Finite Group Databasebeta

← #36355213

Subgroups — #36355213

Order 22144.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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#26011475C211yesmaximal, characteristic, complemented
2#26011465C211yesmaximal, characteristic, complemented
3#26011278C214yesmaximal, complemented
4#535×C173C211yesmaximal, characteristic
5C2×#14583810—21no—
6#19149638—21no—
7#19149598—21no—
8#19149596C2212yes—
9#19149594—21no—
10#19149594C2212yes—
11C2×#41×C173C2211yescharacteristic
12C22×D4×C173C411yescharacteristic, complemented
13#127×C173—21no—
14#124×C173—21no—
15#121×C173—21no—
16#121×C173—21no—
17#120×C173C412yescomplemented
18#120×C173C2212yes—
19#117×C173C411yescharacteristic, complemented
20#14583810—22no—
21#14583810—41no—
22C2×#13063436—41no—
23C4⋊Dic173—41no—
24C22⋊Dic173—22no—
25C4×Dic173—41no—
26C24×C173C2×C412yes—
27#41×C173—41no—
28#41×C173—41no—
29C2×Q8×C173D411yescharacteristic
30C2×D4×C173—22no—
31C2×D4×C173—22no—
32C2×D4×C173—21no—
33C2×D4×C173—21no—
34C2×D4×C173—41no—
35C2×D4×C173—41no—
36C2×D4×C173C2×C411yescharacteristic
37C2×D4×C173D411yescharacteristic
38C2×D4×C173C2311yescharacteristic
39C22×C4×C173C2×C411yescharacteristic
40C22×C4×C173D411yescharacteristic
41C22×C4×C173D411yescharacteristic
42C4⋊C4×C173—21no—
43C22⋊C4×C173—22no—
44C22⋊C4×C173—22no—
45C22⋊C4×C173C2×C412yes—
46C42×C173—21no—
47C2×Dic173—42no—
48#13063436—42no—
49C23×C173—22no—
50C23×C173—42no—
51C23×C173—82no—
52C23×C173C22⋊C411yescharacteristic
53C23×C173C22×C411yescharacteristic
54C23×C173C2×D411yescharacteristic
55Q8×C173—41no—
56D4×C173—41no—
57D4×C173—41no—
58D4×C173—41no—
59D4×C173—42no—
60D4×C173—41no—
61D4×C173—41no—
62D4×C173—82no—
63D4×C173—82no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—22no—
71C2×C4×C173C22⋊C411yescharacteristic
72C2×C4×C173C22⋊C411yescharacteristic
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173—42no—
82C22×C173—42no—
83C22×C173—42no—
84C22×C173—84no—
85C22×C173C2×C22⋊C411yescharacteristic
86C4×C173—21no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—42no—
90C4×C173#9911yescharacteristic
91C4×C173#9911yescharacteristic
92Dic173—82no—
93C2×C173—21no—
94C2×C173—41no—
95C2×C173—41no—
96C2×C173—41no—
97C2×C173—82no—
98C2×C173C2×#9911yescharacteristic
99C173#212211yescharacteristic, complemented, Sylow
100#2122—1731nomaximal, Sylow
101#535D17311yescharacteristic
102#413—1731no—
103#410—1731no—
104AΣL(1,𝔽16)—1734no—
105C2×#41C2×D17311yescharacteristic
106C22×D4Dic17311yescharacteristic, complemented
107C2×M16—3461no—
108#127—21no—
109#124—21no—
110#121—21no—
111#121—21no—
112#120Dic17312yescomplemented
113#120C2×D17312yes—
114#117—3461no—
115#117Dic17311yescharacteristic, complemented
116#101—3461no—
117#100—1732no—
118#99—3461no—
119#99—1732no—
120C24C2×Dic17312yes—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D17311yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic17311yescharacteristic
131C2×D4C22⋊D17311yescharacteristic
132C2×D4C22×D17311yescharacteristic
133C22×C4C2×Dic17311yescharacteristic
134C22×C4C22⋊D17311yescharacteristic
135C22×C4C22⋊D17311yescharacteristic
136M16—3462no—
137M16—6921no—
138C2×C8—6921no—
139C4⋊C4—21no—
140C4⋊C4—6921no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—3462no—
144C22⋊C4C2×Dic17312yes—
145C42—21no—
146C42—6921no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊Dic17311yescharacteristic
151C23C22×Dic17311yescharacteristic
152C23C2×C22⋊D17311yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—6922no—
170C2×C4C22⋊Dic17311yescharacteristic
171C2×C4C22⋊Dic17311yescharacteristic
172C2×C4C22⋊Dic17311yescharacteristic
173C2×C4C2×C22⋊D17311yescharacteristic
174C8—6922no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic17311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—13842no—
191C4#1914959411yescharacteristic
192C4#1914959411yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#1914959411yescharacteristic
subgroupG/H
centerC2C2×#19149594
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
Fitting, prime radical#535×C173C2
socleC2×C173C2×#99
cosocleC2×D4×C173C23
hypercenter#535D173
hypocenterC173#2122
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26011475C2semidirect
2. #26011465C2semidirect
3. #26011278C2semidirect
12. C22×D4×C173C4semidirect
17. #120×C173C4semidirect
19. #117×C173C4semidirect
99. C173#2122semidirect
106. C22×D4Dic173semidirect
112. #120Dic173semidirect
115. #117Dic173semidirect

Maximal subgroups

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

subgroupindex
#260114752
#260114652
#260112782
#535×C1732
#2122173