Finite Group Databasebeta

← #36352830

Subgroups — #36352830

Order 22144.

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

normal subgroups only

This group has 200 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
2#26011768C211yesmaximal, characteristic
3#26011698C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5#26011223C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic
7#498×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9#19149674—22no—
10#19149668—21no—
11#19149659—21no—
12Q8×Dic173—21no—
13C4×Dic346C2211yescharacteristic
14C4×D692C2211yescharacteristic
15#19149637—21no—
16#19149612—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149603—21no—
20#19149599—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#106×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×D692D411yescharacteristic
27C2×C4×D173—21no—
28#14583816—42no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173—22no—
32C4⋊Dic173D411yescharacteristic
33C4⋊Dic173D411yescharacteristic
34#14583805—21no—
35#14583803—22no—
36#14583803—22no—
37#14583803—21no—
38C4×Dic173—22no—
39C4×Dic173—21no—
40C2×Q8×C173—21no—
41C2×Q8×C173—21no—
42C2×C8×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173D412yes—
47C42×C173C2311yescharacteristic
48C22×D173—21no—
49Dic346—21no—
50Dic346—21no—
51D692—21no—
52D692—21no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C4×D173—41no—
57C4×D173—41no—
58#13063436—41no—
59#13063436—41no—
60Q8×C173—41no—
61Q8×C173—41no—
62Q8×C173—41no—
63Q8×C173—41no—
64C2×C4×C173—22no—
65C2×C4×C173—41no—
66C2×C4×C173C2×D411yescharacteristic
67C2×C4×C173C2×D411yescharacteristic
68C2×C4×C173C2×D411yescharacteristic
69C8×C173—22no—
70C22×C173#12011yescharacteristic
71C2×D173—41no—
72C2×D173—41no—
73C2×D173—41no—
74C4×C173—21no—
75C4×C173—42no—
76C4×C173—81no—
77C4×C173#13511yescharacteristic
78C4×C173#13511yescharacteristic
79C4×C173AGL(1,ℤ/8)11yescharacteristic
80C4×C173#13711yescharacteristic
81Dic173—41no—
82Dic173—41no—
83Dic173—41no—
84C2×C173#44911yescharacteristic
85C2×C173#45211yescharacteristic
86C2×C173#45611yescharacteristic
87D173—81no—
88C173#165211yescharacteristic, complemented, Sylow
89#1652—1731nomaximal, Sylow
90#556—1731no—
91#498D17311yescharacteristic, complemented
92#474—1731no—
93#462—1731no—
94#332—1731no—
95Q8⋊C8—1731no—
96D4⋊C8—1731no—
97C2×Q16—3461no—
98C4⋊Q8C2×D17311yescharacteristic
99C4⋊Q8C2×D17311yescharacteristic
100#126—3462no—
101#124—3461no—
102C22⋊Q8—3461no—
103C4×Q8—3461no—
104C4×Q8—1731no—
105C4×D4—1731no—
106#117—3461no—
107#107—3461no—
108#106—22no—
109C4⋊C8—1731no—
110C4⋊C8—1731no—
111#103—3461no—
112#102—3461no—
113C4×C8C2×D17311yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—1731no—
117C2×D4—1731no—
118C22×C4—3461no—
119Q16—6922no—
120C2×C8—21no—
121C2×C8—3461no—
122C2×C8—3461no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3462no—
127C4⋊C4—3461no—
128C4⋊C4—1731no—
129C4⋊C4—1731no—
130C4⋊C4C22⋊D17312yes—
131C22⋊C4—3462no—
132C22⋊C4—3462no—
133C22⋊C4—3461no—
134C42—3462no—
135C42—3461no—
136C42C22×D17311yescharacteristic
137C23—3461no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—3461no—
143Q8—3461no—
144D4—3461no—
145D4—3461no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—3461no—
149C2×C4—3461no—
150C2×C4—3461no—
151C2×C4—6921no—
152C2×C4—6921no—
153C2×C4D4×D17311yescharacteristic
154C2×C4D4×D17311yescharacteristic
155C2×C4C2×C22⋊D17311yescharacteristic
156C8—22no—
157C8—6921no—
158C8—6921no—
159C22—6921no—
160C22—6921no—
161C22—6921no—
162C22#1914964711yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—6921no—
167C4—6921no—
168C4—6921no—
169C4#1914970011yescharacteristic
170C4#1914970011yescharacteristic
171C4#1914970611yescharacteristic
172C4#1914970811yescharacteristic
173C2—13841no—
174C2#2601163511yescharacteristic
175C2#2601165011yescharacteristic
176C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#498×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#498D173
hypocenterC173#1652
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #26011698C2semidirect
4. #26011237C2semidirect
5. #26011223C2semidirect
7. #498×C173C2semidirect
88. C173#1652semidirect
91. #498D173semidirect

Maximal subgroups

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

subgroupindex
#260122152
#260117682
#260116982
#260112372
#260112232
#260112202
#498×C1732
#1652173