Subgroups — C₂²×#19151464 #36712703

Order 22784.

4286 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
2	C₂×#19151464	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#15667919	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₈₉	C₂²	1	4	yes	—
6	C₂²×C₄⋊Dic₈₉	—	2	16	no	—
7	C₂²×C₄⋊Dic₈₉	C₂²	1	24	yes	—
8	#19151464	C₂²	1	64	yes	direct product factor
9	C₂×#15667919	C₂²	1	48	yes	—
10	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₈₉	C₄	1	16	yes	complemented
13	C₂²×C₄²×C₈₉	C₂²	1	12	yes	—
14	C₂³×Dic₈₉	—	2	16	no	—
15	C₂³×Dic₈₉	—	2	32	no	—
16	C₂³×Dic₈₉	C₂³	1	24	yes	—
17	C₂×C₄⋊Dic₈₉	—	2	96	no	—
18	C₂×C₄⋊Dic₈₉	C₂³	1	32	yes	—
19	#15667919	C₂³	1	64	yes	—
20	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₈₉	—	2	16	no	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₈₉	D₄	1	8	yes	—
25	C₂³×C₄×C₈₉	Q₈	1	8	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
27	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	96	yes	complemented
29	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
30	C₂²×Dic₈₉	—	2	48	no	—
31	C₂²×Dic₈₉	—	2	96	no	—
32	C₂²×Dic₈₉	—	4	64	no	—
33	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
34	C₄⋊Dic₈₉	—	2	64	no	—
35	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
36	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
37	C₂⁴×C₈₉	C₂×Q₈	1	4	yes	—
38	C₂⁴×C₈₉	#41	1	16	yes	—
39	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
40	C₂²×C₄×C₈₉	—	2	64	no	—
41	C₂²×C₄×C₈₉	—	2	48	no	—
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	64	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	48	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
45	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
46	C₂²×C₄×C₈₉	C₂×Q₈	1	24	yes	—
47	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
49	C₄²×C₈₉	C₂²×C₄	1	64	yes	complemented
50	C₂³×C₈₉	C₂×C₄⋊C₄	1	16	yes	—
51	C₂³×C₈₉	#117	1	32	yes	—
52	C₂³×C₈₉	#124	1	32	yes	—
53	C₂³×C₈₉	#125	1	32	yes	—
54	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
55	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
56	C₂³×C₈₉	C₂²×Q₈	1	6	yes	—
57	C₂³×C₈₉	C₂×#41	1	24	yes	—
58	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	96	no	—
60	C₂×C₄×C₈₉	—	2	16	no	—
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	96	yes	—
62	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
63	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
64	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
65	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
66	C₂×Dic₈₉	—	2	16	no	—
67	C₂×Dic₈₉	—	2	32	no	—
68	C₂×Dic₈₉	—	4	96	no	—
69	C₂²×C₈₉	#382	1	64	yes	complemented
70	C₂²×C₈₉	C₂²×C₄⋊C₄	1	12	yes	—
71	C₂²×C₈₉	C₂×#117	1	24	yes	—
72	C₂²×C₈₉	C₂×#124	1	24	yes	—
73	C₂²×C₈₉	C₂×#125	1	24	yes	—
74	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
75	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
76	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
77	C₂²×C₈₉	C₂²×#41	1	4	yes	—
78	C₄×C₈₉	—	2	16	no	—
79	C₄×C₈₉	C₂²×C₄⋊C₄	1	16	yes	—
80	Dic₈₉	—	4	16	no	—
81	C₂²×#382	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	C₂×#382	1	24	yes	complemented
83	C₂×C₈₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₈₉	C₂²×#117	1	2	yes	—
85	C₂×C₈₉	C₂²×#124	1	2	yes	—
86	C₂×C₈₉	C₂²×#125	1	2	yes	—
87	C₂³×C₄⋊C₄	—	89	2	no	—
88	C₂³×C₄²	D₈₉	1	1	yes	characteristic
89	C₂×#382	—	89	24	no	—
90	C₂²×#95	—	89	4	no	—
91	C₈₉	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴×C₄	—	89	4	no	—
93	C₂⁴×C₄	C₂×D₈₉	1	2	yes	—
94	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
