Subgroups — #64408⋊(C₄⋊C₁₆) #34034121

Order 18688.

438 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	#22627117	C₂	1	2	yes	maximal
2	C₂×#64408⋊C₁₆	C₂	1	2	yes	maximal
3	#22627002	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556686	C₂²	1	4	yes	—
6	#17989690	C₄	1	4	yes	complemented
7	C₄⋊#11924040	C₄	1	4	yes	complemented
8	#64408⋊C₁₆	—	2	4	no	—
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₄²×#64408	C₄	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
16	#14556686	—	2	4	no	—
17	#14556686	—	2	4	no	—
18	C₂×#11924040	C₂×C₄	1	8	yes	—
19	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924036	C₂³	1	1	yes	characteristic
21	C₂×C₄×#64408	C₈	1	2	yes	—
22	C₂×C₄×#64408	C₈	1	2	yes	—
23	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596833	C₂×C₄	1	2	yes	—
26	C₄×#10596833	C₂×C₄	1	2	yes	—
27	C₈×#64408	—	2	2	no	—
28	C₈×#64408	—	2	2	no	—
29	C₈×#64408	—	2	2	no	—
30	C₈×#64408	D₄	1	2	yes	—
31	C₈×#64408	Q₈	1	2	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₈	1	1	yes	characteristic
36	C₈×Dic₇₃	C₈	1	1	yes	characteristic
37	C₈×Dic₇₃	C₈	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
39	#11924040	—	2	8	no	—
40	C₂²×#64408	C₂×C₈	1	2	yes	—
41	C₂²×#64408	C₂×C₈	1	2	yes	—
42	#11924036	—	2	1	no	—
43	#11924036	—	2	1	no	—
44	#11924036	C₂×D₄	1	1	yes	characteristic
45	#11924036	C₂×Q₈	1	1	yes	characteristic
46	#11924036	#41	1	2	yes	—
47	C₂×#10596833	C₄²	1	2	yes	—
48	C₂×#10596833	C₂²×C₄	1	2	yes	—
49	C₄×#64408	—	2	4	no	—
50	C₄×#64408	—	2	1	no	—
51	C₄×#64408	—	2	1	no	—
52	C₄×#64408	C₁₆	1	8	yes	complemented
53	C₄×#64408	C₄⋊C₄	1	2	yes	—
54	C₄×#64408	M₁₆	1	2	yes	—
55	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
59	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
60	C₈×D₇₃	—	2	1	no	—
61	C₈×D₇₃	—	2	1	no	—
62	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₈×D₇₃	M₁₆	1	2	yes	—
64	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
65	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
66	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×#64408	—	2	2	no	—
72	C₂×#64408	—	2	2	no	—
73	C₂×#64408	C₄⋊C₈	1	4	yes	—
74	C₂×#64408	C₂×C₁₆	1	4	yes	—
75	C₂×#64408	C₂×C₁₆	1	4	yes	—
76	C₂×#64408	M₃₂	1	4	yes	—
77	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
81	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
83	C₄×D₇₃	—	2	2	no	—
84	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
85	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×D₇₃	#117	1	1	yes	characteristic
87	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
88	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
89	#10596833	—	2	2	no	—
90	#10596833	C₄×D₄	1	2	yes	—
91	#10596833	C₄×Q₈	1	2	yes	—
92	C₈×C₇₃	—	2	1	no	—
93	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
94	#10596831	—	2	1	no	—
95	#10596831	C₄⋊C₈	1	2	yes	—
96	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂×D₇₃	#431	1	1	yes	characteristic
99	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
100	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
101	#64408	—	2	4	no	—
102	#64408	C₄⋊C₁₆	1	8	yes	complemented
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	C₄×C₁₆	1	2	yes	complemented
105	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₇₃	#431	1	1	yes	characteristic
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	C₄×C₁₆	1	2	yes	complemented
109	Dic₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
110	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₄⋊C₁₆	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
114	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
115	D₇₃	C₂×C₄⋊C₁₆	1	2	yes	—
116	D₇₃	#2162	1	2	yes	—
117	C₂×C₄⋊C₁₆	—	73	2	no	—
118	C₂×C₄×C₁₆	—	73	2	no	—
119	C₄²×C₈	—	73	1	no	—
120	#1374	—	73	2	no	—
121	C₇₃	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₁₆	—	73	4	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₄³	—	73	1	no	—
130	C₄⋊C₁₆	—	73	4	no	—
131	C₄⋊C₁₆	—	73	4	no	—
132	C₄×C₁₆	—	146	4	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂×C₄²	—	73	1	no	—
138	C₂×C₄²	—	73	2	no	—
139	C₂×C₄²	—	73	1	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	73	2	no	—
142	C₂×C₁₆	—	73	8	no	—
143	C₂×C₁₆	—	146	4	no	—
144	C₂×C₁₆	—	146	4	no	—
145	C₄×C₈	—	73	2	no	—
146	C₄×C₈	—	73	2	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	1	no	—
152	C₄×C₈	—	146	2	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	C₄×C₈	#10596842	1	1	yes	characteristic
156	C₂²×C₄	—	73	2	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	2	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	146	1	no	—
171	C₂×C₈	—	146	1	no	—
172	C₂×C₈	—	146	1	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
175	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
176	C₄²	—	73	8	no	—
177	C₄²	—	73	2	no	—
178	C₄²	—	73	2	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	146	1	no	—
183	C₄²	—	146	1	no	—
184	C₄²	—	146	4	no	—
185	C₄²	C₂×#10596842	1	1	yes	characteristic
186	C₁₆	—	146	8	no	—
187	C₂³	—	73	1	no	—
188	C₂×C₄	—	73	4	no	—
189	C₂×C₄	—	73	4	no	—
190	C₂×C₄	—	73	4	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	4	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	4	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	146	2	no	—
201	C₂×C₄	—	146	2	no	—
202	C₂×C₄	—	146	2	no	—
203	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
204	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
205	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	73	2	no	—
208	C₈	—	73	2	no	—
209	C₈	—	73	2	no	—
210	C₈	—	146	2	no	—
211	C₈	—	146	1	no	—
212	C₈	C₄⋊#10596842	1	2	yes	—
213	C₂²	—	73	2	no	—
214	C₂²	—	73	2	no	—
215	C₂²	—	73	1	no	—
216	C₂²	—	73	1	no	—
217	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	73	8	no	—
220	C₄	—	73	2	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	146	4	no	—
224	C₄	—	146	1	no	—
225	C₄	#64408⋊C₁₆	1	2	yes	complemented
226	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
227	C₄	#17989700	1	1	yes	characteristic
228	C₂	—	73	2	no	—
229	C₂	—	73	2	no	—
230	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic
231	C₂	C₂×#64408⋊C₁₆	1	1	yes	characteristic
232	C₂	#64408⋊M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	C₄×C₄⋊C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989690	C₄	semidirect
7. C₄⋊#11924040	C₄	semidirect
52. C₄×#64408	C₁₆	semidirect
102. #64408	C₄⋊C₁₆	semidirect
104. C₄×C₇₃	C₄×C₁₆	semidirect
108. Dic₇₃	C₄×C₁₆	semidirect
121. C₇₃	C₄×C₄⋊C₁₆	semidirect
225. C₄	#64408⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22627117	2
C₂×#64408⋊C₁₆	2
#22627002	2
C₄×C₈×#64408	2
C₄×C₄⋊C₁₆	73

Subgroups — #64408⋊(C4⋊C16) #34034121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #64408⋊(C₄⋊C₁₆) #34034121