Subgroups — C₂×#376×D₈₉ #36712596

Order 22784.

5150 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×D₈₉	C₂	1	3	yes	maximal
2	C₂×#19151444	C₂	1	3	yes	maximal, complemented
3	C₂×#19151443	C₂	1	4	yes	maximal, complemented
4	#376×D₈₉	C₂	1	16	yes	maximal, direct product factor
5	C₂×#95×D₈₉	C₂	1	4	yes	maximal
6	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₈₉	C₂²	1	6	yes	—
8	C₂³×C₄×D₈₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₈₉	C₂²	1	3	yes	—
11	C₂×C₄²×D₈₉	C₄	1	48	yes	complemented
12	C₂×C₄²×D₈₉	C₂²	1	24	yes	—
13	#19151444	C₂²	1	24	yes	complemented
14	#19151443	C₂²	1	32	yes	complemented
15	C₂×#15667919	C₂²	1	12	yes	—
16	#95×D₈₉	C₂²	1	32	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
18	#376×C₈₉	C₂²	1	8	yes	complemented
19	C₂×#95×C₈₉	C₂²	1	4	yes	—
20	C₂⁴×D₈₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₈₉	C₂³	1	6	yes	—
22	C₂³×Dic₈₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×D₈₉	—	2	48	no	—
24	C₂²×C₄×D₈₉	C₂×C₄	1	48	yes	—
25	C₂²×C₄×D₈₉	C₂×C₄	1	24	yes	—
26	C₂²×C₄×D₈₉	C₂³	1	24	yes	—
27	C₂²×C₄×D₈₉	C₂³	1	4	yes	—
28	C₂×C₄×Dic₈₉	C₂×C₄	1	48	yes	complemented
29	C₂×C₄×Dic₈₉	C₂×C₄	1	24	yes	complemented
30	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
31	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
32	C₄²×D₈₉	C₂×C₄	1	192	yes	complemented
33	#15667919	C₂³	1	48	yes	—
34	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	complemented
37	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
38	#95×C₈₉	C₂³	1	16	yes	—
39	C₂³×D₈₉	C₂²×C₄	1	12	yes	—
40	C₂³×D₈₉	#41	1	16	yes	—
41	C₂³×D₈₉	C₂⁴	1	2	yes	—
42	C₂²×Dic₈₉	—	2	24	no	—
43	C₂²×Dic₈₉	C₂²×C₄	1	24	yes	—
44	C₂²×Dic₈₉	C₂²×C₄	1	12	yes	—
45	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
46	C₂²×Dic₈₉	C₂⁴	1	2	yes	—
47	C₂×C₄×D₈₉	—	2	192	no	—
48	C₂×C₄×D₈₉	—	2	96	no	—
49	C₂×C₄×D₈₉	C₄²	1	64	yes	—
50	C₂×C₄×D₈₉	C₂²×C₄	1	96	yes	—
51	C₂×C₄×D₈₉	C₂²×C₄	1	48	yes	—
52	C₄×Dic₈₉	C₂²×C₄	1	48	yes	complemented
53	C₄×Dic₈₉	C₂²×C₄	1	96	yes	complemented
54	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	—	2	24	no	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
60	C₄²×C₈₉	C₂²×C₄	1	48	yes	complemented
61	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
62	C₂³×C₈₉	C₂×#41	1	8	yes	—
63	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
64	C₂²×D₈₉	C₂×C₄²	1	16	yes	—
65	C₂²×D₈₉	#117	1	96	yes	—
66	C₂²×D₈₉	C₂³×C₄	1	12	yes	—
67	C₂²×D₈₉	C₂×#41	1	16	yes	—
68	C₂×C₄×C₈₉	—	2	48	no	—
69	C₂×C₄×C₈₉	—	2	24	no	—