95	C₂²×C₄⋊C₄	—	89	24	no	—
96	C₂²×C₄⋊C₄	—	178	16	no	—
97	C₂²×C₄²	Dic₈₉	1	16	yes	complemented
98	C₂²×C₄²	C₂×D₈₉	1	12	yes	—
99	#382	—	89	64	no	—
100	C₂×#95	—	89	48	no	—
101	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
102	C₂³×C₄	—	2	16	no	—
103	C₂³×C₄	—	89	24	no	—
104	C₂³×C₄	—	178	16	no	—
105	C₂³×C₄	—	178	32	no	—
106	C₂³×C₄	Dic₁₇₈	1	8	yes	—
107	C₂³×C₄	D₃₅₆	1	8	yes	—
108	C₂³×C₄	C₂×Dic₈₉	1	16	yes	—
109	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
110	C₂³×C₄	C₂²×D₈₉	1	12	yes	—
111	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
112	C₂×C₄⋊C₄	—	89	32	no	—
113	C₂×C₄⋊C₄	—	178	96	no	—
114	C₂×C₄²	C₂×Dic₈₉	1	96	yes	complemented
115	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
116	#95	—	89	64	no	—
117	C₂⁴	C₂²×Dic₈₉	1	4	yes	—
118	C₂⁴	C₂×D₃₅₆	1	4	yes	—
119	C₂⁴	C₂×Dic₁₇₈	1	4	yes	—
120	C₂⁴	#13083574	1	16	yes	—
121	C₂⁴	C₂³×D₈₉	1	3	yes	—
122	C₂²×C₄	—	2	64	no	—
123	C₂²×C₄	—	2	48	no	—
124	C₂²×C₄	—	89	16	no	—
125	C₂²×C₄	—	356	64	no	—
126	C₂²×C₄	—	178	48	no	—
127	C₂²×C₄	—	178	96	no	—
128	C₂²×C₄	C₄⋊Dic₈₉	1	64	yes	—
129	C₂²×C₄	C₂²×Dic₈₉	1	48	yes	—
130	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
131	C₂²×C₄	C₂×D₃₅₆	1	24	yes	—
132	C₂²×C₄	C₂×Dic₁₇₈	1	24	yes	—
133	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
134	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
135	C₄⋊C₄	—	178	64	no	—
136	C₄²	C₂²×Dic₈₉	1	64	yes	complemented
137	C₂³	C₂×C₄⋊Dic₈₉	1	16	yes	—
138	C₂³	#15667971	1	32	yes	—
139	C₂³	#15668000	1	32	yes	—
140	C₂³	#15668004	1	32	yes	—
141	C₂³	C₂³×Dic₈₉	1	6	yes	—
142	C₂³	C₂²×D₃₅₆	1	6	yes	—
143	C₂³	C₂²×Dic₁₇₈	1	6	yes	—
144	C₂³	C₂×#13083574	1	24	yes	—
145	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
146	C₂×C₄	—	2	16	no	—
147	C₂×C₄	—	2	96	no	—
148	C₂×C₄	—	356	96	no	—
149	C₂×C₄	—	178	16	no	—
150	C₂×C₄	—	178	32	no	—
151	C₂×C₄	C₂×C₄⋊Dic₈₉	1	96	yes	—
152	C₂×C₄	C₂³×Dic₈₉	1	16	yes	—
153	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
154	C₂×C₄	C₂²×D₃₅₆	1	8	yes	—
155	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
156	C₂²	#19151464	1	64	yes	direct product factor
157	C₂²	C₂²×C₄⋊Dic₈₉	1	12	yes	—
158	C₂²	C₂×#15667971	1	24	yes	—
159	C₂²	C₂×#15668000	1	24	yes	—
160	C₂²	C₂×#15668004	1	24	yes	—
161	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
162	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
163	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
164	C₂²	C₂²×#13083574	1	4	yes	—
165	C₄	—	2	16	no	—
166	C₄	—	356	16	no	—
167	C₄	C₂²×C₄⋊Dic₈₉	1	16	yes	—
168	C₂	C₂×#19151464	1	24	yes	direct product factor
169	C₂	C₂³×C₄⋊Dic₈₉	1	1	yes	characteristic
170	C₂	C₂²×#15667971	1	2	yes	—
171	C₂	C₂²×#15668000	1	2	yes	—
172	C₂	C₂²×#15668004	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄²×C₈₉	C₂
socle, prime radical	C₂⁵×C₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂³×C₄²	D₈₉
hypocenter	C₈₉	C₂²×#382
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#19151464	C₂	direct (N × H)
8. #19151464	C₂²	direct (N × H)
12. C₂²×C₄²×C₈₉	C₄	semidirect
28. C₂×C₄²×C₈₉	C₂×C₄	semidirect
49. C₄²×C₈₉	C₂²×C₄	semidirect
69. C₂²×C₈₉	#382	semidirect
82. C₂×C₈₉	C₂×#382	semidirect
91. C₈₉	C₂²×#382	semidirect
97. C₂²×C₄²	Dic₈₉	semidirect
114. C₂×C₄²	C₂×Dic₈₉	semidirect
136. C₄²	C₂²×Dic₈₉	semidirect
156. C₂²	#19151464	direct (N × H)
168. C₂	C₂×#19151464	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₈₉	2
C₂×#19151464	2
C₂²×#15667919	2
C₂³×C₄²×C₈₉	2
C₂²×#382	89

Subgroups — C22×#19151464 #36712703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151464 #36712703