Subgroups — C₄×#19151356 #36459595

Order 22784.

858 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083558	C₂	1	2	yes	maximal
2	C₂×#19151356	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	C₂²	1	4	yes	—
5	C₄×#13083558	—	2	4	no	—
6	C₄×#13083558	—	2	4	no	—
7	C₂×#15667923	C₄	1	4	yes	—
8	C₂×C₄×#10623494	C₄	1	4	yes	—
9	#19151356	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	—
14	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
15	C₂×#13083558	—	2	4	no	—
16	C₂×#13083558	—	2	4	no	—
17	C₂×#13083558	—	2	8	no	—
18	C₂×#13083558	C₂×C₄	1	16	yes	—
19	C₂²×#10623494	C₂×C₄	1	8	yes	—
20	#15667923	—	2	8	no	—
21	C₄×#10623494	—	2	8	no	—
22	C₂³×C₄×C₈₉	Q₈	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	2	yes	—
33	C₂×C₄²×C₈₉	D₄	1	4	yes	—
34	#97×C₈₉	—	2	4	no	—
35	C₄×C₈×C₈₉	—	2	4	no	—
36	#13083558	—	2	16	no	—
37	#13083558	—	2	16	no	—
38	C₂×#10623494	—	2	8	no	—
39	C₂×#10623494	—	2	8	no	—
40	C₂×#10623494	C₄²	1	16	yes	—
41	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	—	4	4	no	—
43	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	4	yes	—
53	M₁₆×C₈₉	—	2	16	no	—
54	C₂×C₈×C₈₉	—	2	4	no	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	C₄²	1	8	yes	—
57	C₄²×C₈₉	C₄²	1	2	yes	—
58	C₄²×C₈₉	C₄²	1	1	yes	characteristic
59	C₄²×C₈₉	C₄²	1	1	yes	characteristic
60	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₈₉	—	4	2	no	—
65	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂³×C₈₉	C₄×Q₈	1	4	yes	—
68	C₂×C₄×C₈₉	—	2	8	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	—	4	4	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	#95	1	4	yes	—
73	C₂×C₄×C₈₉	#95	1	2	yes	—
74	C₂×C₄×C₈₉	#95	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#117	1	4	yes	—
81	C₂×C₄×C₈₉	#117	1	2	yes	—
82	C₂×C₄×C₈₉	C₄×D₄	1	8	yes	—
83	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
84	C₈×C₈₉	—	2	8	no	—
85	#10623494	—	2	16	no	—
86	C₂²×C₈₉	—	2	4	no	—
87	C₂²×C₈₉	—	2	2	no	—
88	C₂²×C₈₉	—	4	4	no	—
89	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
90	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
91	C₂²×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	#338	1	4	yes	complemented
95	C₄×C₈₉	C₄³	1	1	yes	characteristic
96	C₄×C₈₉	#376	1	1	yes	characteristic
97	C₄×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
98	C₄×C₈₉	C₄×C₂²⋊C₄	1	4	yes	—
99	C₄×#338	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	2	no	—
101	C₂×C₈₉	—	2	4	no	—
102	C₂×C₈₉	C₄×#95	1	1	yes	characteristic
103	C₂×C₈₉	C₂×#338	1	2	yes	—
104	C₂×C₄×M₁₆	—	89	2	no	—
105	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
106	C₂×#338	—	89	4	no	—
107	C₈₉	C₄×#338	1	1	yes	characteristic, complemented, Sylow
108	C₂²×M₁₆	—	89	4	no	—
109	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
110	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
111	C₄×M₁₆	—	2	4	no	—
112	C₄×M₁₆	—	178	4	no	—
113	C₄×M₁₆	—	178	4	no	—
114	C₂×#97	—	89	4	no	—
115	C₂×#97	Dic₈₉	1	2	yes	—
116	C₂×C₄×C₈	—	89	4	no	—
117	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
118	#338	—	89	16	no	—
119	C₂³×C₄	Dic₁₇₈	1	2	yes	—
120	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	89	16	no	—
124	C₂×M₁₆	—	178	4	no	—
125	C₂×M₁₆	—	178	4	no	—
126	C₂×M₁₆	—	178	8	no	—
127	C₂×M₁₆	C₄×D₈₉	1	8	yes	—
128	C₂²×C₈	—	89	8	no	—
129	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
130	C₂×C₄²	C₄×D₈₉	1	2	yes	—
131	C₂×C₄²	C₄×D₈₉	1	2	yes	—
132	C₂×C₄²	D₃₅₆	1	2	yes	—
133	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₈₉	1	4	yes	—
136	#97	—	2	4	no	—
137	#97	—	178	8	no	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	178	8	no	—
140	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₂²×C₄	—	4	4	no	—
142	C₂²×C₄	#13083553	1	4	yes	—
143	C₂²×C₄	#13083553	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
148	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
150	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
151	C₂²×C₄	#13083574	1	4	yes	—
152	M₁₆	—	2	16	no	—
153	M₁₆	—	178	16	no	—
154	M₁₆	—	178	16	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	89	16	no	—
158	C₂×C₈	—	178	8	no	—
159	C₂×C₈	—	178	8	no	—
160	C₂×C₈	C₄×Dic₈₉	1	8	yes	—
161	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
162	C₄²	C₄×Dic₈₉	1	2	yes	—
163	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
164	C₄²	#13083551	1	4	yes	—
165	C₄²	#13083551	1	4	yes	—
166	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
167	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
168	C₂³	—	4	2	no	—
169	C₂³	C₂×#13083553	1	2	yes	—
170	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
171	C₂³	C₄×Dic₁₇₈	1	4	yes	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#15667919	1	4	yes	—
177	C₂×C₄	#15667919	1	2	yes	—
178	C₂×C₄	#15667919	1	2	yes	—
179	C₂×C₄	C₄²×D₈₉	1	4	yes	—
180	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
182	C₂×C₄	C₂×#13083551	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
184	C₂×C₄	#15667969	1	4	yes	—
185	C₂×C₄	#15667971	1	2	yes	—
186	C₂×C₄	C₄×C₂²⋊D₈₉	1	8	yes	—
187	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
188	C₈	—	2	8	no	—
189	C₈	—	178	16	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂×#15667919	1	1	yes	characteristic
194	C₂²	C₄×C₄⋊Dic₈₉	1	2	yes	—
195	C₂²	C₄×#13083553	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	#19151356	1	4	yes	direct product factor
199	C₄	C₄²×Dic₈₉	1	1	yes	characteristic
200	C₄	#19151444	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊Dic₈₉	1	2	yes	—
202	C₄	C₄×#13083551	1	4	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	2	4	no	—
205	C₂	C₄×#15667919	1	1	yes	characteristic
206	C₂	C₂×#19151356	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×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	C₄×#338
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
9. #19151356	C₄	direct (N × H)
94. C₄×C₈₉	#338	semidirect
107. C₈₉	C₄×#338	semidirect
198. C₄	#19151356	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×C₄×#13083558	2
C₂×#19151356	2
C₂×C₄×M₁₆×C₈₉	2
C₄×#338	89

Subgroups — C4×#19151356 #36459595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151356 #36459595