Finite Group Databasebeta

← #36364458

Subgroups — #36364458

Order 22144.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012062C212yesmaximal
2#26011833C214yesmaximal, complemented
3#26011749C212yesmaximal, complemented
4#26011732C212yesmaximal, complemented
5#26011649C214yesmaximal, complemented
6#408×C173C211yesmaximal, characteristic
7C22×Dic346C2212yes—
8C2×Dic692C2212yes—
9C2×Dic692C2212yes—
10#19149677—22no—
11#19149666—22no—
12#19149660—22no—
13#19149660—22no—
14#19149658C2214yes—
15#19149658C2214yes—
16#19149638C2212yes—
17#19149613C2212yes—
18#19149611C2212yes—
19#19149604C2214yes—
20#19149604C2214yes—
21C2×#41×C173C2211yescharacteristic
22C2×M16×C173C2211yescharacteristic
23C22×C8×C173C2211yescharacteristic
24C22⋊C8×C173C2214yes—
25C2×Dic346—22no—
26C2×Dic346—42no—
27C2×Dic346—42no—
28C2×Dic346C2314yes—
29C22×Dic173—22no—
30Dic692—42no—
31Dic692—42no—
32C4⋊Dic173C2314yes—
33#14583805—24no—
34#14583805—24no—
35C22⋊Dic173—22no—
36C22⋊Dic173—24no—
37C22⋊Dic173—24no—
38C4×Dic173—22no—
39#41×C173—41no—
40#41×C173—41no—
41C2×Q8×C173D411yescharacteristic
42C2×D4×C173D411yescharacteristic
43C2×D4×C173D412yes—
44C22×C4×C173C2311yescharacteristic
45C22×C4×C173C2312yes—
46M16×C173—41no—
47C2×C8×C173—41no—
48C2×C8×C173C2312yes—
49C2×C8×C173C2312yes—
50Dic346—24no—
51Dic346—24no—
52Dic346—44no—
53C2×Dic173—24no—
54C2×Dic173—24no—
55C2×Dic173—44no—
56C23×C173C2×D412yes—
57C23×C173C2×D411yescharacteristic
58Q8×C173—41no—
59D4×C173—41no—
60D4×C173—42no—
61C2×C4×C173—22no—
62C2×C4×C173—22no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173C2×D412yes—
66C2×C4×C173C2×D411yescharacteristic
67C2×C4×C173C2411yescharacteristic
68C8×C173—22no—
69C8×C173—22no—
70C22×C173—21no—
71C22×C173—22no—
72C22×C173—24no—
73C22×C173—22no—
74C22×C173C22×D411yescharacteristic
75C4×C173—21no—
76C4×C173—22no—
77C4×C173H(5,𝔽2)11yescharacteristic
78C4×C173H(5,𝔽2)11yescharacteristic
79Dic173—44no—
80Dic173—44no—
81C2×C173—41no—
82C2×C173—42no—
83C2×C173#53411yescharacteristic
84C2×C173#57812yes—
85C173#320211yescharacteristic, complemented, Sylow
86#3202—1731nomaximal, Sylow
87#536—1732no—
88#485—1734no—
89#470—1732no—
90#467—1732no—
91#451—1734no—
92#408D17311yescharacteristic
93C2×#41C2×D17311yescharacteristic
94C22×Q8—1732no—
95C2×Q16—1732no—
96C2×Q16—1732no—
97C2×M16C2×D17311yescharacteristic
98C22×C8C2×D17311yescharacteristic
99C4⋊Q8—3462no—
100#124—3462no—
101#123—3462no—
102#123—3462no—
103C22⋊Q8—1734no—
104C22⋊Q8—1734no—
105#117—1732no—
106#107—1732no—
107#106—1732no—
108#103—1734no—
109#103—1734no—
110C22⋊C8C2×D17314yes—
111#41—41no—
112#41—41no—
113C2×Q8—3462no—
114C2×Q8—1734no—
115C2×Q8—6922no—
116C2×Q8—6922no—
117C2×Q8D69211yescharacteristic
118C2×D4D69211yescharacteristic
119C2×D4D69212yes—
120C22×C4—3462no—
121C22×C4C22×D17312yes—
122C22×C4C22×D17311yescharacteristic
123Q16—6922no—
124Q16—6922no—
125M16—41no—
126C2×C8—41no—
127C2×C8C22×D17312yes—
128C2×C8C22×D17312yes—
129C4⋊C4—3464no—
130C4⋊C4—3464no—
131C4⋊C4—1734no—
132C22⋊C4—3462no—
133C22⋊C4—3464no—
134C22⋊C4—3464no—
135C42—3462no—
136C23C2×D69211yescharacteristic
137C23C2×D69212yes—
138Q8—41no—
139Q8—3464no—
140Q8—3464no—
141Q8—6924no—
142D4—42no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—3464no—
149C2×C4—3464no—
150C2×C4—6924no—
151C2×C4C2×D69212yes—
152C2×C4C2×D69211yescharacteristic
153C2×C4C23×D17311yescharacteristic
154C8—22no—
155C8—22no—
156C22—22no—
157C22—24no—
158C22—22no—
159C22—21no—
160C22C22×D69211yescharacteristic
161C4—22no—
162C4—21no—
163C4—6924no—
164C4—6924no—
165C4#1914972411yescharacteristic
166C4#1914972411yescharacteristic
167C2—42no—
168C2—41no—
169C2#2601205311yescharacteristic
170C2#2601230812yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#408×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×C173C22
hypercenter#408D173
hypocenterC173#3202
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #26011833C2semidirect
3. #26011749C2semidirect
4. #26011732C2semidirect
5. #26011649C2semidirect
85. C173#3202semidirect

Maximal subgroups

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

subgroupindex
#260120622
#260118332
#260117492
#260117322
#260116492
#408×C1732
#3202173