. C₂×M₁₆×C₃₁	C₂×C₄	direct (N × H)
31. C₂²×C₈×C₃₁	C₂×C₄	semidirect
40. C₃₁×#97	C₂³	semidirect
41. C₄×C₈×C₃₁	C₂³	semidirect
50. C₂²×C₄×C₃₁	M₁₆	direct (N × H)
61. M₁₆×C₃₁	C₂²×C₄	direct (N × H)
62. C₂×C₈×C₃₁	C₂²×C₄	semidirect
69. C₂²×C₄×M₁₆	C₃₁	direct (N × H)
82. C₂×C₄×C₃₁	C₂×M₁₆	direct (N × H)
90. C₈×C₃₁	C₂³×C₄	semidirect
93. C₂×C₄×M₁₆	C₂×C₃₁	direct (N × H)
94. C₂²×#97	C₂×C₃₁	semidirect
95. C₂²×C₄×C₈	C₂×C₃₁	semidirect
98. C₂²×C₃₁	C₄×M₁₆	direct (N × H)
106. C₄×C₃₁	C₂²×M₁₆	direct (N × H)
110. C₂²×M₁₆	C₄×C₃₁	direct (N × H)
118. C₄×M₁₆	C₂²×C₃₁	direct (N × H)
119. C₂×#97	C₂²×C₃₁	semidirect
120. C₂×C₄×C₈	C₂²×C₃₁	semidirect
122. C₂×C₃₁	C₂×C₄×M₁₆	direct (N × H)
135. C₂×M₁₆	C₂×C₄×C₃₁	direct (N × H)
137. C₂²×C₈	C₂×C₄×C₃₁	semidirect
146. #97	C₂³×C₃₁	semidirect
147. C₄×C₈	C₂³×C₃₁	semidirect
148. C₃₁	C₂²×C₄×M₁₆	direct (N × H)
157. C₂²×C₄	M₁₆×C₃₁	direct (N × H)
168. M₁₆	C₂²×C₄×C₃₁	direct (N × H)
169. C₂×C₈	C₂²×C₄×C₃₁	semidirect
188. C₂×C₄	C₂×M₁₆×C₃₁	direct (N × H)
196. C₈	C₂³×C₄×C₃₁	semidirect
199. C₂²	C₄×M₁₆×C₃₁	direct (N × H)
207. C₄	C₂²×M₁₆×C₃₁	direct (N × H)
210. C₂	C₂×C₄×M₁₆×C₃₁	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×M₁₆×C₃₁	2
C₂³×C₄²×C₃₁	2
C₂×C₄×M₁₆×C₃₁	2
C₂²×C₃₁×#97	2
C₂²×C₄×C₈×C₃₁	2
C₂²×C₄×M₁₆	31

Subgroups — C22×C4×M16×C31 #21516811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×M₁₆×C₃₁ #21516811