Subgroups — C₄×C₄₁×C₄⋊C₁₆ #23788853

Order 10496.

438 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₄²×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1374	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₄₁	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₄³×C₄₁	C₄	1	1	yes	characteristic
13	C₄₁×C₄⋊C₁₆	C₄	1	8	yes	direct product factor
14	C₄×C₁₆×C₄₁	—	2	4	no	—
15	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₄₁	C₈	1	2	yes	—
20	C₂×C₄²×C₄₁	C₈	1	2	yes	—
21	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₁	—	2	4	no	—
25	C₂×C₁₆×C₄₁	—	2	4	no	—
26	C₂×C₁₆×C₄₁	C₂×C₄	1	8	yes	—
27	C₄×C₈×C₄₁	—	2	2	no	—
28	C₄×C₈×C₄₁	—	2	2	no	—
29	C₄×C₈×C₄₁	—	2	2	no	—
30	C₄×C₈×C₄₁	C₈	1	4	yes	—
31	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₄₁	D₄	1	2	yes	—
33	C₄×C₈×C₄₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₁	—	2	1	no	—
40	C₂×C₈×C₄₁	—	2	1	no	—
41	C₂×C₈×C₄₁	—	2	1	no	—
42	C₂×C₈×C₄₁	—	2	1	no	—
43	C₂×C₈×C₄₁	C₄²	1	2	yes	—
44	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
48	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₁	#41	1	2	yes	—
52	C₄²×C₄₁	—	2	4	no	—
53	C₄²×C₄₁	—	2	1	no	—
54	C₄²×C₄₁	—	2	1	no	—
55	C₄²×C₄₁	C₁₆	1	8	yes	complemented
56	C₄²×C₄₁	C₄²	1	2	yes	—
57	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
58	C₄²×C₄₁	C₂×C₈	1	2	yes	—
59	C₄²×C₄₁	M₁₆	1	2	yes	—
60	C₁₆×C₄₁	—	2	8	no	—
61	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₄₁	—	2	2	no	—
63	C₂×C₄×C₄₁	—	2	2	no	—
64	C₂×C₄×C₄₁	—	2	2	no	—
65	C₂×C₄×C₄₁	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₄₁	C₄×C₈	1	2	yes	—
67	C₂×C₄×C₄₁	C₄⋊C₈	1	4	yes	—
68	C₂×C₄×C₄₁	C₂×C₁₆	1	4	yes	—
69	C₂×C₄×C₄₁	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₄₁	C₂×C₁₆	1	4	yes	—
71	C₂×C₄×C₄₁	M₃₂	1	4	yes	—
72	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
78	C₈×C₄₁	—	2	2	no	—
79	C₈×C₄₁	—	2	2	no	—
80	C₈×C₄₁	C₄⋊C₈	1	4	yes	—
81	C₈×C₄₁	C₄×D₄	1	2	yes	—
82	C₈×C₄₁	C₄×Q₈	1	2	yes	—
83	C₄×C₄⋊C₁₆	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂²×C₄₁	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₄₁	#431	1	1	yes	characteristic
87	C₂²×C₄₁	C₂²×C₁₆	1	2	yes	—
88	C₂²×C₄₁	C₂×M₃₂	1	2	yes	—
89	C₄×C₄₁	—	2	2	no	—
90	C₄×C₄₁	—	2	4	no	—
91	C₄×C₄₁	C₄×C₁₆	1	4	yes	complemented
92	C₄×C₄₁	C₄⋊C₁₆	1	8	yes	direct product factor
93	C₄×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₄₁	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₄₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₄₁	#431	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	C₂×C₄₁	1	2	yes	—
98	C₂×C₄×C₁₆	C₂×C₄₁	1	2	yes	—
99	C₄²×C₈	C₂×C₄₁	1	1	yes	characteristic
100	#1374	C₂×C₄₁	1	2	yes	—
101	C₂×C₄₁	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₄₁	C₂×C₄×C₁₆	1	1	yes	characteristic
103	C₂×C₄₁	C₄×M₃₂	1	1	yes	characteristic
104	C₂×C₄₁	C₂×C₄⋊C₁₆	1	2	yes	—
105	C₂×C₄₁	#2162	1	2	yes	—
106	C₂²×C₁₆	C₂²×C₄₁	1	4	yes	—
107	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
113	C₄³	C₄×C₄₁	1	1	yes	characteristic
114	C₄⋊C₁₆	C₄×C₄₁	1	8	yes	direct product factor
115	C₄×C₁₆	—	2	4	no	—
116	C₄₁	C₄×C₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
