Finite Group Databasebeta

← #36365009

Subgroups — #36365009

Order 22144.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012103C211yesmaximal, characteristic, complemented
2#26011795C212yesmaximal, complemented
3#26011791C212yesmaximal, complemented
4#26011735C212yesmaximal, complemented
5#26011718C212yesmaximal, complemented
6#26011696C212yesmaximal, complemented
7#26011688C212yesmaximal, complemented
8#26011518C211yesmaximal, characteristic, complemented
9#540×C173C211yesmaximal, characteristic
10C2×#14583814C2212yes—
11C2×#14583814C2212yes—
12C2×#14583810C2211yescharacteristic
13C22×#13063436C2211yescharacteristic
14#19149679C2211yescharacteristic
15#19149672—22no—
16#19149671C2211yescharacteristic
17#19149669—22no—
18#19149658C2212yes—
19C4×Dic346C2212yes—
20#19149612C2212yes—
21#19149610C2212yes—
22C4⋊#13063436C2212yes—
23C4⋊#13063436C2212yes—
24#19149602C2212yes—
25#19149602C2212yes—
26#19149600C2212yes—
27#19149600C2212yes—
28#127×C173C2211yescharacteristic
29#125×C173C2211yescharacteristic
30#123×C173—22no—
31#121×C173—22no—
32#121×C173C2212yes—
33C4×D4×C173C2212yes—
34#117×C173C2211yescharacteristic
35C2×Dic346C2312yes—
36#14583814—42no—
37#14583814—42no—
38#14583810—41no—
39C2×#13063436—41no—
40C2×#13063436C2312yes—
41C2×#13063436C2312yes—
42C4⋊Dic173—22no—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583805—22no—
46#14583805—22no—
47C22⋊Dic173—22no—
48C4×Dic173—22no—
49C2×D4×C173—22no—
50C2×D4×C173—22no—
51C2×D4×C173C2312yes—
52C22×C4×C173—22no—
53C22×C4×C173C2311yescharacteristic
54C4⋊C4×C173—22no—
55C4⋊C4×C173D412yes—
56C4⋊C4×C173C2312yes—
57C22⋊C4×C173—22no—
58C22⋊C4×C173—22no—
59C22⋊C4×C173D412yes—
60C42×C173C2311yescharacteristic
61C42×C173C2311yescharacteristic
62Dic346—22no—
63Dic346—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66#13063436—22no—
67#13063436—22no—
68C23×C173—22no—
69C23×C173C2×D411yescharacteristic
70D4×C173—22no—
71D4×C173—22no—
72D4×C173—82no—
73D4×C173—82no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—22no—
77C2×C4×C173—82no—
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2411yescharacteristic
82C22×C173—21no—
83C22×C173—22no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173—42no—
87C22×C173C22×D411yescharacteristic
88C4×C173—21no—
89C4×C173—22no—
90C4×C173—22no—
91C4×C173—42no—
92C4×C173H(5,𝔽2)11yescharacteristic
93C4×C1732−1+411yescharacteristic
94Dic173—42no—
95Dic173—42no—
96C2×C173—41no—
97C2×C173—82no—
98C2×C173#53711yescharacteristic
99C2×C173#57712yes—
100C173#326011yescharacteristic, complemented, Sylow
101#3260—1731nomaximal, Sylow
102#541—1731no—
103#540D17311yescharacteristic
104#479—1732no—
105#478—1732no—
106#468—1732no—
107#465—1732no—
108#461—1732no—
109#460—1732no—
110#424—1731no—
111C2×SD16—1732no—
112C2×SD16—1732no—
113C2×M16—1731no—
114C22×C8—1731no—
115C4⋊Q8—1731no—
116#127C2×D17311yescharacteristic
117#126—3462no—
118#125—3462no—
119#125—1731no—
120#125C2×D17311yescharacteristic
121#123—22no—
122C22⋊Q8—1732no—
123#121—22no—
124#121C2×D17312yes—
125C4×Q8—1732no—
126C4×D4C2×D17312yes—
127#117C2×D17311yescharacteristic
128#107—1732no—
129#106—1732no—
130C4⋊C8—1732no—
131C4⋊C8—1732no—
132#103—1732no—
133#103—1732no—
134#102—1732no—
135#102—1732no—
136C2×Q8—1732no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4C22×D17312yes—
140C22×C4—22no—
141C22×C4C22×D17311yescharacteristic
142SD16—6922no—
143SD16—6922no—
144M16—6921no—
145C2×C8—1732no—
146C2×C8—1732no—
147C2×C8—6921no—
148C4⋊C4—22no—
149C4⋊C4—3462no—
150C4⋊C4—3462no—
151C4⋊C4—3462no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4C22⋊D17312yes—
155C4⋊C4C22×D17312yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—3462no—
159C22⋊C4C22⋊D17312yes—
160C42—3462no—
161C42C22×D17311yescharacteristic
162C42C22×D17311yescharacteristic
163C23—22no—
164C23C2×C22⋊D17311yescharacteristic
165Q8—3462no—
166Q8—3462no—
167D4—22no—
168D4—22no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—82no—
175C2×C4—3462no—
176C2×C4—3462no—
177C2×C4C2×C22⋊D17312yes—
178C2×C4C2×C22⋊D17312yes—
179C2×C4C2×C22⋊D17311yescharacteristic
180C2×C4C23×D17311yescharacteristic
181C8—3462no—
182C8—3462no—
183C22—22no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22C22×C22⋊D17311yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—6922no—
194C4—6922no—
195C4#1914972511yescharacteristic
196C4#1914972611yescharacteristic
197C2—41no—
198C2—82no—
199C2#2601206911yescharacteristic
200C2#2601230212yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#540×C173C2
socleC22×C173C22×D4
hypercenter#540D173
hypocenterC173#3260
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #26012103C2semidirect
2. #26011795C2semidirect
3. #26011791C2semidirect
4. #26011735C2semidirect
5. #26011718C2semidirect
6. #26011696C2semidirect
7. #26011688C2semidirect
8. #26011518C2semidirect
100. C173#3260semidirect

Maximal subgroups

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

subgroupindex
#260121032
#260117952
#260117912
#260117352
#260117182
#260116962
#260116882
#260115182
#540×C1732
#3260173