Subgroups — C₄×#19151355 #36459593

Order 22784.

954 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	C₄×#15667971	C₂	1	2	yes	maximal
2	C₂×#19151355	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667971	C₂²	1	4	yes	—
5	C₄×C₄⋊Dic₈₉	—	2	4	no	—
6	C₄×C₂²⋊Dic₈₉	—	2	4	no	—
7	#19151444	C₄	1	4	yes	complemented
8	C₄²×Dic₈₉	C₄	1	4	yes	complemented
9	#19151355	C₄	1	16	yes	direct product factor
10	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₈₉	—	2	4	no	—
13	C₂×#97×C₈₉	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	complemented
15	#15667971	—	2	8	no	—
16	#15667971	C₂×C₄	1	16	yes	—
17	C₂×C₄⋊Dic₈₉	—	2	4	no	—
18	C₂×C₂²⋊Dic₈₉	—	2	4	no	—
19	C₂×C₄×Dic₈₉	—	2	8	no	—
20	C₂×C₄×Dic₈₉	C₂×C₄	1	8	yes	—
21	#15667919	—	2	8	no	—
22	C₂³×C₄×C₈₉	D₄	1	2	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₈₉	—	2	4	no	—
25	C₂×M₁₆×C₈₉	—	2	4	no	—
26	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	D₄	1	4	yes	—
33	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
34	#97×C₈₉	—	2	4	no	—
35	C₄×C₈×C₈₉	—	2	4	no	—
36	C₂²×Dic₈₉	—	2	8	no	—
37	C₄⋊Dic₈₉	—	2	16	no	—
38	C₂²⋊Dic₈₉	—	2	16	no	—
39	C₄×Dic₈₉	—	2	8	no	—
40	C₄×Dic₈₉	—	4	16	no	—
41	C₄×Dic₈₉	C₄²	1	16	yes	complemented
42	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	—	4	4	no	—
44	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	#41	1	4	yes	—
54	M₁₆×C₈₉	—	2	16	no	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	—	2	4	no	—
57	C₂×C₈×C₈₉	C₄²	1	8	yes	complemented
58	C₄²×C₈₉	C₄²	1	2	yes	—
59	C₄²×C₈₉	C₄²	1	1	yes	characteristic
60	C₄²×C₈₉	C₄²	1	1	yes	characteristic
61	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
64	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
65	C₂³×C₈₉	—	4	2	no	—
66	C₂³×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₈₉	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₈₉	—	2	8	no	—
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	—	4	4	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	#95	1	2	yes	—
74	C₂×C₄×C₈₉	#95	1	4	yes	—
75	C₂×C₄×C₈₉	#95	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₈₉	#117	1	4	yes	—
82	C₂×C₄×C₈₉	#117	1	2	yes	—
83	C₂×C₄×C₈₉	C₄×D₄	1	8	yes	—
84	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
85	C₂×Dic₈₉	—	2	16	no	—
86	C₂×Dic₈₉	—	4	16	no	—
87	C₈×C₈₉	—	2	8	no	—
88	C₂²×C₈₉	—	2	4	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	—	4	4	no	—
91	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
92	C₂²×C₈₉	C₄×C₂²⋊C₄	1	4	yes	—
93	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	H(3,ℤ/4)	1	4	yes	complemented
97	C₄×C₈₉	C₄³	1	1	yes	characteristic
98	C₄×C₈₉	#376	1	1	yes	characteristic
99	C₄×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
100	C₄×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
101	Dic₈₉	—	4	16	no	—
102	C₄×H(3,ℤ/4)	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	2	4	no	—
104	C₂×C₈₉	—	2	2	no	—
105	C₂×C₈₉	C₄×#95	1	1	yes	characteristic
106	C₂×C₈₉	C₂×H(3,ℤ/4)	1	2	yes	—
107	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
108	C₄×#117	—	89	2	no	—
109	C₂×H(3,ℤ/4)	—	89	4	no	—
110	C₈₉	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
111	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
112	C₂×#117	—	89	4	no	—
113	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
114	C₄×M₁₆	—	2	4	no	—
115	C₂×#97	Dic₈₉	1	2	yes	complemented
116	C₂×C₄×C₈	Dic₈₉	1	2	yes	complemented
117	C₄×C₄⋊C₄	—	178	4	no	—
118	C₄×C₂²⋊C₄	—	178	4	no	—
