Finite Group Databasebeta

← #36369571

Subgroups — #36369571

Order 22144.

344 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012559C211yesmaximal, characteristic, complemented
2#26012483C212yesmaximal
3D173⋊#101C211yesmaximal, characteristic, complemented
4#26012362C212yesmaximal, complemented
5#124×D173C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7Q8×#10621924—24no—
8#19149758—22no—
9#19149758C2212yes—
10#19149741—24no—
11#19149740—22no—
12C2×Q8×D173C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14#19149668C411yescharacteristic, complemented
15#19149667C411yescharacteristic, complemented
16#19149666C411yescharacteristic, complemented
17#19149665—22no—
18C22⋊C4×D173—22no—
19C42×D173C2211yescharacteristic
20#124×C173C411yescharacteristic, complemented
21#14583836—22no—
22#14583835—22no—
23D173⋊C8—22no—
24C4⋊#10621924—24no—
25C4⋊#10621924D414yes—
26C22⋊#10621924—22no—
27C4×#10621924—24no—
28C4×#10621924—22no—
29C23×D173—21no—
30C2×Dic346C2×C411yescharacteristic
31C2×Dic346D412yes—
32Q8×D173—22no—
33Q8×D173—22no—
34C2×D692C2×C411yescharacteristic
35C2×C22⋊D173—21no—
36D4×D173—22no—
37D4×D173—22no—
38C2×C4×D173—21no—
39C2×C4×D173D412yes—
40C2×C4×D173C2311yescharacteristic
41C22⋊Dic173—22no—
42#14583803—22no—
43#14583803—22no—
44C4×Dic173—21no—
45C4×Dic173C2×C411yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—22no—
49C42×C173C2×C411yescharacteristic
50C2×#10621924—24no—
51#13063443—42no—
52C22×D173—21no—
53C22×D173—44no—
54C22×D173—41no—
55C22×D173—41no—
56C22×D173C2×D411yescharacteristic
57Dic346—22no—
58Dic346—24no—
59D692—22no—
60C22⋊D173—42no—
61C2×Dic173—21no—
62C2×Dic173C22⋊C412yes—
63C2×Dic173C2×D411yescharacteristic
64C4×D173—22no—
65C4×D173—22no—
66C4×D173—42no—
67C4×D173C2×D412yes—
68C4×D173#4112yes—
69C23×C173—21no—
70Q8×C173—22no—
71D4×C173—22no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C412yes—
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—42no—
76C22×C173—41no—
77C22×C173C2×C22⋊C411yescharacteristic
78C2×D173—22no—
79C2×D173—42no—
80C2×D173—41no—
81C2×D173—82no—
82C2×D173—82no—
83C2×D173#12011yescharacteristic
84C2×D173#12311yescharacteristic
85#10621924—44no—
86C4×C173—22no—
87C4×C173—41no—
88C4×C173C4×D412yes—
89Dic173—22no—
90Dic173—41no—
91Dic173#12112yes—
92C2×C173—21no—
93C2×C173—81no—
94C2×C173#38611yescharacteristic
95D173—21no—
96D173—81no—
97D173#45412yes—
98C173#188811yescharacteristic, complemented, Sylow
99#1888—1731nomaximal, Sylow
100C2×#124—1731no—
101#519—1731no—
102#418—1732no—
103C2×#101—1731no—
104#339—1732no—
105C22×Q8—1731no—
106C22×D4—1731no—
107C2×M16—1732no—
108#124—3462no—
109#124—1731no—
110#124—1731no—
111#124—1731no—
112#124#1062192411yescharacteristic, complemented
113C4×Q8—3464no—
114#117—3462no—
115#117—1732no—
116C2×C22⋊C4—3462no—
117C2×C42—1731no—
118#102—3464no—
119#101—3462no—
120C24—3461no—
121C2×Q8—3462no—
122C2×Q8—3462no—
123C2×Q8—1732no—
124C2×Q8—1731no—
125C2×Q8C2×#1062192411yescharacteristic
126C2×D4—3462no—
127C2×D4—3462no—
128C2×D4—3461no—
129C2×D4—1731no—
130C2×D4C2×#1062192411yescharacteristic
131C22×C4—3461no—
132C22×C4—1732no—
133C22×C4—1731no—
134M16—3462no—
135M16—3462no—
136C2×C8—3462no—
137C4⋊C4—3464no—
138C4⋊C4—1734no—
139C22⋊C4—22no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C22⋊C4—3462no—
143C22⋊C4—3462no—
144C42—3461no—
145C42—3464no—
146C42—3462no—
147C42—1731no—
148C42C2×#1062192411yescharacteristic
149C23—21no—
150C23—3461no—
151C23—1731no—
152C23—6924no—
153C23—6921no—
154C23—6921no—
155Q8—22no—
156Q8—3464no—
157Q8—3462no—
158D4—22no—
159D4—3462no—
160D4—6922no—
161C2×C4—21no—
162C2×C4—3462no—
163C2×C4—3462no—
164C2×C4—3464no—
165C2×C4—3461no—
166C2×C4—1732no—
167C2×C4—1731no—
168C2×C4—1732no—
169C2×C4—1732no—
170C2×C4—6922no—
171C2×C4C22⋊#1062192412yes—
172C2×C4C22×#1062192411yescharacteristic
173C8—6922no—
174C22—42no—
175C22—41no—
176C22—3462no—
177C22—13842no—
178C22—13842no—
179C22—1731no—
180C22—1731no—
181C22—6922no—
182C22—6921no—
183C22C2×C22⋊#1062192411yescharacteristic
184C4—22no—
185C4—41no—
186C4—3462no—
187C4—1732no—
188C4—6921no—
189C4—6924no—
190C4D4×#1062192412yes—
191C2—21no—
192C2—81no—
193C2—3461no—
194C2—13841no—
195C2—1732no—
196C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC2×C173#386
prime radicalC2×D4×D173C22
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1888
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012559C2semidirect
3. D173⋊#101C2semidirect
4. #26012362C2semidirect
14. #19149668C4semidirect
15. #19149667C4semidirect
16. #19149666C4semidirect
20. #124×C173C4semidirect
98. C173#1888semidirect
112. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125592
#260124832
D173⋊#1012
#260123622
#124×D1732
#1888173