Finite Group Databasebeta

← #36365004

Subgroups — #36365004

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#26012096C211yesmaximal, characteristic
2#26011798C212yesmaximal, complemented
3#26011788C212yesmaximal, complemented
4#26011739C212yesmaximal, complemented
5#26011722C212yesmaximal, complemented
6#26011693C212yesmaximal, complemented
7#26011691C212yesmaximal, complemented
8#26011518C211yesmaximal, characteristic, complemented
9#541×C173C211yesmaximal, characteristic, complemented
10C2×#14583813C2212yes—
11C2×#14583813C2212yes—
12C2×#14583810C2211yescharacteristic
13C22×#13063436C2211yescharacteristic
14#19149676C2211yescharacteristic
15#19149671C2211yescharacteristic
16#19149664—22no—
17#19149653C2212yes—
18#19149652—22no—
19C4×D692C2212yes—
20#19149612C2212yes—
21#19149610C2212yes—
22C4⋊#13063436C2212yes—
23C4⋊#13063436C2212yes—
24#19149603C2212yes—
25#19149603C2212yes—
26#19149599C2212yes—
27#19149599C2212yes—
28C4⋊Q8×C173C2211yescharacteristic
29#126×C173—22no—
30#125×C173—22no—
31#125×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33C4×Q8×C173C2212yes—
34#117×C173C2211yescharacteristic
35C2×D692—22no—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39#14583813—42no—
40#14583813—42no—
41#14583810—41no—
42C2×#13063436—41no—
43C2×#13063436C2312yes—
44C2×#13063436C2312yes—
45C4⋊Dic173C2312yes—
46#14583805—22no—
47#14583803—22no—
48#14583803—22no—
49C2×Q8×C173C2312yes—
50C22×C4×C173C2311yescharacteristic
51C4⋊C4×C173—22no—
52C4⋊C4×C173—22no—
53C4⋊C4×C173—22no—
54C4⋊C4×C173—22no—
55C4⋊C4×C173D412yes—
56C4⋊C4×C173C2312yes—
57C22⋊C4×C173—22no—
58C22⋊C4×C173D412yes—
59C42×C173—22no—
60C42×C173C2311yescharacteristic
61C42×C173C2311yescharacteristic
62C22×D173—22no—
63D692—22no—
64D692—22no—
65D692—82no—
66C22⋊D173—82no—
67C2×Dic173—22no—
68C4×D173—82no—
69#13063436—22no—
70#13063436—22no—
71C23×C173C2×D411yescharacteristic
72Q8×C173—22no—
73Q8×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2411yescharacteristic
82C22×C173—21no—
83C22×C173—22no—
84C22×C173C22×D411yescharacteristic
85C2×D173—42no—
86C2×D173—42no—
87C2×D173—42no—
88C4×C173—21no—
89C4×C173—22no—
90C4×C173—22no—
91C4×C173—42no—
92C4×C173—42no—
93C4×C173H(5,𝔽2)11yescharacteristic
94C4×C1732−1+411yescharacteristic
95Dic173—42no—
96C2×C173—41no—
97C2×C173#53711yescharacteristic
98C2×C173#57712yes—
99D173—82no—
100C173#326011yescharacteristic, complemented, Sylow
101#3260—1731nomaximal, Sylow
102#541D17311yescharacteristic, complemented
103#540—1731no—
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⋊Q8C2×D17311yescharacteristic
116#127—1731no—
117#126—22no—
118#125—22no—
119#125—1731no—
120#125C2×D17311yescharacteristic
121#123—3462no—
122C22⋊Q8C2×D17312yes—
123#121—3462no—
124#121—1732no—
125C4×Q8C2×D17312yes—
126C4×D4—1732no—
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×Q8C22×D17312yes—
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—1732no—
140C22×C4—3462no—
141C22×C4C22×D17311yescharacteristic
142SD16—6922no—
143SD16—6922no—
144M16—6921no—
145C2×C8—1732no—
146C2×C8—1732no—
147C2×C8—6921no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4C22⋊D17312yes—
155C4⋊C4C22×D17312yes—
156C22⋊C4—22no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C22⋊C4C22⋊D17312yes—
160C42—22no—
161C42C22×D17311yescharacteristic
162C42C22×D17311yescharacteristic
163C23—3462no—
164C23C2×C22⋊D17311yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—3462no—
168D4—3462no—
169D4—13842no—
170D4—13842no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—3462no—
176C2×C4—13842no—
177C2×C4C2×C22⋊D17312yes—
178C2×C4C2×C22⋊D17312yes—
179C2×C4C2×C22⋊D17311yescharacteristic
180C2×C4C23×D17311yescharacteristic
181C8—3462no—
182C8—3462no—
183C22—22no—
184C22—21no—
185C22—6922no—
186C22—6922no—
187C22—6922no—
188C22C22×C22⋊D17311yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—6922no—
195C4#1914972511yescharacteristic
196C4#1914972611yescharacteristic
197C2—41no—
198C2—13842no—
199C2#2601206911yescharacteristic
200C2#2601229912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#541×C173C2
socleC22×C173C22×D4
prime radical#26012096C2
hypercenter#541D173
hypocenterC173#3260
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #26011798C2semidirect
3. #26011788C2semidirect
4. #26011739C2semidirect
5. #26011722C2semidirect
6. #26011693C2semidirect
7. #26011691C2semidirect
8. #26011518C2semidirect
9. #541×C173C2semidirect
100. C173#3260semidirect
102. #541D173semidirect

Maximal subgroups

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

subgroupindex
#260120962
#260117982
#260117882
#260117392
#260117222
#260116932
#260116912
#260115182
#541×C1732
#3260173