Subgroups — #36554579

Order 22784.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	2	yes	maximal
2	#26020369	C₂	1	8	yes	maximal, complemented
3	#26019946	C₂	1	4	yes	maximal
4	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	4	yes	—
6	#19151622	—	2	8	no	—
7	C₂×C₄⋊#10623494	C₂²	1	8	yes	—
8	#19151353	C₂²	1	16	yes	—
9	C₂²×C₄²×C₈₉	C₄	1	4	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₄³×C₈₉	C₄	1	2	yes	—
13	C₄³×C₈₉	C₄	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	4	yes	—
15	C₂×#13083558	—	2	8	no	—
16	C₂×#13083558	—	2	16	no	—
17	C₂²×#10623494	C₂³	1	8	yes	—
18	C₄⋊#10623494	—	2	16	no	—
19	C₂³×C₄×C₈₉	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	—	2	8	no	—
23	C₂×C₄²×C₈₉	—	2	16	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₈₉	D₄	1	4	yes	—
31	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
34	#13083558	—	4	16	no	—
35	C₂×#10623494	—	2	8	no	—
36	C₂×#10623494	—	2	16	no	—
37	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	—	2	8	no	—
39	C₂²×C₄×C₈₉	—	2	16	no	—
40	C₂²×C₄×C₈₉	—	2	8	no	—
41	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
48	C₂²×C₄×C₈₉	#41	1	8	yes	—
49	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₄²×C₈₉	—	2	8	no	—
51	C₄²×C₈₉	—	2	16	no	—
52	C₄²×C₈₉	—	2	8	no	—
53	C₄²×C₈₉	—	2	8	no	—
54	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
55	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
56	C₄²×C₈₉	M₁₆	1	16	yes	complemented
57	C₄²×C₈₉	C₂×D₄	1	4	yes	—
58	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
59	C₂³×C₈₉	—	2	4	no	—
60	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₂³×C₈₉	#117	1	4	yes	—
62	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	32	no	—
64	C₂×C₄×C₈₉	—	2	8	no	—
65	C₂×C₄×C₈₉	—	2	4	no	—
66	C₂×C₄×C₈₉	—	2	8	no	—
67	C₂×C₄×C₈₉	—	2	16	no	—
68	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₈₉	#117	1	4	yes	—
72	C₂×C₄×C₈₉	#124	1	8	yes	—
73	C₂×C₄×C₈₉	#125	1	8	yes	—
74	C₂×C₄×C₈₉	C₂×M₁₆	1	16	yes	—
75	C₂×C₄×C₈₉	C₂×M₁₆	1	8	yes	—
76	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
79	#10623494	—	4	8	no	—
80	C₂²×C₈₉	—	2	4	no	—
81	C₂²×C₈₉	—	2	8	no	—
82	C₂²×C₈₉	#382	1	4	yes	—
83	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×#117	1	2	yes	—
85	C₂²×C₈₉	C₂²×M₁₆	1	4	yes	—
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	—	2	16	no	—
88	C₄×C₈₉	#382	1	4	yes	—
89	C₄×C₈₉	#423	1	8	yes	—
90	C₄×C₈₉	C₂×#124	1	2	yes	—
91	C₄×C₈₉	C₂×#125	1	2	yes	—
92	#20752	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	—	2	4	no	—
94	C₂×C₈₉	C₂×#382	1	1	yes	characteristic
95	C₂×C₈₉	C₂×#423	1	2	yes	—
96	C₂×C₈₉	#2920	1	4	yes	—
97	C₂×#423	—	89	2	no	—
98	C₂×C₄³	D₈₉	1	1	yes	characteristic
99	#1831	—	89	8	no	—
100	#1728	—	89	4	no	—
101	C₈₉	#20752	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	89	4	no	—
103	C₂²×C₄²	Dic₈₉	1	4	yes	—
104	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
105	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
106	#423	—	178	8	no	—
107	C₂×C₄⋊C₈	—	89	8	no	—
108	C₄³	Dic₈₉	1	2	yes	—
109	C₄³	Dic₈₉	1	2	yes	—
110	C₄³	C₂×D₈₉	1	4	yes	—
111	#336	—	89	16	no	—
112	C₂³×C₄	C₂×Dic₈₉	1	4	yes	—
113	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
114	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
115	C₂×M₁₆	—	178	8	no	—
116	C₂×M₁₆	—	178	16	no	—
117	C₂²×C₈	—	89	8	no	—
118	C₂×C₄²	—	2	8	no	—
119	C₂×C₄²	—	2	16	no	—
120	C₂×C₄²	Dic₁₇₈	1	4	yes	—
121	C₂×C₄²	D₃₅₆	1	4	yes	—
122	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
123	C₂×C₄²	C₂×Dic₈₉	1	8	yes	—
124	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
125	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
128	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
129	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
130	C₄⋊C₈	—	178	16	no	—
131	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	—	2	16	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₄⋊Dic₈₉	1	8	yes	—
136	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
137	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
140	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
141	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
142	C₂²×C₄	#13083574	1	8	yes	—
143	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
144	M₁₆	—	356	16	no	—
145	C₂×C₈	—	178	8	no	—
146	C₂×C₈	—	178	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	16	no	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₄⋊Dic₈₉	1	4	yes	—
152	C₄²	C₄⋊Dic₈₉	1	4	yes	—
153	C₄²	#13083558	1	16	yes	complemented
154	C₄²	C₂×D₃₅₆	1	4	yes	—
155	C₄²	C₂×Dic₁₇₈	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
158	C₂³	#15667971	1	4	yes	—
159	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
160	C₂×C₄	—	2	32	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
167	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
168	C₂×C₄	#15667971	1	4	yes	—
169	C₂×C₄	#15668000	1	8	yes	—
170	C₂×C₄	#15668004	1	8	yes	—
171	C₂×C₄	C₂×#13083558	1	16	yes	—
172	C₂×C₄	C₂×#13083558	1	8	yes	—
173	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
174	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
175	C₂×C₄	C₂×#13083574	1	4	yes	—
176	C₈	—	356	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	#19151464	1	4	yes	—
180	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
181	C₂²	C₂×#15667971	1	2	yes	—
182	C₂²	C₂²×#13083558	1	4	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	16	no	—
185	C₄	#19151464	1	4	yes	—
186	C₄	#19151622	1	8	yes	—
187	C₄	C₂×#15668000	1	2	yes	—
188	C₄	C₂×#15668004	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×#19151464	1	1	yes	characteristic
191	C₂	C₂×#19151622	1	2	yes	—
192	C₂	#26028885	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle	C₂³×C₈₉	C₂³×C₄
Fitting	C₂×C₄³×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	#20752
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #26020369	C₂	semidirect
56. C₄²×C₈₉	M₁₆	semidirect
101. C₈₉	#20752	semidirect
153. C₄²	#13083558	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
#26020369	2
#26019946	2
C₂×C₄³×C₈₉	2
#20752	89