Finite Group Databasebeta

← #28373284

Subgroups — #28373284

Order 13952.

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

normal subgroups only

This group has 394 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#20295638C211yesmaximal, characteristic, complemented
2#20295367C212yesmaximal, complemented
3#20295326C212yesmaximal, complemented
4#20295231C212yesmaximal, complemented
5#20295177C212yesmaximal
6#20295126C212yesmaximal, complemented
7C4×#13182017C212yesmaximal, complemented
8#20295087C211yesmaximal, characteristic, complemented
9#518×C109C211yesmaximal, characteristic
10C2×#13182017C2212yes—
11C2×#13182017C2212yes—
12#16812699C2211yescharacteristic, complemented
13#16812692—22no—
14#16812691C2211yescharacteristic, complemented
15#16812689—22no—
16#16812678C2212yes—
17C4×Dic218C2212yescomplemented
18#16812632C2212yescomplemented
19#16812630C2212yescomplemented
20C4⋊#11724784C2212yescomplemented
21#16812622C2212yescomplemented
22#16812622C2212yescomplemented
23#16812620C2212yes—
24#16812620C2212yes—
25C22⋊#11724784C2212yes—
26#16812610C2211yescharacteristic, complemented
27C4×#11724784C2211yescharacteristic, complemented
28C22×D4×C109C2211yescharacteristic
29C4×D4×C109—22no—
30C4×D4×C109C2212yes—
31C4×D4×C109C2212yes—
32#117×C109C2211yescharacteristic
33#117×C109C2211yescharacteristic
34C2×C22⋊C4×C109—22no—
35C2×Dic218C2312yes—
36#13182017—24no—
37#13182017—24no—
38C2×#11724784C2312yes—
39C2×#11724784C2312yes—
40C4⋊Dic109—22no—
41C4⋊Dic109C2312yes—
42#13182008—22no—
43#13182008—22no—
44#13182008—22no—
45C22⋊Dic109—22no—
46C4×Dic109—22no—
47C24×C109—21no—
48C2×D4×C109—24no—
49C2×D4×C109—21no—
50C2×D4×C109C2312yes—
51C22×C4×C109—22no—
52C22×C4×C109—22no—
53C22×C4×C109C2311yescharacteristic
54C4⋊C4×C109D412yes—
55C4⋊C4×C109C2312yes—
56C22⋊C4×C109—21no—
57C22⋊C4×C109—22no—
58C22⋊C4×C109—22no—
59C22⋊C4×C109D412yes—
60C42×C109—21no—
61C42×C109C2311yescharacteristic
62C42×C109C2311yescharacteristic
63C23×C109—24no—
64C23×C109—22no—
65C23×C109—82no—
66C23×C109C2×D411yescharacteristic
67D4×C109—24no—
68D4×C109—22no—
69D4×C109#4114yes—
70C2×C4×C109—22no—
71C2×C4×C109—22no—
72C2×C4×C109—82no—
73C2×C4×C109—82no—
74C2×C4×C109C2×D411yescharacteristic
75C2×C4×C109C2×D412yes—
76C2×C4×C109C2×D412yes—
77C2×C4×C109C2411yescharacteristic
78C2×Dic109—22no—
79C2×Dic109—22no—
80Dic218—22no—
81Dic218—22no—
82#11724784—22no—
83#11724784—22no—
84C22×C109—21no—
85C22×C109—24no—
86C22×C109—22no—
87C22×C109—42no—
88C22×C109—42no—
89C22×C109—84no—
90C22×C109C22×D411yescharacteristic
91C4×C109—21no—
92C4×C109—22no—
93C4×C109—22no—
94C4×C109—42no—
95C4×C109C2×#4112yes—
96Dic109—42no—
97Dic109—42no—
98C2×C109—41no—
99C2×C109—44no—
100C2×C109#52511yescharacteristic
101C2×C109#57712yes—
102#3123—1091nomaximal, Sylow
103C109#312311yescharacteristic, complemented, Sylow
104#541—1091no—
105#518D10911yescharacteristic
106#485—1092no—
107#478—1092no—
108#461—1092no—
109#450—1092no—
110#440—1092no—
111C4×SD16—1092no—
112#433—1091no—
113C22×D4C2×D10911yescharacteristic
114C2×SD16—1092no—
115C2×SD16—1092no—
116C4⋊Q8—1091no—
117#126—2182no—
118#125—1091no—
119#125—2182no—
120C22⋊Q8—1092no—
121C4×Q8—1092no—
122C4×D4—22no—
123C4×D4C2×D10912yes—
124C4×D4C2×D10912yes—
125#117C2×D10911yescharacteristic
126#117C2×D10911yescharacteristic
127C2×C22⋊C4—22no—
128#107—1092no—
129#106—1092no—
130C4⋊C8—1092no—
131#103—1092no—
132#103—1092no—
133#102—1092no—
134#102—1092no—
135C22⋊C8—1092no—
136#97—1091no—
137C4×C8—1091no—
138C24—21no—
139C2×Q8—1092no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D10912yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D10911yescharacteristic
146SD16—2184no—
147SD16—2184no—
148C2×C8—1092no—
149C2×C8—1092no—
150C4⋊C4—1092no—
151C4⋊C4—2182no—
152C4⋊C4—2182no—
153C4⋊C4—2182no—
154C4⋊C4—2182no—
155C4⋊C4C22⋊D10912yes—
156C4⋊C4C22×D10912yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—2182no—
161C22⋊C4C22⋊D10912yes—
162C42—21no—
163C42—2182no—
164C42C22×D10911yescharacteristic
165C42C22×D10911yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D10911yescharacteristic
170Q8—2182no—
171Q8—2182no—
172D4—24no—
173D4—22no—
174D4#1318202914yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—2182no—
180C2×C4—2182no—
181C2×C4C2×C22⋊D10911yescharacteristic
182C2×C4C2×C22⋊D10912yes—
183C2×C4C2×C22⋊D10912yes—
184C2×C4C23×D10911yescharacteristic
185C8—2182no—
186C8—2182no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D10911yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—4362no—
199C4—4362no—
200C4C2×#1318202912yes—
201C2—41no—
202C2—44no—
203C2#2029553711yescharacteristic
204C2#2029583712yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#518×C109C2
socleC22×C109C22×D4
prime radicalC22×D4×C109C22
hypercenter#518D109
hypocenterC109#3123
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20295638C2semidirect
2. #20295367C2semidirect
3. #20295326C2semidirect
4. #20295231C2semidirect
6. #20295126C2semidirect
7. C4×#13182017C2semidirect
8. #20295087C2semidirect
12. #16812699C22semidirect
14. #16812691C22semidirect
17. C4×Dic218C22semidirect
18. #16812632C22semidirect
19. #16812630C22semidirect
20. C4⋊#11724784C22semidirect
21. #16812622C22semidirect
22. #16812622C22semidirect
26. #16812610C22semidirect
27. C4×#11724784C22semidirect
103. C109#3123semidirect

Maximal subgroups

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

subgroupindex
#202956382
#202953672
#202953262
#202952312
#202951772
#202951262
C4×#131820172
#202950872
#518×C1092
#3123109