Finite Group Databasebeta

← #36354101

Subgroups — #36354101

Order 22144.

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

normal subgroups only

This group has 314 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#26011969C211yesmaximal, characteristic, complemented
2#26011508C212yesmaximal
3C2×#19149598C211yesmaximal, characteristic, complemented
4#26011247C212yesmaximal, complemented
5#536×C173C211yesmaximal, characteristic
6C2×#14583810C2212yes—
7Q8×Dic173—24no—
8#19149638—22no—
9#19149638C2212yes—
10#19149606—24no—
11#19149598—22no—
12C2×#41×C173C2211yescharacteristic
13C22×Q8×C173C2211yescharacteristic
14C4⋊Q8×C173C412yescomplemented
15#124×C173C412yescomplemented
16#123×C173—22no—
17C22⋊Q8×C173—22no—
18#117×C173C2211yescharacteristic
19#14583810—22no—
20#14583810—22no—
21C2×#13063436—22no—
22C4⋊Dic173—24no—
23C4⋊Dic173—22no—
24C22⋊Dic173—22no—
25C4×Dic173—24no—
26C4×Dic173D414yes—
27#41×C173—22no—
28#41×C173—22no—
29C2×Q8×C173—22no—
30C2×Q8×C173—22no—
31C2×Q8×C173C2×C411yescharacteristic
32C2×Q8×C173C2×C412yes—
33C2×Q8×C173D412yes—
34C2×D4×C173—21no—
35C2×D4×C173C2×C411yescharacteristic
36C22×C4×C173—21no—
37C22×C4×C173D412yes—
38C22×C4×C173C2311yescharacteristic
39C4⋊C4×C173—24no—
40C4⋊C4×C173—21no—
41C22⋊C4×C173—21no—
42C22⋊C4×C173—24no—
43C42×C173C2×C412yes—
44C2×Dic173—24no—
45#13063436—42no—
46C23×C173—21no—
47C23×C173C2×D411yescharacteristic
48Q8×C173—24no—
49Q8×C173—24no—
50Q8×C173—22no—
51D4×C173—22no—
52D4×C173—41no—
53D4×C173—41no—
54C2×C4×C173—22no—
55C2×C4×C173—22no—
56C2×C4×C173—22no—
57C2×C4×C173—21no—
58C2×C4×C173—42no—
59C2×C4×C173C22⋊C414yes—
60C2×C4×C173C22×C411yescharacteristic
61C2×C4×C173C2×D412yes—
62C2×C4×C173C2×D411yescharacteristic
63C2×C4×C173#4112yes—
64C22×C173—41no—
65C22×C173—42no—
66C22×C173—41no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173#12011yescharacteristic
69C22×C173#12311yescharacteristic
70C4×C173—24no—
71C4×C173—42no—
72C4×C173—41no—
73C4×C173C4×D412yes—
74C4×C173#12112yes—
75Dic173—44no—
76C2×C173—21no—
77C2×C173—21no—
78C2×C173—21no—
79C2×C173—81no—
80C2×C173#38611yescharacteristic
81C173#191211yescharacteristic, complemented, Sylow
82#1912—1731nomaximal, Sylow
83#536D17311yescharacteristic
84#519—1731no—
85#421—1732no—
86C2×#101—1731no—
87H(3,ℤ/4)—1732no—
88C2×#41C2×D17311yescharacteristic
89C22×Q8C2×D17311yescharacteristic
90C2×M16—1732no—
91C4⋊Q8Dic17312yescomplemented
92#124Dic17312yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—3464no—
96#117—3462no—
97#117—1732no—
98#117C2×D17311yescharacteristic
99#104—3464no—
100#101—3462no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic17311yescharacteristic
106C2×Q8C2×Dic17312yes—
107C2×Q8C22⋊D17312yes—
108C2×D4—21no—
109C2×D4C2×Dic17311yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D17312yes—
112C22×C4C22×D17311yescharacteristic
113M16—3462no—
114M16—3462no—
115C2×C8—3462no—
116C4⋊C4—21no—
117C4⋊C4—24no—
118C4⋊C4—3464no—
119C4⋊C4—3462no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—3462no—
123C42—3464no—
124C42—1734no—
125C42C2×Dic17312yes—
126C23—21no—
127C23C2×C22⋊D17311yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3464no—
140C2×C4C22⋊Dic17314yes—
141C2×C4C22×Dic17311yescharacteristic
142C2×C4D4×D17312yes—
143C2×C4C2×C22⋊D17311yescharacteristic
144C2×C4#1458382512yes—
145C8—6922no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic17311yescharacteristic
150C22#1914964711yescharacteristic
151C22#1914966011yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—6924no—
156C4D4×Dic17312yes—
157C4#1914965112yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2601137111yescharacteristic
subgroupG/H
centerC2#26011371
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#536×C173C2
socleC2×C173#386
prime radicalC2×#41×C173C22
cosocleC22×C4×C173C23
hypercenter#536D173
hypocenterC173#1912
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26011969C2semidirect
3. C2×#19149598C2semidirect
4. #26011247C2semidirect
14. C4⋊Q8×C173C4semidirect
15. #124×C173C4semidirect
81. C173#1912semidirect
91. C4⋊Q8Dic173semidirect
92. #124Dic173semidirect

Maximal subgroups

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

subgroupindex
#260119692
#260115082
C2×#191495982
#260112472
#536×C1732
#1912173