Finite Group Databasebeta

← #36369568

Subgroups — #36369568

Order 22144.

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

normal subgroups only

This group has 360 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#26012558C211yesmaximal, characteristic
2#26012484C212yesmaximal, complemented
3D173⋊#100C211yesmaximal, characteristic
4#26012362C212yesmaximal, complemented
5#124×D173C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7D4×#10621924—24no—
8#19149758C2212yes—
9C2×C22⋊#10621924—22no—
10#19149742—24no—
11#19149739—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
21C22×#10621924—24no—
22#14583836—22no—
23#14583835—22no—
24D173⋊C8—22no—
25C4⋊#10621924D414yes—
26C22⋊#10621924—24no—
27C22⋊#10621924—22no—
28C4×#10621924—22no—
29C23×D173D412yes—
30C2×Dic346—21no—
31C2×Dic346C2×C411yescharacteristic, complemented
32Q8×D173—22no—
33Q8×D173—22no—
34C2×D692C2×C411yescharacteristic
35C2×C22⋊D173D412yes—
36D4×D173—22no—
37D4×D173—22no—
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40C2×C4×D173C2311yescharacteristic
41C22⋊Dic173—22no—
42#14583803—22no—
43#14583803—22no—
44C4×Dic173—21no—
45C4×Dic173C2×C411yescharacteristic, complemented
46C2×Q8×C173C2×C411yescharacteristic, complemented
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173—22no—
49C42×C173C2×C411yescharacteristic, complemented
50C2×#10621924—24no—
51C2×#10621924—84no—
52#13063443—42no—
53C22×D173—22no—
54C22×D173—22no—
55C22×D173—44no—
56C22×D173C22⋊C412yes—
57C22×D173C2×D411yescharacteristic
58Dic346—22no—
59Dic346—42no—
60D692—22no—
61C22⋊D173—24no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173C2×D411yescharacteristic
65C4×D173—42no—
66C4×D173—41no—
67C4×D173—41no—
68C4×D173C2×D412yes—
69C4×D173#4112yes—
70C23×C173C22⋊C412yes—
71Q8×C173—22no—
72D4×C173—22no—
73C2×C4×C173—21no—
74C2×C4×C173—21no—
75C2×C4×C173C22×C411yescharacteristic
76C22×C173—22no—
77C22×C173—42no—
78C22×C173C2×C22⋊C411yescharacteristic
79C2×D173—22no—
80C2×D173—22no—
81C2×D173—44no—
82C2×D173—42no—
83C2×D173—82no—
84C2×D173#12011yescharacteristic
85C2×D173#12311yescharacteristic
86#10621924—44no—
87C4×C173—41no—
88C4×C173—41no—
89C4×C173C4×D412yes—
90Dic173—41no—
91Dic173—41no—
92Dic173#12112yes—
93C2×C173—21no—
94C2×C173—42no—
95C2×C173#38611yescharacteristic
96D173—21no—
97D173—42no—
98D173#45512yes—
99C173#188511yescharacteristic, complemented, Sylow
100#1885—1731nomaximal, Sylow
101C2×#124—1731no—
102#518—1731no—
103#419—1732no—
104C2×#100—1731no—
105#339—1732no—
106C22×Q8—1731no—
107C22×D4—1731no—
108C2×M16—1732no—
109#124—3462no—
110#124—1731no—
111#124—1731no—
112#124—1731no—
113#124#1062192411yescharacteristic, complemented
114C4×D4—3464no—
115#117—1732no—
116C2×C22⋊C4—3462no—
117C2×C22⋊C4—3462no—
118C2×C42—1731no—
119#103—3464no—
120#100—3462no—
121C24—1732no—
122C2×Q8—3461no—
123C2×Q8—3462no—
124C2×Q8—3462no—
125C2×Q8—1731no—
126C2×Q8C2×#1062192411yescharacteristic, complemented
127C2×D4—3462no—
128C2×D4—3462no—
129C2×D4—1732no—
130C2×D4—1731no—
131C2×D4C2×#1062192411yescharacteristic
132C22×C4—3461no—
133C22×C4—3461no—
134C22×C4—3464no—
135C22×C4—1731no—
136M16—3462no—
137M16—3462no—
138C2×C8—3462no—
139C4⋊C4—1734no—
140C22⋊C4—22no—
141C22⋊C4—3462no—
142C22⋊C4—3462no—
143C22⋊C4—3464no—
144C22⋊C4—3462no—
145C22⋊C4—3462no—
146C42—3461no—
147C42—3462no—
148C42—1731no—
149C42C2×#1062192411yescharacteristic, complemented
150C23—3462no—
151C23—3462no—
152C23—1731no—
153C23—1732no—
154C23—6924no—
155C23C22⋊#1062192412yes—
156Q8—22no—
157Q8—3462no—
158Q8—6922no—
159D4—22no—
160D4—3464no—
161D4—3462no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—3464no—
165C2×C4—3461no—
166C2×C4—3461no—
167C2×C4—13844no—
168C2×C4—1731no—
169C2×C4—1732no—
170C2×C4—1732no—
171C2×C4—6922no—
172C2×C4—6921no—
173C2×C4—6921no—
174C2×C4C22×#1062192411yescharacteristic
175C8—6922no—
176C22—22no—
177C22—42no—
178C22—3462no—
179C22—3462no—
180C22—13842no—
181C22—1731no—
182C22—1731no—
183C22—6922no—
184C22—6924no—
185C22C2×C22⋊#1062192411yescharacteristic
186C4—41no—
187C4—41no—
188C4—1732no—
189C4—6921no—
190C4—6924no—
191C4—6921no—
192C4D4×#1062192412yes—
193C2—21no—
194C2—42no—
195C2—3461no—
196C2—1732no—
197C2—6922no—
198C2#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#1885
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
2. #26012484C2semidirect
4. #26012362C2semidirect
14. #19149668C4semidirect
15. #19149667C4semidirect
16. #19149666C4semidirect
20. #124×C173C4semidirect
31. C2×Dic346C2×C4semidirect
45. C4×Dic173C2×C4semidirect
46. C2×Q8×C173C2×C4semidirect
49. C42×C173C2×C4semidirect
99. C173#1885semidirect
113. #124#10621924semidirect
126. C2×Q8C2×#10621924semidirect
149. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125582
#260124842
D173⋊#1002
#260123622
#124×D1732
#1885173