Finite Group Databasebeta

← #36353946

Subgroups — #36353946

Order 22144.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#26012037C211yesmaximal, characteristic
2#26011507C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic, complemented
4#26011252C212yesmaximal, complemented
5#568×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7#19149677C411yescharacteristic, complemented
8#19149675C411yescharacteristic, complemented
9#19149668C411yescharacteristic, complemented
10#19149666C411yescharacteristic, complemented
11#19149651—22no—
12C4×C22⋊D173—22no—
13#19149638C2212yes—
14C2×C4×Dic173C2211yescharacteristic
15#19149605—24no—
16#19149594—22no—
17C2×#41×C173C2211yescharacteristic
18#131×C173—24no—
19C2×M16×C173—22no—
20C2×M16×C173C2212yes—
21#14583826—22no—
22#14583826—22no—
23C2×Dic346C2×C411yescharacteristic
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27C22×Dic173—21no—
28C2×C4×D173—21no—
29C4⋊Dic173—22no—
30#14583805—22no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33#14583803—22no—
34C4×Dic173—21no—
35C4×Dic173—22no—
36C4×Dic173C2×C411yescharacteristic
37C4×Dic173C2×C411yescharacteristic
38#41×C173—22no—
39#41×C173—22no—
40C2×Q8×C173C2×C411yescharacteristic
41C2×D4×C173C2×C411yescharacteristic
42C2×D4×C173D412yes—
43C22×C4×C173D412yes—
44C22×C4×C173C2311yescharacteristic
45M16×C173—24no—
46M16×C173—22no—
47M16×C173D414yescomplemented
48C2×C8×C173—24no—
49C2×C8×C173—22no—
50C22×D173—21no—
51Dic346—22no—
52D692—22no—
53C22⋊D173—44no—
54C22⋊D173—42no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C2×Dic173—42no—
59C2×Dic173—42no—
60C4×D173—41no—
61C4×D173—41no—
62C23×C173C22⋊C412yes—
63C23×C173C2×D411yescharacteristic
64Q8×C173—22no—
65D4×C173—24no—
66D4×C173—22no—
67C2×C4×C173—22no—
68C2×C4×C173—22no—
69C2×C4×C173C22⋊C412yes—
70C2×C4×C173C22×C411yescharacteristic
71C2×C4×C173C2×D412yes—
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173#4112yes—
74C8×C173—24no—
75C22×C173—22no—
76C22×C173—41no—
77C22×C173—42no—
78C22×C173C2×C22⋊C411yescharacteristic
79C22×C173#12112yes—
80C2×D173—42no—
81C2×D173—41no—
82C4×C173—22no—
83C4×C173C4×D412yes—
84C4×C173#12011yescharacteristic
85C4×C173#12311yescharacteristic
86Dic173—22no—
87Dic173—41no—
88Dic173—41no—
89Dic173—82no—
90C2×C173—21no—
91C2×C173—22no—
92C2×C173—42no—
93C2×C173#38611yescharacteristic
94D173—81no—
95C173#188411yescharacteristic, complemented, Sylow
96#1884—1731nomaximal, Sylow
97#568D17311yescharacteristic, complemented
98#532—1731no—
99#421—1732no—
100#410—1731no—
101#339—1732no—
102C2×#41—1731no—
103C2×#41C2×D17311yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D17312yes—
107C4⋊Q8—1731no—
108#127—1731no—
109#124—1731no—
110#124—1731no—
111#121—3462no—
112C4×D4—3462no—
113#117—1732no—
114C2×C42—1731no—
115#104—3464no—
116#99—3462no—
117#41—22no—
118#41—22no—
119#41—3462no—
120#41—3462no—
121C2×Q8—1731no—
122C2×Q8C4×D17311yescharacteristic
123C2×D4—3461no—
124C2×D4—3462no—
125C2×D4—1731no—
126C2×D4C4×D17311yescharacteristic
127C2×D4D69212yes—
128C22×C4—3461no—
129C22×C4—3461no—
130C22×C4C22⋊D17312yes—
131C22×C4C22×D17311yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D17314yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—3462no—
138C4⋊C4—3462no—
139C22⋊C4—3462no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C42—3461no—
143C42—3462no—
144C42—1731no—
145C42—1731no—
146C23—3461no—
147C23#1458380312yes—
148C23C2×D69211yescharacteristic
149Q8—22no—
150Q8—3462no—
151D4—24no—
152D4—22no—
153D4—3462no—
154D4—6924no—
155D4—6922no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—3461no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—6922no—
162C2×C4—6922no—
163C2×C4—6921no—
164C2×C4—6921no—
165C2×C4#1458380312yes—
166C2×C4C2×C4×D17311yescharacteristic
167C2×C4C2×C22⋊D17312yes—
168C2×C4C2×C22⋊D17311yescharacteristic
169C2×C4#1458382612yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—6922no—
175C22—6921no—
176C22C2×#1458380311yescharacteristic
177C22#1914965312yes—
178C4—22no—
179C4—3462no—
180C4—13842no—
181C4—6921no—
182C4—6921no—
183C4C4×C22⋊D17312yes—
184C4#1914964811yescharacteristic
185C4#1914966211yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—13841no—
190C2#2601137011yescharacteristic
subgroupG/H
centerC4#19149648
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#568×C173C2
socleC2×C173#386
prime radical#26012037C2
cosocleC22×C4×C173C23
hypercenter#568D173
hypocenterC173#1884
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26011507C2semidirect
3. #26011465C2semidirect
4. #26011252C2semidirect
5. #568×C173C2semidirect
7. #19149677C4semidirect
8. #19149675C4semidirect
9. #19149668C4semidirect
10. #19149666C4semidirect
47. M16×C173D4semidirect
95. C173#1884semidirect
97. #568D173semidirect
134. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120372
#260115072
#260114652
#260112522
#568×C1732
#1884173