Finite Group Databasebeta

← #36352185

Subgroups — #36352185

Order 22144.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D173C211yesmaximal, characteristic
2#26011546C212yesmaximal, complemented
3#26011231C211yesmaximal, characteristic, complemented
4#26011230C212yesmaximal, complemented
5#329×C173C211yesmaximal, characteristic, complemented
6C2×Q8×D173C411yescharacteristic
7C2×D4×D173C411yescharacteristic
8#19149668C411yescharacteristic
9#19149667C2211yescharacteristic
10#19149666C411yescharacteristic
11#19149665—22no—
12C22⋊C4×D173—22no—
13C42×D173C2211yescharacteristic
14#19149608—22no—
15#19149591—22no—
16#19149590C2212yescomplemented
17#124×C173C2211yescharacteristic
18#97×C173C2212yescomplemented
19C23×D173—21no—
20C2×Dic346—21no—
21C2×Dic346C2×C411yescharacteristic
22Q8×D173—42no—
23C2×D692C2×C411yescharacteristic
24C2×C22⋊D173—21no—
25D4×D173—42no—
26C2×C4×D173—21no—
27C2×C4×D173C2×C411yescharacteristic
28C2×C4×D173D412yes—
29C2×#13063436—22no—
30C22⋊Dic173—22no—
31#14583803—22no—
32#14583803—22no—
33C4×Dic173C2×C411yescharacteristic
34C4×Dic173D412yes—
35C2×Q8×C173C2×C411yescharacteristic
36C2×D4×C173C2×C411yescharacteristic
37C2×C8×C173—22no—
38C22⋊C4×C173—22no—
39C42×C173C2311yescharacteristic
40C22×D173—21no—
41C22×D173—44no—
42C22×D173—42no—
43C22×D173C22⋊C411yescharacteristic
44Dic346—42no—
45Dic346—41no—
46D692—41no—
47C22⋊D173—42no—
48C2×Dic173—21no—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C22⋊C412yes—
51C4×D173—22no—
52C4×D173—42no—
53C4×D173—42no—
54#13063436—42no—
55C23×C173—21no—
56Q8×C173—41no—
57D4×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173C22×C411yescharacteristic
60C2×C4×C173C2×D412yes—
61C8×C173—42no—
62C22×C173—41no—
63C22×C173—42no—
64C22×C173C2×C22⋊C411yescharacteristic
65C2×D173—22no—
66C2×D173—41no—
67C2×D173—42no—
68C2×D173—84no—
69C2×D173#10012yes—
70C4×C173—21no—
71C4×C173—22no—
72C4×C173—41no—
73Dic173—21no—
74Dic173—22no—
75Dic173—41no—
76C2×C173—81no—
77C2×C173C2×#10011yescharacteristic
78C2×C173#42112yes—
79D173—22no—
80D173—81no—
81C173#153311yescharacteristic, complemented, Sylow
82#1533—1731nomaximal, Sylow
83C2×#124—1731no—
84#432—1732no—
85#329—1731no—
86#329—1732no—
87#329D17311yescharacteristic, complemented
88C22×Q8—1731no—
89C22×D4—1731no—
90#124—3462no—
91#124—1731no—
92#124—1731no—
93#124—1731no—
94#124C2×D17311yescharacteristic
95C2×C22⋊C4—3462no—
96C2×C42—1731no—
97C4⋊C8—3462no—
98C22⋊C8—3462no—
99#97—1732no—
100#97C2×D17312yescomplemented
101C24—3461no—
102C2×Q8—3461no—
103C2×Q8—1731no—
104C2×Q8—6922no—
105C2×Q8C4×D17311yescharacteristic
106C2×D4—3461no—
107C2×D4—1731no—
108C2×D4—6922no—
109C2×D4C4×D17311yescharacteristic
110C22×C4—3461no—
111C22×C4—1731no—
112C22×C4—1732no—
113C2×C8—22no—
114C2×C8—3462no—
115C22⋊C4—22no—
116C22⋊C4—3462no—
117C22⋊C4—3462no—
118C22⋊C4—3462no—
119C42—1731no—
120C42—1732no—
121C42C22×D17311yescharacteristic
122C23—21no—
123C23—3461no—
124C23—1731no—
125C23—6924no—
126C23—6922no—
127Q8—41no—
128Q8—6921no—
129Q8—6922no—
130D4—41no—
131D4—6921no—
132D4—6922no—
133C2×C4—21no—
134C2×C4—3462no—
135C2×C4—3461no—
136C2×C4—1732no—
137C2×C4—1731no—
138C2×C4—6922no—
139C2×C4—6922no—
140C2×C4C2×C4×D17311yescharacteristic
141C2×C4D4×D17312yes—
142C8—42no—
143C8—6922no—
144C22—41no—
145C22—42no—
146C22—3462no—
147C22—13844no—
148C22—1732no—
149C22—6921no—
150C22—6922no—
151C22C22⋊C4×D17311yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—3461no—
156C4—3462no—
157C4—6921no—
158C2—81no—
159C2—3462no—
160C2—13841no—
161C2#100×D17311yescharacteristic
162C2#2601150612yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC42×C173C23
hypercenter#329D173
hypocenterC173#1533
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26011546C2semidirect
3. #26011231C2semidirect
4. #26011230C2semidirect
5. #329×C173C2semidirect
16. #19149590C22semidirect
18. #97×C173C22semidirect
81. C173#1533semidirect
87. #329D173semidirect
100. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#124×D1732
#260115462
#260112312
#260112302
#329×C1732
#1533173