70	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	—
71	C₂×C₄×C₈₉	C₂³×C₄	1	24	yes	—
72	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
73	C₂×Dic₈₉	—	2	48	no	—
74	C₂×Dic₈₉	—	2	24	no	—
75	C₂×Dic₈₉	C₂×C₄²	1	16	yes	—
76	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
77	C₂×Dic₈₉	C₂³×C₄	1	12	yes	—
78	C₄×D₈₉	—	2	192	no	—
79	C₄×D₈₉	C₂×C₄²	1	64	yes	—
80	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
81	C₂²×C₈₉	C₂×#117	1	24	yes	—
82	C₂²×C₈₉	C₂⁴×C₄	1	3	yes	—
83	C₂²×C₈₉	C₂²×#41	1	4	yes	—
84	C₂×D₈₉	#376	1	64	yes	direct product factor
85	C₂×D₈₉	C₂²×C₄²	1	8	yes	—
86	C₂×D₈₉	C₂×#117	1	48	yes	—
87	C₄×C₈₉	—	2	24	no	—
88	C₄×C₈₉	C₂²×C₄²	1	8	yes	—
89	Dic₈₉	—	2	24	no	—
90	Dic₈₉	C₂²×C₄²	1	8	yes	—
91	C₂²×#376	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	C₂×#376	1	8	yes	complemented
93	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
94	C₂×C₈₉	C₂²×#117	1	6	yes	—
95	D₈₉	C₂×#376	1	16	yes	direct product factor
96	C₂³×C₄²	—	89	3	no	—
97	C₂×#376	—	89	3	no	—
98	C₂×#376	—	89	16	no	—
99	C₂×#376	—	89	4	no	—
100	C₂×#376	D₈₉	1	1	yes	characteristic, direct product factor
101	C₂²×#95	—	89	4	no	—
102	C₈₉	C₂²×#376	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴×C₄	—	89	1	no	—
104	C₂⁴×C₄	—	89	6	no	—
105	C₂²×C₄²	—	89	6	no	—
106	C₂²×C₄²	—	89	3	no	—
107	C₂²×C₄²	—	89	48	no	—
108	C₂²×C₄²	—	89	24	no	—
109	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
110	#376	—	89	24	no	—
111	#376	—	89	32	no	—
112	#376	C₂×D₈₉	1	8	yes	direct product factor
113	C₂×#95	—	89	12	no	—
114	C₂×#95	—	89	32	no	—
115	C₂×#95	C₂×D₈₉	1	4	yes	—
116	C₂⁵	—	89	1	no	—
117	C₂³×C₄	—	89	1	no	—
118	C₂³×C₄	—	89	24	no	—
119	C₂³×C₄	—	89	4	no	—
120	C₂³×C₄	—	89	6	no	—
121	C₂³×C₄	—	89	48	no	—
122	C₂³×C₄	—	89	24	no	—
123	C₂³×C₄	—	178	48	no	—
124	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
125	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
126	C₂×C₄²	—	89	48	no	—
127	C₂×C₄²	—	89	24	no	—
128	C₂×C₄²	—	89	24	no	—
129	C₂×C₄²	—	89	12	no	—
130	C₂×C₄²	—	89	192	no	—
131	C₂×C₄²	C₄×D₈₉	1	24	yes	complemented
132	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
133	#95	—	89	48	no	—
134	#95	C₂²×D₈₉	1	16	yes	—
135	C₂⁴	—	89	12	no	—
136	C₂⁴	—	89	16	no	—
137	C₂⁴	—	89	2	no	—
138	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
139	C₂²×C₄	—	2	24	no	—
140	C₂²×C₄	—	89	12	no	—
141	C₂²×C₄	—	89	2	no	—
142	C₂²×C₄	—	89	48	no	—
143	C₂²×C₄	—	89	64	no	—
144	C₂²×C₄	—	89	24	no	—
145	C₂²×C₄	—	89	12	no	—
146	C₂²×C₄	—	89	96	no	—
147	C₂²×C₄	—	178	96	no	—
148	C₂²×C₄	—	178	192	no	—
