Finite Group Databasebeta

← #36352670

Subgroups — #36352670

Order 22144.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012215C211yesmaximal, characteristic, complemented
2#26011688C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic
4#26011237C211yesmaximal, characteristic, complemented
5#26011223C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#486×C173C211yesmaximal, characteristic, complemented
8C2×#14583814—21no—
9C2×#14583811—21no—
10#19149674—22no—
11#19149668—21no—
12#19149659—21no—
13Q8×Dic173—21no—
14C4×Dic346C2211yescharacteristic
15C4×D692C2211yescharacteristic
16#19149637—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149600—21no—
20#19149600—21no—
21C4⋊Q8×C173C2211yescharacteristic
22#127×C173C2211yescharacteristic
23#102×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×D692D411yescharacteristic
27C2×C4×D173—21no—
28#14583814—42no—
29#14583811—42no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C4⋊Dic173—22no—
33C4⋊Dic173D411yescharacteristic
34C4⋊Dic173D411yescharacteristic
35#14583805—21no—
36#14583803—22no—
37#14583803—22no—
38#14583803—21no—
39C4×Dic173—22no—
40C4×Dic173—21no—
41C2×Q8×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173—21no—
45C2×C8×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D412yes—
48C42×C173C2311yescharacteristic
49C22×D173—21no—
50Dic346—21no—
51Dic346—21no—
52D692—21no—
53D692—21no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C4×D173—41no—
58C4×D173—41no—
59#13063436—41no—
60#13063436—41no—
61C23×C173—41no—
62Q8×C173—41no—
63Q8×C173—41no—
64D4×C173—42no—
65D4×C173—41no—
66D4×C173—41no—
67D4×C173—41no—
68D4×C173—41no—
69C2×C4×C173—22no—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—41no—
74C22×C173—81no—
75C22×C173—81no—
76C22×C173—81no—
77C22×C173#12011yescharacteristic
78C2×D173—41no—
79C2×D173—41no—
80C2×D173—41no—
81C4×C173—21no—
82C4×C173—42no—
83C4×C173#13511yescharacteristic
84C4×C173#13511yescharacteristic
85C4×C173AGL(1,ℤ/8)11yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87Dic173—41no—
88Dic173—41no—
89Dic173—41no—
90C2×C173—161no—
91C2×C173#44911yescharacteristic
92C2×C173#44911yescharacteristic
93C2×C173#45411yescharacteristic
94D173—81no—
95C173#162811yescharacteristic, complemented, Sylow
96#1628—1731nomaximal, Sylow
97#556—1731no—
98#486D17311yescharacteristic, complemented
99#460—1731no—
100#459—1731no—
101#332—1731no—
102Q8⋊C8—1731no—
103D4⋊C8—1731no—
104C2×SD16—3461no—
105C2×D8—3461no—
106C4⋊Q8C2×D17311yescharacteristic
107#127C2×D17311yescharacteristic
108#126—3462no—
109#124—3461no—
110C22⋊Q8—3461no—
111C4×Q8—3461no—
112C4×Q8—1731no—
113C4×D4—1731no—
114#117—3461no—
115C4⋊C8—1731no—
116C4⋊C8—1731no—
117#102—22no—
118#102—3461no—
119#102—3461no—
120C4×C8C2×D17311yescharacteristic
121C2×Q8—21no—
122C2×Q8—1731no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1731no—
127C22×C4—3461no—
128SD16—6922no—
129D8—6922no—
130C2×C8—21no—
131C2×C8—3461no—
132C2×C8—3461no—
133C4⋊C4—21no—
134C4⋊C4—3462no—
135C4⋊C4—3461no—
136C4⋊C4—1731no—
137C4⋊C4—1731no—
138C4⋊C4C22⋊D17312yes—
139C22⋊C4—3462no—
140C22⋊C4—3462no—
141C22⋊C4—3461no—
142C42—3462no—
143C42—3461no—
144C42C22×D17311yescharacteristic
145C23—41no—
146C23—3461no—
147Q8—41no—
148Q8—41no—
149Q8—3461no—
150Q8—3461no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3461no—
157D4—3461no—
158C2×C4—22no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—3461no—
162C2×C4—6921no—
163C2×C4—6921no—
164C2×C4D4×D17311yescharacteristic
165C2×C4D4×D17311yescharacteristic
166C2×C4C2×C22⋊D17311yescharacteristic
167C8—41no—
168C8—6921no—
169C8—6921no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—6921no—
174C22—6921no—
175C22—6921no—
176C22#1914964711yescharacteristic
177C4—21no—
178C4—42no—
179C4—6921no—
180C4—6921no—
181C4—6921no—
182C4#1914969811yescharacteristic
183C4#1914970011yescharacteristic
184C4#1914970311yescharacteristic
185C4#1914970611yescharacteristic
186C2—161no—
187C2—13841no—
188C2#2601163311yescharacteristic
189C2#2601163511yescharacteristic
190C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#486×C173C2
socleC22×C173#120
prime radical#26011683C2
hypercenter#486D173
hypocenterC173#1628
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #26012215C2semidirect
2. #26011688C2semidirect
4. #26011237C2semidirect
5. #26011223C2semidirect
6. #26011220C2semidirect
7. #486×C173C2semidirect
95. C173#1628semidirect
98. #486D173semidirect

Maximal subgroups

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

subgroupindex
#260122152
#260116882
#260116832
#260112372
#260112232
#260112202
#486×C1732
#1628173