Subgroups — #36442128

Order 22784.

190 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	C₂²×#15667918	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
3	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667946	—	2	1	no	—
5	C₂×#15667918	—	2	1	no	—
6	C₂×#15667918	—	2	1	no	—
7	C₂×#15667918	C₄	1	2	yes	complemented
8	#19151424	—	2	1	no	—
9	#19151424	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₈₉	—	2	1	no	—
12	#366×C₈₉	—	2	1	no	—
13	#366×C₈₉	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₈₉	—	4	1	no	—
15	#15667946	—	2	2	no	—
16	#15667946	—	4	1	no	—
17	#15667918	—	2	2	no	—
18	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
20	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
26	#107×C₈₉	—	2	2	no	—
27	#107×C₈₉	—	4	1	no	—
28	C₂²×Dic₈₉	—	8	1	no	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	C₄⋊Dic₈₉	—	8	1	no	—
31	C₂²×C₄×C₈₉	—	8	1	no	—
32	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
38	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
39	C₄⋊C₄×C₈₉	—	4	2	no	—
40	C₄⋊C₄×C₈₉	—	8	1	no	—
41	C₁₆×C₈₉	C₄²	1	1	yes	characteristic
42	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	—	16	1	no	—
48	C₂×C₄×C₈₉	—	16	1	no	—
49	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
52	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₈₉	Q₃₂	1	1	yes	characteristic
55	C₂×Dic₈₉	—	8	2	no	—
56	C₂×Dic₈₉	—	16	1	no	—
57	C₂×Dic₈₉	—	16	1	no	—
58	C₈×C₈₉	#95	1	1	yes	characteristic
59	C₈×C₈₉	#103	1	1	yes	characteristic
60	C₈×C₈₉	#103	1	1	yes	characteristic
61	C₈×C₈₉	#106	1	1	yes	characteristic
62	C₂²×C₈₉	#357	1	1	yes	characteristic
63	C₂²×C₈₉	#357	1	1	yes	characteristic
64	C₂²×C₈₉	#366	1	1	yes	characteristic
65	C₂²×C₈₉	D₃₂	1	1	yes	characteristic
66	C₂²×C₈₉	SD₆₄	1	1	yes	characteristic
67	C₂²×C₈₉	SD₆₄	1	1	yes	characteristic
68	C₂²×C₈₉	Q₆₄	1	1	yes	characteristic
69	C₄×C₈₉	—	16	2	no	—
70	C₄×C₈₉	#340	1	1	yes	characteristic
71	C₄×C₈₉	#358	1	1	yes	characteristic
72	C₄×C₈₉	#358	1	1	yes	characteristic
73	C₄×C₈₉	#367	1	1	yes	characteristic
74	Dic₈₉	—	16	2	no	—
75	#7471	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	#1380	1	1	yes	characteristic
77	C₂×C₈₉	#1415	1	1	yes	characteristic
78	C₂×C₈₉	#1415	1	1	yes	characteristic
79	C₂×C₈₉	#1416	1	1	yes	characteristic
80	C₂×C₈₉	#1416	1	1	yes	characteristic
81	C₂×C₈₉	#1423	1	1	yes	characteristic
82	C₂×C₈₉	#1424	1	1	yes	characteristic
83	C₂²×C₃₂	—	89	1	no	—
84	C₂×#366	—	89	1	no	—
85	C₂×#366	D₈₉	1	1	yes	characteristic
86	C₈₉	#7471	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	C₂×#107	—	2	1	no	—
89	C₂×#107	—	178	1	no	—
90	C₂×C₃₂	—	89	2	no	—
91	C₂×C₃₂	—	178	1	no	—
92	C₂×C₃₂	—	178	1	no	—
93	#366	—	2	1	no	—
94	#366	—	89	2	no	—
95	#366	—	178	1	no	—
96	#366	Dic₈₉	1	2	yes	complemented
97	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	356	1	no	—
100	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
103	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
106	#107	—	2	2	no	—
107	#107	—	4	1	no	—
108	#107	—	356	1	no	—
109	#107	—	178	2	no	—
110	C₃₂	—	178	2	no	—
111	C₂²×C₄	—	8	1	no	—
112	C₂²×C₄	—	712	1	no	—
113	C₂²×C₄	#13083559	1	1	yes	characteristic
114	C₂×C₈	#13083551	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
116	C₂×C₈	#13083553	1	1	yes	characteristic
117	C₂×C₈	#13083561	1	1	yes	characteristic
118	C₂×C₈	#13083562	1	1	yes	characteristic
119	C₂×C₈	#13083564	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	356	2	no	—
123	C₄⋊C₄	—	712	1	no	—
124	C₁₆	C₄×Dic₈₉	1	1	yes	characteristic
125	C₁₆	#13083551	1	1	yes	characteristic
126	C₁₆	#13083553	1	1	yes	characteristic
127	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
128	C₂³	#15667953	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	1424	1	no	—
133	C₂×C₄	—	1424	1	no	—
134	C₂×C₄	—	712	2	no	—
135	C₂×C₄	#15667932	1	1	yes	characteristic
136	C₂×C₄	#15667933	1	1	yes	characteristic
137	C₂×C₄	#15667945	1	1	yes	characteristic
138	C₂×C₄	#15667955	1	1	yes	characteristic
139	C₂×C₄	#15667956	1	1	yes	characteristic
140	C₂×C₄	#15667958	1	1	yes	characteristic
141	C₈	#15667919	1	1	yes	characteristic
142	C₈	#15667935	1	1	yes	characteristic
143	C₈	#15667936	1	1	yes	characteristic
144	C₈	#15667943	1	1	yes	characteristic
145	C₂²	#19151398	1	1	yes	characteristic
146	C₂²	#19151399	1	1	yes	characteristic
147	C₂²	#19151423	1	1	yes	characteristic
148	C₂²	#19151433	1	1	yes	characteristic
149	C₂²	#19151435	1	1	yes	characteristic
150	C₂²	#19151436	1	1	yes	characteristic
151	C₂²	#19151438	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	1424	2	no	—
154	C₄	#19151360	1	1	yes	characteristic
155	C₄	#19151401	1	1	yes	characteristic
156	C₄	#19151402	1	1	yes	characteristic
157	C₄	#19151425	1	1	yes	characteristic
158	C₂	#26018343	1	1	yes	characteristic
159	C₂	#26018429	1	1	yes	characteristic
160	C₂	#26018430	1	1	yes	characteristic
161	C₂	#26018432	1	1	yes	characteristic
162	C₂	#26018433	1	1	yes	characteristic
163	C₂	#26018451	1	1	yes	characteristic
164	C₂	#26018453	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667953
commutator	C₁₆×C₈₉	C₄²
Frattini	C₂²×C₁₆	C₂×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	D₁₆
cosocle	C₂²×C₁₆×C₈₉	C₂²
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	#7471
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
7. C₂×#15667918	C₄	semidirect
9. #19151424	C₄	semidirect
13. #366×C₈₉	C₄	semidirect
86. C₈₉	#7471	semidirect
96. #366	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667918	2
C₂×#19151424	2
C₂×#366×C₈₉	2
#7471	89