Finite Group Databasebeta

← #36363855

Subgroups — #36363855

Order 22144.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#26011968C211yesmaximal, characteristic, complemented
3#26011816C212yesmaximal, complemented
4#26011799C212yesmaximal, complemented
5#26011692C212yesmaximal, complemented
6#26011632C212yesmaximal, complemented
7#26011593C212yesmaximal, complemented
8C4×#14583815C212yesmaximal, complemented
9#433×C173C211yesmaximal, characteristic, complemented
10C22×Dic346C2211yescharacteristic
11C2×#14583815C2212yes—
12C2×#14583815C2212yes—
13#19149676C2211yescharacteristic
14#19149671C2211yescharacteristic
15#19149664—22no—
16#19149653C2212yes—
17#19149652—22no—
18C4×Dic346C2212yes—
19C4×Dic346C2212yes—
20#19149644—22no—
21C4×D692C2212yes—
22#19149638C2211yescharacteristic
23#19149635—22no—
24#19149613C2212yes—
25#19149611C2212yes—
26#19149604C2212yes—
27#19149604C2212yes—
28#19149601C2212yes—
29#19149601C2212yes—
30#117×C173C2211yescharacteristic
31C4⋊C8×C173C2212yes—
32C22⋊C8×C173C2212yes—
33#97×C173C2211yescharacteristic
34C4×C8×C173C2211yescharacteristic
35C2×Dic346—24no—
36C2×Dic346—21no—
37C2×Dic346C2312yes—
38C2×D692—22no—
39C2×D692C2312yes—
40C2×C22⋊D173—22no—
41C22×Dic173—21no—
42C2×C4×D173—22no—
43#14583815—24no—
44#14583815—24no—
45C4⋊Dic173C2312yes—
46C4⋊Dic173C2312yes—
47#14583805—22no—
48#14583805—22no—
49#14583805—22no—
50C22⋊Dic173—21no—
51#14583803—22no—
52#14583803—22no—
53C4×Dic173—22no—
54C4×Dic173—22no—
55C4×Dic173—21no—
56C22×C4×C173C2311yescharacteristic
57C2×C8×C173C2312yes—
58C2×C8×C173C2312yes—
59C4⋊C4×C173D412yes—
60C22⋊C4×C173D412yes—
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63C22×D173—22no—
64Dic346—24no—
65Dic346—22no—
66Dic346#4114yes—
67D692—22no—
68D692—22no—
69D692—82no—
70C22⋊D173—82no—
71C2×Dic173—24no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C4×D173—82no—
76C23×C173C2×D411yescharacteristic
77C2×C4×C173—22no—
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2411yescharacteristic
82C8×C173—22no—
83C8×C173—22no—
84C22×C173—21no—
85C22×C173—22no—
86C22×C173C22×D411yescharacteristic
87C2×D173—42no—
88C2×D173—42no—
89C2×D173—42no—
90C4×C173—21no—
91C4×C173—22no—
92C4×C173—22no—
93C4×C173C2×#4112yes—
94Dic173—24no—
95Dic173—42no—
96Dic173—42no—
97C2×C173—41no—
98C2×C173#52511yescharacteristic
99C2×C173#57712yes—
100D173—82no—
101C173#312611yescharacteristic, complemented, Sylow
102#3126—1731nomaximal, Sylow
103#540—1731no—
104#519—1731no—
105#482—1732no—
106#479—1732no—
107#460—1732no—
108#448—1732no—
109#440—1732no—
110C4×SD16—1732no—
111#433D17311yescharacteristic, complemented
112C22×Q8—1731no—
113C2×SD16—1732no—
114C2×SD16—1732no—
115#127—1731no—
116#125—1731no—
117#123—3462no—
118#121—3462no—
119#121—1732no—
120C4×Q8—3462no—
121C4×Q8—1732no—
122C4×Q8—1732no—
123C4×D4—1732no—
124#117—3462no—
125#117—1731no—
126#117C2×D17311yescharacteristic
127#107—1732no—
128#106—1732no—
129C4⋊C8C2×D17312yes—
130#103—1732no—
131#103—1732no—
132#102—1732no—
133#102—1732no—
134C22⋊C8C2×D17312yes—
135#97C2×D17311yescharacteristic
136C4×C8C2×D17311yescharacteristic
137C2×Q8—3464no—
138C2×Q8—3461no—
139C2×Q8—1732no—
140C2×D4—3462no—
141C2×D4—3462no—
142C2×D4—1732no—
143C22×C4—3461no—
144C22×C4—3462no—
145C22×C4C22×D17311yescharacteristic
146SD16—3464no—
147SD16—3464no—
148C2×C8C22×D17312yes—
149C2×C8C22×D17312yes—
150C4⋊C4—3462no—
151C4⋊C4—3462no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4—1732no—
155C4⋊C4D69212yes—
156C22⋊C4—3461no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C22⋊C4D69212yes—
160C42—3462no—
161C42—3462no—
162C42—3461no—
163C42C22×D17311yescharacteristic
164C42C22×D17311yescharacteristic
165C23—3462no—
166C23C2×D69211yescharacteristic
167Q8—3464no—
168Q8—3462no—
169Q8—1734no—
170D4—3462no—
171D4—3462no—
172D4—13842no—
173D4—13842no—
174C2×C4—22no—
175C2×C4—3464no—
176C2×C4—3462no—
177C2×C4—3462no—
178C2×C4—3462no—
179C2×C4—13842no—
180C2×C4C2×D69212yes—
181C2×C4C2×D69212yes—
182C2×C4C2×D69211yescharacteristic
183C2×C4C23×D17311yescharacteristic
184C8—22no—
185C8—22no—
186C22—21no—
187C22—22no—
188C22—6922no—
189C22—6922no—
190C22—6922no—
191C22C22×D69211yescharacteristic
192C4—21no—
193C4—22no—
194C4—22no—
195C4—3464no—
196C4—6922no—
197C4—6922no—
198C4#41×D17312yes—
199C2—41no—
200C2—13842no—
201C2#2601200311yescharacteristic
202C2#2601230312yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#433×C173C2
socleC22×C173C22×D4
prime radical#26012096C2
hypercenter#433D173
hypocenterC173#3126
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #26011968C2semidirect
3. #26011816C2semidirect
4. #26011799C2semidirect
5. #26011692C2semidirect
6. #26011632C2semidirect
7. #26011593C2semidirect
8. C4×#14583815C2semidirect
9. #433×C173C2semidirect
101. C173#3126semidirect
111. #433D173semidirect

Maximal subgroups

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

subgroupindex
#260120962
#260119682
#260118162
#260117992
#260116922
#260116322
#260115932
C4×#145838152
#433×C1732
#3126173