Finite Group Databasebeta

← #36353947

Subgroups — #36353947

Order 22144.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#26012258C211yesmaximal, characteristic, complemented
2#26011508C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic, complemented
4#26011251C212yesmaximal, complemented
5#532×C173C211yesmaximal, characteristic
6#19149686—24no—
7C2×#14583810—22no—
8C2×#14583810C2212yes—
9#19149638C2212yes—
10#19149606—24no—
11#19149594—22no—
12C2×#41×C173C2211yescharacteristic
13C2×#41×C173C2211yescharacteristic
14C4⋊Q8×C173C411yescharacteristic, complemented
15#127×C173C411yescharacteristic, complemented
16#124×C173C411yescharacteristic, complemented
17#124×C173C411yescharacteristic, complemented
18#121×C173—22no—
19C4×D4×C173—22no—
20C2×C42×C173C2211yescharacteristic
21#14583810—24no—
22#14583810—22no—
23#14583810D414yescomplemented
24C2×#13063436—24no—
25C2×#13063436—22no—
26C4⋊Dic173—22no—
27C22⋊Dic173—22no—
28C4×Dic173—22no—
29#41×C173—22no—
30#41×C173—22no—
31#41×C173—22no—
32#41×C173—22no—
33C2×Q8×C173C2×C411yescharacteristic
34C2×Q8×C173C2×C411yescharacteristic
35C2×D4×C173—22no—
36C2×D4×C173—21no—
37C2×D4×C173C2×C411yescharacteristic
38C2×D4×C173C2×C411yescharacteristic
39C2×D4×C173D412yes—
40C22×C4×C173—21no—
41C22×C4×C173—21no—
42C22×C4×C173D412yes—
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—22no—
45C22⋊C4×C173—22no—
46C22⋊C4×C173—22no—
47C42×C173—21no—
48C42×C173C2×C411yescharacteristic
49C42×C173C2×C411yescharacteristic
50C2×Dic173—42no—
51#13063436—24no—
52C23×C173—21no—
53C23×C173C22⋊C412yes—
54C23×C173C2×D411yescharacteristic
55Q8×C173—22no—
56Q8×C173—22no—
57D4×C173—24no—
58D4×C173—22no—
59D4×C173—22no—
60D4×C173—44no—
61D4×C173—42no—
62C2×C4×C173—21no—
63C2×C4×C173—22no—
64C2×C4×C173—22no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—42no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173C22⋊C412yes—
71C2×C4×C173C22×C411yescharacteristic
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173#4112yes—
75C22×C173—22no—
76C22×C173—41no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—41no—
80C22×C173C2×C22⋊C411yescharacteristic
81C22×C173#12112yes—
82C4×C173—22no—
83C4×C173—22no—
84C4×C173—41no—
85C4×C173—41no—
86C4×C173C4×D412yes—
87C4×C173#12011yescharacteristic
88C4×C173#12311yescharacteristic
89Dic173—82no—
90C2×C173—22no—
91C2×C173—21no—
92C2×C173—42no—
93C2×C173—81no—
94C2×C173#38611yescharacteristic
95C173#188411yescharacteristic, complemented, Sylow
96#1884—1731nomaximal, Sylow
97#568—1731no—
98#532D17311yescharacteristic
99#421—1732no—
100#410—1731no—
101#339—1732no—
102C2×#41C2×D17311yescharacteristic
103C2×#41C2×D17311yescharacteristic
104#131—3464no—
105C2×M16—3462no—
106C2×M16—1732no—
107C4⋊Q8Dic17311yescharacteristic, complemented
108#127Dic17311yescharacteristic, complemented
109#124Dic17311yescharacteristic, complemented
110#124Dic17311yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—1732no—
114C2×C42C2×D17311yescharacteristic
115#104—3464no—
116#99—3462no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic17311yescharacteristic
122C2×Q8C2×Dic17311yescharacteristic
123C2×D4—21no—
124C2×D4—22no—
125C2×D4C2×Dic17311yescharacteristic
126C2×D4C2×Dic17311yescharacteristic
127C2×D4C22⋊D17312yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D17312yes—
131C22×C4C22×D17311yescharacteristic
132M16—3464no—
133M16—3462no—
134M16—1734no—
135C2×C8—3464no—
136C2×C8—3462no—
137C4⋊C4—22no—
138C4⋊C4—3462no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—3462no—
142C42—21no—
143C42—3462no—
144C42C2×Dic17311yescharacteristic
145C42C2×Dic17311yescharacteristic
146C23—21no—
147C23C22⋊Dic17312yes—
148C23C2×C22⋊D17311yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—6922no—
165C2×C4C22⋊Dic17312yes—
166C2×C4C22×Dic17311yescharacteristic
167C2×C4D4×D17312yes—
168C2×C4C2×C22⋊D17311yescharacteristic
169C2×C4#1458382512yes—
170C8—3464no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic17311yescharacteristic
177C22#1914965112yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—13842no—
183C4D4×Dic17312yes—
184C4#1914964711yescharacteristic
185C4#1914966011yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#2601137111yescharacteristic
subgroupG/H
centerC4#19149647
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#532×C173C2
socleC2×C173#386
cosocleC22×C4×C173C23
hypercenter#532D173
hypocenterC173#1884
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26012258C2semidirect
2. #26011508C2semidirect
3. #26011465C2semidirect
4. #26011251C2semidirect
14. C4⋊Q8×C173C4semidirect
15. #127×C173C4semidirect
16. #124×C173C4semidirect
17. #124×C173C4semidirect
23. #14583810D4semidirect
95. C173#1884semidirect
107. C4⋊Q8Dic173semidirect
108. #127Dic173semidirect
109. #124Dic173semidirect
110. #124Dic173semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260115082
#260114652
#260112512
#532×C1732
#1884173