Finite Group Databasebeta

← #36363854

Subgroups — #36363854

Order 22144.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#26011815C212yesmaximal, complemented
3#26011798C212yesmaximal, complemented
4#26011691C212yesmaximal, complemented
5#26011630C212yesmaximal, complemented
6#26011589C212yesmaximal, complemented
7C4×#14583813C212yesmaximal, complemented
8#26011552C211yesmaximal, characteristic, complemented
9#519×C173C211yesmaximal, characteristic, complemented
10C2×#14583813C2212yes—
11C2×#14583813C2212yes—
12#19149676C2211yescharacteristic
13#19149671C2211yescharacteristic
14#19149664—22no—
15#19149653C2212yes—
16#19149652—22no—
17C4×D692C2212yes—
18#19149612C2212yes—
19#19149610C2212yes—
20C4⋊#13063436C2212yes—
21#19149603C2212yes—
22#19149603C2212yes—
23#19149599C2212yes—
24#19149599C2212yes—
25C22⋊#13063436C2212yes—
26#19149590C2211yescharacteristic
27C4×#13063436C2211yescharacteristic
28C22×Q8×C173C2211yescharacteristic
29C4×Q8×C173—22no—
30C4×Q8×C173C2212yes—
31C4×Q8×C173C2212yes—
32#117×C173—22no—
33#117×C173C2211yescharacteristic
34#117×C173C2211yescharacteristic
35C2×D692—22no—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39#14583813—24no—
40#14583813—24no—
41C2×#13063436C2312yes—
42C2×#13063436C2312yes—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583803—22no—
46#14583803—22no—
47C2×Q8×C173—24no—
48C2×Q8×C173—21no—
49C2×Q8×C173C2312yes—
50C22×C4×C173—21no—
51C22×C4×C173C2311yescharacteristic
52C4⋊C4×C173—22no—
53C4⋊C4×C173—22no—
54C4⋊C4×C173D412yes—
55C4⋊C4×C173C2312yes—
56C22⋊C4×C173—21no—
57C22⋊C4×C173D412yes—
58C42×C173—22no—
59C42×C173—22no—
60C42×C173—21no—
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63C22×D173—22no—
64D692—22no—
65D692—22no—
66D692—82no—
67C22⋊D173—82no—
68C2×Dic173—22no—
69C4×D173—82no—
70#13063436—22no—
71#13063436—22no—
72C23×C173C2×D411yescharacteristic
73Q8×C173—24no—
74Q8×C173—22no—
75Q8×C173#4114yes—
76C2×C4×C173—24no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—22no—
80C2×C4×C173C2×D412yes—
81C2×C4×C173C2×D412yes—
82C2×C4×C173C2×D411yescharacteristic
83C2×C4×C173C2411yescharacteristic
84C22×C173—21no—
85C22×C173—22no—
86C22×C173C22×D411yescharacteristic
87C2×D173—42no—
88C2×D173—42no—
89C2×D173—42no—
90C4×C173—24no—
91C4×C173—21no—
92C4×C173—22no—
93C4×C173—22no—
94C4×C173—42no—
95C4×C173C2×#4112yes—
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#519D17311yescharacteristic, complemented
105#482—1732no—
106#479—1732no—
107#460—1732no—
108#448—1732no—
109#440—1732no—
110C4×SD16—1732no—
111#433—1731no—
112C22×Q8C2×D17311yescharacteristic
113C2×SD16—1732no—
114C2×SD16—1732no—
115#127—1731no—
116#125—1731no—
117#123—3462no—
118#121—3462no—
119#121—1732no—
120C4×Q8—22no—
121C4×Q8C2×D17312yes—
122C4×Q8C2×D17312yes—
123C4×D4—1732no—
124#117—22no—
125#117C2×D17311yescharacteristic
126#117C2×D17311yescharacteristic
127#107—1732no—
128#106—1732no—
129C4⋊C8—1732no—
130#103—1732no—
131#103—1732no—
132#102—1732no—
133#102—1732no—
134C22⋊C8—1732no—
135#97—1731no—
136C4×C8—1731no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D17312yes—
140C2×D4—3462no—
141C2×D4—3462no—
142C2×D4—1732no—
143C22×C4—21no—
144C22×C4—3462no—
145C22×C4C22×D17311yescharacteristic
146SD16—3464no—
147SD16—3464no—
148C2×C8—1732no—
149C2×C8—1732no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4C22⋊D17312yes—
155C4⋊C4C22×D17312yes—
156C22⋊C4—21no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C22⋊C4C22⋊D17312yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D17311yescharacteristic
164C42C22×D17311yescharacteristic
165C23—3462no—
166C23C2×C22⋊D17311yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1458382614yes—
170D4—3462no—
171D4—3462no—
172D4—13842no—
173D4—13842no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3462no—
179C2×C4—13842no—
180C2×C4C2×C22⋊D17312yes—
181C2×C4C2×C22⋊D17312yes—
182C2×C4C2×C22⋊D17311yescharacteristic
183C2×C4C23×D17311yescharacteristic
184C8—3462no—
185C8—3462no—
186C22—21no—
187C22—22no—
188C22—6922no—
189C22—6922no—
190C22—6922no—
191C22C22×C22⋊D17311yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—6922no—
198C4C2×#1458382612yes—
199C2—41no—
200C2—13842no—
201C2#2601200211yescharacteristic
202C2#2601229912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#519×C173C2
socleC22×C173C22×D4
prime radical#26012096C2
hypercenter#519D173
hypocenterC173#3126
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #26011815C2semidirect
3. #26011798C2semidirect
4. #26011691C2semidirect
5. #26011630C2semidirect
6. #26011589C2semidirect
7. C4×#14583813C2semidirect
8. #26011552C2semidirect
9. #519×C173C2semidirect
101. C173#3126semidirect
104. #519D173semidirect

Maximal subgroups

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

subgroupindex
#260120962
#260118152
#260117982
#260116912
#260116302
#260115892
C4×#145838132
#260115522
#519×C1732
#3126173