Finite Group Databasebeta

← #36352746

Subgroups — #36352746

Order 22144.

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

normal subgroups only

This group has 212 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#26012227C211yesmaximal, characteristic
2#26011773C211yesmaximal, characteristic, complemented
3#26011698C211yesmaximal, characteristic, complemented
4#26011235C211yesmaximal, characteristic
5#26011223C211yesmaximal, characteristic, complemented
6#26011223C211yesmaximal, characteristic, complemented
7#486×C173C211yesmaximal, characteristic
8C2×#14583816—21no—
9#19149666—21no—
10#19149666—21no—
11#19149660—22no—
12C4×Dic346C2211yescharacteristic
13C4×Dic346C2211yescharacteristic
14D4×Dic173—21no—
15D4×Dic173—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149603—21no—
20#19149602—21no—
21C4⋊Q8×C173C2211yescharacteristic
22#127×C173C2211yescharacteristic
23#102×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×Dic346D411yescharacteristic
26C2×Dic346D411yescharacteristic
27C22×Dic173—22no—
28#14583816—42no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173D411yescharacteristic
32C4⋊Dic173D411yescharacteristic
33#14583805—21no—
34#14583805—21no—
35C22⋊Dic173—22no—
36C22⋊Dic173—22no—
37C22⋊Dic173—22no—
38C4×Dic173—21no—
39C4×Dic173—21no—
40C2×Q8×C173—21no—
41C2×D4×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173D412yes—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C42×C173C2311yescharacteristic
48Dic346—21no—
49Dic346—21no—
50Dic346—21no—
51Dic346—21no—
52C2×Dic173—21no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—82no—
57#13063436—41no—
58#13063436—41no—
59C23×C173—22no—
60Q8×C173—41no—
61Q8×C173—41no—
62D4×C173—42no—
63D4×C173—41no—
64D4×C173—41no—
65D4×C173—41no—
66D4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—41no—
72C22×C173—42no—
73C22×C173—42no—
74C22×C173—42no—
75C22×C173#12011yescharacteristic
76C4×C173—21no—
77C4×C173—81no—
78C4×C173#13511yescharacteristic
79C4×C173#13511yescharacteristic
80C4×C173AGL(1,ℤ/8)11yescharacteristic
81C4×C173#13711yescharacteristic
82Dic173—41no—
83Dic173—41no—
84Dic173—41no—
85Dic173—41no—
86C2×C173—82no—
87C2×C173#44911yescharacteristic
88C2×C173#45211yescharacteristic
89C2×C173#45511yescharacteristic
90C173#164011yescharacteristic, complemented, Sylow
91#1640—1731nomaximal, Sylow
92#559—1731no—
93#486D17311yescharacteristic
94#475—1731no—
95#462—1731no—
96#331—1731no—
97Q8⋊C8—1731no—
98Q8⋊C8—1731no—
99C2×Q16—3461no—
100C4⋊Q8C2×D17311yescharacteristic
101#127C2×D17311yescharacteristic
102#124—3461no—
103#124—3461no—
104#123—3462no—
105C4×Q8—1731no—
106C4×Q8—1731no—
107C4×D4—3461no—
108C4×D4—3461no—
109#106—3461no—
110C4⋊C8—1731no—
111C4⋊C8—1731no—
112#103—3461no—
113#103—3461no—
114#102—22no—
115C4×C8C2×D17311yescharacteristic
116C2×Q8—21no—
117C2×Q8—1731no—
118C2×Q8—1731no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D17312yes—
122C22×C4—3462no—
123Q16—6922no—
124C2×C8—21no—
125C2×C8—3461no—
126C2×C8—3461no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—1731no—
132C4⋊C4—1731no—
133C22⋊C4—3462no—
134C22⋊C4—3462no—
135C22⋊C4—3462no—
136C42—3461no—
137C42—3461no—
138C42C22×D17311yescharacteristic
139C23—22no—
140Q8—41no—
141Q8—41no—
142Q8—3461no—
143Q8—3461no—
144Q8—3461no—
145Q8—3461no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—41no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—13842no—
157C2×C4D4×D17311yescharacteristic
158C2×C4D4×D17311yescharacteristic
159C2×C4C2×C22⋊D17311yescharacteristic
160C8—41no—
161C8—6921no—
162C8—6921no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1914964711yescharacteristic
167C4—21no—
168C4—81no—
169C4—6921no—
170C4—6921no—
171C4—6921no—
172C4—6921no—
173C4#1914969811yescharacteristic
174C4#1914970011yescharacteristic
175C4#1914970311yescharacteristic
176C4#1914970811yescharacteristic
177C2—82no—
178C2#2601163311yescharacteristic
179C2#2601165011yescharacteristic
180C2#2601166811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#486×C173C2
socleC22×C173#120
prime radical#127×C173C22
hypercenter#486D173
hypocenterC173#1640
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #26011773C2semidirect
3. #26011698C2semidirect
5. #26011223C2semidirect
6. #26011223C2semidirect
90. C173#1640semidirect

Maximal subgroups

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

subgroupindex
#260122272
#260117732
#260116982
#260112352
#260112232
#486×C1732
#1640173