149	C₂²×C₄	—	178	24	no	—
150	C₂²×C₄	C₂×C₄×D₈₉	1	24	yes	—
151	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
152	C₂²×C₄	C₂³×D₈₉	1	12	yes	—
153	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
154	C₄²	—	89	96	no	—
155	C₄²	—	89	48	no	—
156	C₄²	C₂×C₄×D₈₉	1	48	yes	complemented
157	C₂³	—	89	12	no	—
158	C₂³	—	89	16	no	—
159	C₂³	—	89	16	no	—
160	C₂³	—	89	96	no	—
161	C₂³	C₂²×C₄×D₈₉	1	6	yes	—
162	C₂³	#41×D₈₉	1	8	yes	—
163	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
164	C₂×C₄	—	2	24	no	—
165	C₂×C₄	—	2	48	no	—
166	C₂×C₄	—	89	12	no	—
167	C₂×C₄	—	89	16	no	—
168	C₂×C₄	—	89	64	no	—
169	C₂×C₄	—	89	24	no	—
170	C₂×C₄	—	178	192	no	—
171	C₂×C₄	—	178	24	no	—
172	C₂×C₄	—	178	48	no	—
173	C₂×C₄	C₄²×D₈₉	1	16	yes	—
174	C₂×C₄	C₂²×C₄×D₈₉	1	24	yes	—
175	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
176	C₂²	—	89	48	no	—
177	C₂²	—	89	64	no	—
178	C₂²	—	89	8	no	—
179	C₂²	C₂×C₄²×D₈₉	1	4	yes	—
180	C₂²	#117×D₈₉	1	24	yes	—
181	C₂²	C₂³×C₄×D₈₉	1	3	yes	—
182	C₂²	C₂×#41×D₈₉	1	4	yes	—
183	C₄	—	2	24	no	—
184	C₄	—	89	8	no	—
185	C₄	—	178	24	no	—
186	C₄	C₂×C₄²×D₈₉	1	8	yes	—
187	C₂	—	89	16	no	—
188	C₂	#376×D₈₉	1	8	yes	direct product factor
189	C₂	C₂²×C₄²×D₈₉	1	1	yes	characteristic
190	C₂	C₂×#117×D₈₉	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×#376×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂⁴×D₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	C₂²×#376
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. C₂×#19151444	C₂	semidirect
3. C₂×#19151443	C₂	semidirect
4. #376×D₈₉	C₂	direct (N × H)
6. C₂×#376×C₈₉	C₂	semidirect
11. C₂×C₄²×D₈₉	C₄	semidirect
13. #19151444	C₂²	semidirect
14. #19151443	C₂²	semidirect
18. #376×C₈₉	C₂²	semidirect
28. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
29. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
32. C₄²×D₈₉	C₂×C₄	semidirect
36. C₂×C₄²×C₈₉	C₂×C₄	semidirect
52. C₄×Dic₈₉	C₂²×C₄	semidirect
53. C₄×Dic₈₉	C₂²×C₄	semidirect
60. C₄²×C₈₉	C₂²×C₄	semidirect
84. C₂×D₈₉	#376	direct (N × H)
92. C₂×C₈₉	C₂×#376	semidirect
95. D₈₉	C₂×#376	direct (N × H)
100. C₂×#376	D₈₉	direct (N × H)
102. C₈₉	C₂²×#376	semidirect
112. #376	C₂×D₈₉	direct (N × H)
131. C₂×C₄²	C₄×D₈₉	semidirect
156. C₄²	C₂×C₄×D₈₉	semidirect
188. C₂	#376×D₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×D₈₉	2
C₂×#19151444	2
C₂×#19151443	2
#376×D₈₉	2
C₂×#95×D₈₉	2
C₂×#376×C₈₉	2
C₂²×#376	89

Subgroups — C2×#376×D89 #36712596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#376×D₈₉ #36712596