Subgroups — #36554583

Order 22784.

926 conjugacy classes of subgroups.

The subgroup lattice has 166 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	#26020372	C₂	1	8	yes	maximal, complemented
3	#26019946	C₂	1	4	yes	maximal
4	C₂×#376×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	C₂²	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	#376×C₈₉	C₄	1	4	yes	—
12	#376×C₈₉	C₂²	1	4	yes	—
13	C₂×#95×C₈₉	C₄	1	4	yes	—
14	C₂×#13083558	—	2	8	no	—
15	C₂×#13083558	—	2	16	no	—
16	C₂²×#10623494	C₂³	1	8	yes	—
17	C₄⋊#10623494	—	2	16	no	—
18	C₂³×C₄×C₈₉	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	—	2	8	no	—
22	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₈₉	D₄	1	4	yes	—
25	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
26	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
27	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
28	#95×C₈₉	C₂×C₄	1	16	yes	—
29	#13083558	—	4	16	no	—
30	C₂×#10623494	—	2	8	no	—
31	C₂×#10623494	—	2	16	no	—
32	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	—	2	8	no	—
34	C₂²×C₄×C₈₉	—	4	4	no	—
35	C₂²×C₄×C₈₉	—	4	8	no	—
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₈₉	#41	1	8	yes	—
43	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
44	C₄²×C₈₉	—	2	8	no	—
45	C₄²×C₈₉	—	2	8	no	—
46	C₄²×C₈₉	C₄⋊C₄	1	8	yes	—
47	C₄²×C₈₉	C₂×D₄	1	4	yes	—
48	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
49	C₂³×C₈₉	—	2	4	no	—
50	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
51	C₂³×C₈₉	#117	1	4	yes	—
52	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	2	8	no	—
54	C₂×C₄×C₈₉	—	2	8	no	—
55	C₂×C₄×C₈₉	—	2	16	no	—
56	C₂×C₄×C₈₉	—	4	8	no	—
57	C₂×C₄×C₈₉	—	4	16	no	—
58	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
59	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₈₉	#117	1	4	yes	—
61	C₂×C₄×C₈₉	#124	1	8	yes	—
62	C₂×C₄×C₈₉	#125	1	8	yes	—
63	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
66	#10623494	—	4	8	no	—
67	C₂²×C₈₉	—	2	4	no	—
68	C₂²×C₈₉	—	2	8	no	—
69	C₂²×C₈₉	#382	1	4	yes	—
70	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈₉	C₂×#117	1	2	yes	—
72	C₂²×C₈₉	#568	1	4	yes	—
73	C₄×C₈₉	—	2	8	no	—
74	C₄×C₈₉	—	4	8	no	—
75	C₄×C₈₉	#382	1	4	yes	—
76	C₄×C₈₉	C₂×#124	1	2	yes	—
77	C₄×C₈₉	C₂×#125	1	2	yes	—
78	#20753	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	—	2	4	no	—
80	C₂×C₈₉	C₂×#382	1	1	yes	characteristic
81	C₂×C₈₉	#2905	1	2	yes	—
82	C₂×C₈₉	#2921	1	4	yes	—
83	C₂×#423	—	89	2	no	—
84	C₂×#376	D₈₉	1	1	yes	characteristic
85	#1832	—	89	8	no	—
86	#1728	—	89	4	no	—
87	C₈₉	#20753	1	1	yes	characteristic, complemented, Sylow
88	C₂²×M₁₆	—	89	4	no	—
89	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
90	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
91	#423	—	178	8	no	—
92	C₂×C₄⋊C₈	—	89	8	no	—
93	#376	Dic₈₉	1	4	yes	—
94	#376	C₂×D₈₉	1	4	yes	—
95	C₂×#95	Dic₈₉	1	4	yes	—
96	#336	—	89	16	no	—
97	C₂³×C₄	C₂×Dic₈₉	1	4	yes	—
98	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
99	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	C₂×M₁₆	—	178	8	no	—
101	C₂×M₁₆	—	178	16	no	—
102	C₂²×C₈	—	89	8	no	—
103	C₂×C₄²	—	2	8	no	—
104	C₂×C₄²	Dic₁₇₈	1	4	yes	—
105	C₂×C₄²	D₃₅₆	1	4	yes	—
106	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
107	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
108	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
109	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
110	C₄⋊C₈	—	178	16	no	—
111	#95	C₂×Dic₈₉	1	16	yes	—
112	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	—	4	8	no	—
116	C₂²×C₄	C₄⋊Dic₈₉	1	8	yes	—
117	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
119	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
122	C₂²×C₄	#13083574	1	8	yes	—
123	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
124	M₁₆	—	356	16	no	—
125	C₂×C₈	—	178	8	no	—
126	C₂×C₈	—	178	16	no	—
127	C₄²	—	2	8	no	—
128	C₄²	—	2	8	no	—
129	C₄²	C₄⋊Dic₈₉	1	8	yes	—
130	C₄²	C₂×D₃₅₆	1	4	yes	—
131	C₄²	C₂×Dic₁₇₈	1	4	yes	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
134	C₂³	#15667971	1	4	yes	—
135	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	—	4	16	no	—
141	C₂×C₄	C₂×C₄⋊Dic₈₉	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
143	C₂×C₄	#15667971	1	4	yes	—
144	C₂×C₄	#15668000	1	8	yes	—
145	C₂×C₄	#15668004	1	8	yes	—
146	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#13083574	1	4	yes	—
149	C₈	—	356	8	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	#19151464	1	4	yes	—
153	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
154	C₂²	C₂×#15667971	1	2	yes	—
155	C₂²	#19152365	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	#19151464	1	4	yes	—
159	C₄	C₂×#15668000	1	2	yes	—
160	C₄	C₂×#15668004	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	C₂×#19151464	1	1	yes	characteristic
163	C₂	#26028816	1	2	yes	—
164	C₂	#26028890	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle	C₂³×C₈₉	C₂³×C₄
Fitting	C₂×#376×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#20753
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #26020372	C₂	semidirect
87. C₈₉	#20753	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
#26020372	2
#26019946	2
C₂×#376×C₈₉	2
#20753	89