119	#376	—	89	4	no	—
120	C₄³	—	89	4	no	—
121	H(3,ℤ/4)	—	89	16	no	—
122	C₂³×C₄	D₃₅₆	1	2	yes	—
123	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	C₄×D₈₉	1	8	yes	—
127	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
128	#117	—	89	16	no	—
129	#117	—	178	8	no	—
130	C₂×C₄⋊C₄	—	178	4	no	—
131	C₂×C₂²⋊C₄	—	178	4	no	—
132	C₂×C₄²	—	89	8	no	—
133	C₂×C₄²	—	178	8	no	—
134	C₂×C₄²	Dic₁₇₈	1	2	yes	—
135	C₂×C₄²	C₄×D₈₉	1	2	yes	—
136	C₂×C₄²	C₄×D₈₉	1	2	yes	—
137	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₈₉	1	4	yes	—
140	#97	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	#95	—	178	8	no	—
143	C₂⁴	C₂×D₃₅₆	1	1	yes	characteristic
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	178	8	no	—
146	C₂²×C₄	#13083551	1	4	yes	—
147	C₂²×C₄	#13083551	1	4	yes	—
148	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
149	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
150	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
154	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
155	C₂²×C₄	#13083574	1	4	yes	—
156	M₁₆	—	2	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₄×Dic₈₉	1	8	yes	complemented
160	C₄⋊C₄	—	178	16	no	—
161	C₂²⋊C₄	—	178	16	no	—
162	C₄²	—	89	16	no	—
163	C₄²	—	356	16	no	—
164	C₄²	—	178	8	no	—
165	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
166	C₄²	C₄×Dic₈₉	1	2	yes	—
167	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
168	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
169	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
170	C₄²	#13083553	1	4	yes	—
171	C₄²	#13083553	1	4	yes	—
172	C₂³	—	4	2	no	—
173	C₂³	C₂×#13083551	1	2	yes	—
174	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
175	C₂³	C₄×D₃₅₆	1	4	yes	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	356	16	no	—
181	C₂×C₄	—	178	16	no	—
182	C₂×C₄	#15667919	1	2	yes	—
183	C₂×C₄	#15667919	1	4	yes	—
184	C₂×C₄	#15667919	1	2	yes	—
185	C₂×C₄	C₄²×D₈₉	1	4	yes	—
186	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
189	C₂×C₄	C₂×#13083553	1	2	yes	—
190	C₂×C₄	#15667969	1	4	yes	—
191	C₂×C₄	#15667971	1	2	yes	—
192	C₂×C₄	C₄×C₂²⋊D₈₉	1	8	yes	—
193	C₂×C₄	C₄×Dic₁₇₈	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#15667919	1	1	yes	characteristic
199	C₂²	C₄×#13083551	1	4	yes	—
200	C₂²	C₄×C₄⋊Dic₈₉	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	356	16	no	—
204	C₄	#19151355	1	4	yes	direct product factor
205	C₄	C₄²×Dic₈₉	1	1	yes	characteristic
206	C₄	#19151444	1	1	yes	characteristic
207	C₄	C₄×C₂²⋊Dic₈₉	1	2	yes	—
208	C₄	C₄×#13083553	1	4	yes	—
209	C₂	—	2	4	no	—
210	C₂	—	2	2	no	—
211	C₂	C₄×#15667919	1	1	yes	characteristic
212	C₂	C₂×#19151355	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₈₉
commutator	C₄×C₈₉	C₄³
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×C₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
7. #19151444	C₄	semidirect
8. C₄²×Dic₈₉	C₄	semidirect
9. #19151355	C₄	direct (N × H)
13. C₂×#97×C₈₉	C₄	semidirect
14. C₂×C₄×C₈×C₈₉	C₄	semidirect
41. C₄×Dic₈₉	C₄²	semidirect
57. C₂×C₈×C₈₉	C₄²	semidirect
96. C₄×C₈₉	H(3,ℤ/4)	semidirect
110. C₈₉	C₄×H(3,ℤ/4)	semidirect
115. C₂×#97	Dic₈₉	semidirect
116. C₂×C₄×C₈	Dic₈₉	semidirect
159. C₂×C₈	C₄×Dic₈₉	semidirect
204. C₄	#19151355	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#15667971	2
C₂×#19151355	2
C₂×C₄×M₁₆×C₈₉	2
C₄×H(3,ℤ/4)	89

Subgroups — C4×#19151355 #36459593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151355 #36459593