Finite Group Databasebeta

← #31791041

Subgroups — #31791041

Order 17536.

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#22578015C211yesmaximal, characteristic, complemented
2#22577744C212yesmaximal, complemented
3#22577703C212yesmaximal, complemented
4#22577608C212yesmaximal, complemented
5#22577554C212yesmaximal
6#22577503C212yesmaximal, complemented
7C4×#14555498C212yesmaximal, complemented
8#22577464C211yesmaximal, characteristic, complemented
9#518×C137C211yesmaximal, characteristic
10C2×#14555498C2212yes—
11C2×#14555498C2212yes—
12#17984088C2211yescharacteristic, complemented
13#17984081—22no—
14#17984080C2211yescharacteristic, complemented
15#17984078—22no—
16#17984067C2212yes—
17C4×Dic274C2212yescomplemented
18#17984021C2212yescomplemented
19#17984019C2212yescomplemented
20C4⋊#11764040C2212yescomplemented
21#17984011C2212yescomplemented
22#17984011C2212yescomplemented
23#17984009C2212yes—
24#17984009C2212yes—
25C22⋊#11764040C2212yes—
26#17983999C2211yescharacteristic, complemented
27C4×#11764040C2211yescharacteristic, complemented
28C22×D4×C137C2211yescharacteristic
29C4×D4×C137—22no—
30C4×D4×C137C2212yes—
31C4×D4×C137C2212yes—
32#117×C137C2211yescharacteristic
33#117×C137C2211yescharacteristic
34C2×C22⋊C4×C137—22no—
35C2×Dic274C2312yes—
36#14555498—24no—
37#14555498—24no—
38C2×#11764040C2312yes—
39C2×#11764040C2312yes—
40C4⋊Dic137—22no—
41C4⋊Dic137C2312yes—
42#14555489—22no—
43#14555489—22no—
44#14555489—22no—
45C22⋊Dic137—22no—
46C4×Dic137—22no—
47C24×C137—21no—
48C2×D4×C137—24no—
49C2×D4×C137—21no—
50C2×D4×C137C2312yes—
51C22×C4×C137—22no—
52C22×C4×C137—22no—
53C22×C4×C137C2311yescharacteristic
54C4⋊C4×C137D412yes—
55C4⋊C4×C137C2312yes—
56C22⋊C4×C137—21no—
57C22⋊C4×C137—22no—
58C22⋊C4×C137—22no—
59C22⋊C4×C137D412yes—
60C42×C137—21no—
61C42×C137C2311yescharacteristic
62C42×C137C2311yescharacteristic
63Dic274—22no—
64Dic274—22no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67#11764040—22no—
68#11764040—22no—
69C23×C137—24no—
70C23×C137—22no—
71C23×C137—82no—
72C23×C137C2×D411yescharacteristic
73D4×C137—24no—
74D4×C137—22no—
75D4×C137#4114yes—
76C2×C4×C137—22no—
77C2×C4×C137—22no—
78C2×C4×C137—82no—
79C2×C4×C137—82no—
80C2×C4×C137C2×D411yescharacteristic
81C2×C4×C137C2×D412yes—
82C2×C4×C137C2×D412yes—
83C2×C4×C137C2411yescharacteristic
84C22×C137—21no—
85C22×C137—24no—
86C22×C137—22no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137—84no—
90C22×C137C22×D411yescharacteristic
91C4×C137—21no—
92C4×C137—22no—
93C4×C137—22no—
94C4×C137—42no—
95C4×C137C2×#4112yes—
96Dic137—42no—
97Dic137—42no—
98C2×C137—41no—
99C2×C137—44no—
100C2×C137#52511yescharacteristic
101C2×C137#57712yes—
102C137#312311yescharacteristic, complemented, Sylow
103#3123—1371nomaximal, Sylow
104#541—1371no—
105#518D13711yescharacteristic
106#485—1372no—
107#478—1372no—
108#461—1372no—
109#450—1372no—
110#440—1372no—
111C4×SD16—1372no—
112#433—1371no—
113C22×D4C2×D13711yescharacteristic
114C2×SD16—1372no—
115C2×SD16—1372no—
116C4⋊Q8—1371no—
117#126—2742no—
118#125—2742no—
119#125—1371no—
120C22⋊Q8—1372no—
121C4×Q8—1372no—
122C4×D4—22no—
123C4×D4C2×D13712yes—
124C4×D4C2×D13712yes—
125#117C2×D13711yescharacteristic
126#117C2×D13711yescharacteristic
127C2×C22⋊C4—22no—
128#107—1372no—
129#106—1372no—
130C4⋊C8—1372no—
131#103—1372no—
132#103—1372no—
133#102—1372no—
134#102—1372no—
135C22⋊C8—1372no—
136#97—1371no—
137C4×C8—1371no—
138C24—21no—
139C2×Q8—1372no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D13712yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D13711yescharacteristic
146SD16—2744no—
147SD16—2744no—
148C2×C8—1372no—
149C2×C8—1372no—
150C4⋊C4—2742no—
151C4⋊C4—2742no—
152C4⋊C4—2742no—
153C4⋊C4—2742no—
154C4⋊C4—1372no—
155C4⋊C4C22⋊D13712yes—
156C4⋊C4C22×D13712yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—2742no—
161C22⋊C4C22⋊D13712yes—
162C42—21no—
163C42—2742no—
164C42C22×D13711yescharacteristic
165C42C22×D13711yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D13711yescharacteristic
170Q8—2742no—
171Q8—2742no—
172D4—24no—
173D4—22no—
174D4#1455551014yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—2742no—
180C2×C4—2742no—
181C2×C4C2×C22⋊D13711yescharacteristic
182C2×C4C2×C22⋊D13712yes—
183C2×C4C2×C22⋊D13712yes—
184C2×C4C23×D13711yescharacteristic
185C8—2742no—
186C8—2742no—
187C22—22no—
188C22—21no—
189C22—24no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D13711yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—5482no—
199C4—5482no—
200C4C2×#1455551012yes—
201C2—41no—
202C2—44no—
203C2#2257791411yescharacteristic
204C2#2257821412yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#518×C137C2
socleC22×C137C22×D4
prime radicalC22×D4×C137C22
hypercenter#518D137
hypocenterC137#3123
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #22578015C2semidirect
2. #22577744C2semidirect
3. #22577703C2semidirect
4. #22577608C2semidirect
6. #22577503C2semidirect
7. C4×#14555498C2semidirect
8. #22577464C2semidirect
12. #17984088C22semidirect
14. #17984080C22semidirect
17. C4×Dic274C22semidirect
18. #17984021C22semidirect
19. #17984019C22semidirect
20. C4⋊#11764040C22semidirect
21. #17984011C22semidirect
22. #17984011C22semidirect
26. #17983999C22semidirect
27. C4×#11764040C22semidirect
102. C137#3123semidirect

Maximal subgroups

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

subgroupindex
#225780152
#225777442
#225777032
#225776082
#225775542
#225775032
C4×#145554982
#225774642
#518×C1372
#3123137