121	C₂×C₄²	C₈×C₄₁	1	2	yes	—
122	C₂×C₄²	C₈×C₄₁	1	2	yes	—
123	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
126	C₂×C₁₆	—	2	4	no	—
127	C₂×C₁₆	—	2	4	no	—
128	C₂×C₁₆	C₂×C₄×C₄₁	1	8	yes	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₈×C₄₁	1	4	yes	—
133	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
134	C₄×C₈	D₄×C₄₁	1	2	yes	—
135	C₄×C₈	Q₈×C₄₁	1	2	yes	—
136	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
138	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
140	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	C₄²×C₄₁	1	2	yes	—
146	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
147	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
148	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
149	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
151	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
153	C₂×C₈	#41×C₄₁	1	2	yes	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₁₆×C₄₁	1	8	yes	complemented
158	C₄²	C₄²×C₄₁	1	2	yes	—
159	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
160	C₄²	C₂×C₈×C₄₁	1	2	yes	—
161	C₄²	M₁₆×C₄₁	1	2	yes	—
162	C₁₆	—	2	8	no	—
163	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₄×C₈×C₄₁	1	2	yes	—
168	C₂×C₄	C₄×C₈×C₄₁	1	2	yes	—
169	C₂×C₄	C₄⋊C₈×C₄₁	1	4	yes	—
170	C₂×C₄	C₂×C₁₆×C₄₁	1	4	yes	—
171	C₂×C₄	C₂×C₁₆×C₄₁	1	4	yes	—
172	C₂×C₄	C₂×C₁₆×C₄₁	1	4	yes	—
173	C₂×C₄	M₃₂×C₄₁	1	4	yes	—
174	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
176	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
178	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
179	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	C₄⋊C₈×C₄₁	1	4	yes	—
183	C₈	C₄×D₄×C₄₁	1	2	yes	—
184	C₈	C₄×Q₈×C₄₁	1	2	yes	—
185	C₂²	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊C₈×C₄₁	1	1	yes	characteristic
187	C₂²	C₄₁×#431	1	1	yes	characteristic
188	C₂²	C₂²×C₁₆×C₄₁	1	2	yes	—
189	C₂²	C₂×M₃₂×C₄₁	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	C₄×C₁₆×C₄₁	1	4	yes	complemented
193	C₄	C₄₁×C₄⋊C₁₆	1	8	yes	direct product factor
194	C₄	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
195	C₄	C₄×M₁₆×C₄₁	1	1	yes	characteristic
196	C₄	C₂×C₄⋊C₈×C₄₁	1	1	yes	characteristic
197	C₄	C₄₁×#431	1	1	yes	characteristic
198	C₂	C₄×C₄⋊C₈×C₄₁	1	1	yes	characteristic
199	C₂	C₂×C₄×C₁₆×C₄₁	1	1	yes	characteristic
200	C₂	C₄×M₃₂×C₄₁	1	1	yes	characteristic
201	C₂	C₂×C₄₁×C₄⋊C₁₆	1	2	yes	—
202	C₂	C₄₁×#2162	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₄₁	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₁×C₄⋊C₁₆	C₄	direct (N × H)
55. C₄²×C₄₁	C₁₆	semidirect
83. C₄×C₄⋊C₁₆	C₄₁	direct (N × H)
91. C₄×C₄₁	C₄×C₁₆	semidirect
92. C₄×C₄₁	C₄⋊C₁₆	direct (N × H)
114. C₄⋊C₁₆	C₄×C₄₁	direct (N × H)
116. C₄₁	C₄×C₄⋊C₁₆	direct (N × H)
157. C₄²	C₁₆×C₄₁	semidirect
192. C₄	C₄×C₁₆×C₄₁	semidirect
193. C₄	C₄₁×C₄⋊C₁₆	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×C₄₁×C₄⋊C₁₆	2
C₂×C₄×C₁₆×C₄₁	2
C₄²×C₈×C₄₁	2
C₄₁×#1374	2
C₄×C₄⋊C₁₆	41

Subgroups — C4×C41×C4⋊C16 #23788853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×C₄⋊C₁₆ #23788853