Finite Group Databasebeta

← #31791032

Subgroups — #31791032

Order 17536.

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

normal subgroups only

This group has 418 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#22578008C211yesmaximal, characteristic, complemented
2#22577727C212yesmaximal, complemented
3#22577703C212yesmaximal, complemented
4#22577598C212yesmaximal, complemented
5#22577537C212yesmaximal, complemented
6#22577513C212yesmaximal, complemented
7C4×#14555495C212yesmaximal, complemented
8#22577464C211yesmaximal, characteristic, complemented
9#518×C137C211yesmaximal, characteristic, complemented
10C2×#14555495C2212yes—
11C2×#14555495C2212yes—
12#17984085C2211yescharacteristic, complemented
13#17984080C2211yescharacteristic, complemented
14#17984073—22no—
15#17984062C2212yes—
16#17984061—22no—
17C4×D548C2212yescomplemented
18#17984021C2212yescomplemented
19#17984019C2212yescomplemented
20C4⋊#11764040C2212yescomplemented
21#17984009C2212yes—
22#17984009C2212yes—
23#17984008C2212yescomplemented
24#17984008C2212yescomplemented
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×D548—22no—
36C2×D548C2312yes—
37C2×C22⋊D137—22no—
38C2×C4×D137—22no—
39#14555495—24no—
40#14555495—24no—
41C2×#11764040C2312yes—
42C2×#11764040C2312yes—
43C4⋊Dic137C2312yes—
44#14555489—22no—
45#14555487—22no—
46#14555487—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
63C22×D137—22no—
64D548—22no—
65D548—22no—
66D548—82no—
67C22⋊D137—82no—
68C2×Dic137—22no—
69C4×D137—82no—
70#11764040—22no—
71#11764040—22no—
72C23×C137—24no—
73C23×C137—22no—
74C23×C137—82no—
75C23×C137C2×D411yescharacteristic
76D4×C137—24no—
77D4×C137—22no—
78D4×C137#4114yes—
79C2×C4×C137—22no—
80C2×C4×C137—22no—
81C2×C4×C137—82no—
82C2×C4×C137—82no—
83C2×C4×C137C2×D411yescharacteristic
84C2×C4×C137C2×D412yes—
85C2×C4×C137C2×D412yes—
86C2×C4×C137C2411yescharacteristic
87C22×C137—21no—
88C22×C137—24no—
89C22×C137—22no—
90C22×C137—42no—
91C22×C137—42no—
92C22×C137—84no—
93C22×C137C22×D411yescharacteristic
94C2×D137—42no—
95C2×D137—42no—
96C2×D137—42no—
97C4×C137—21no—
98C4×C137—22no—
99C4×C137—22no—
100C4×C137—42no—
101C4×C137C2×#4112yes—
102Dic137—42no—
103C2×C137—41no—
104C2×C137—44no—
105C2×C137#52511yescharacteristic
106C2×C137#57612yes—
107D137—82no—
108C137#312211yescharacteristic, complemented, Sylow
109#3122—1371nomaximal, Sylow
110#540—1371no—
111#518D13711yescharacteristic, complemented
112#482—1372no—
113#478—1372no—
114#459—1372no—
115#447—1372no—
116#442—1372no—
117C4×D8—1372no—
118#433—1371no—
119C22×D4C2×D13711yescharacteristic
120C2×D8—1372no—
121C2×D8—1372no—
122#127—1371no—
123#125—1371no—
124#123—2742no—
125#121—2742no—
126#121—1372no—
127C4×D4—22no—
128C4×D4—1372no—
129C4×D4C2×D13712yes—
130C4×D4C2×D13712yes—
131#117C2×D13711yescharacteristic
132#117C2×D13711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1372no—
135#106—1372no—
136C4⋊C8—1372no—
137#102—1372no—
138#102—1372no—
139#102—1372no—
140#102—1372no—
141C22⋊C8—1372no—
142#97—1371no—
143C4×C8—1371no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—2742no—
148C2×D4—2742no—
149C2×D4—1372no—
150C2×D4C22×D13712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—2742no—
154C22×C4C22×D13711yescharacteristic
155D8—2744no—
156D8—2744no—
157C2×C8—1372no—
158C2×C8—1372no—
159C4⋊C4—2742no—
160C4⋊C4—1372no—
161C4⋊C4C22⋊D13712yes—
162C4⋊C4C22×D13712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—2742no—
167C22⋊C4—2742no—
168C22⋊C4C22⋊D13712yes—
169C42—21no—
170C42C22×D13711yescharacteristic
171C42C22×D13711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—2742no—
176C23C2×C22⋊D13711yescharacteristic
177D4—24no—
178D4—22no—
179D4—2742no—
180D4—2742no—
181D4—10962no—
182D4—10962no—
183D4#1455551014yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—2742no—
189C2×C4—10962no—
190C2×C4C2×C22⋊D13711yescharacteristic
191C2×C4C2×C22⋊D13712yes—
192C2×C4C2×C22⋊D13712yes—
193C2×C4C23×D13711yescharacteristic
194C8—2742no—
195C8—2742no—
196C22—22no—
197C22—21no—
198C22—24no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—5482no—
203C22—5482no—
204C22—5482no—
205C22C22×C22⋊D13711yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—42no—
210C4—5482no—
211C4C2×#1455551012yes—
212C2—41no—
213C2—44no—
214C2—10962no—
215C2#2257791411yescharacteristic
216C2#2257820612yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#518×C137C2
socleC22×C137C22×D4
hypercenter#518D137
hypocenterC137#3122
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. #22578008C2semidirect
2. #22577727C2semidirect
3. #22577703C2semidirect
4. #22577598C2semidirect
5. #22577537C2semidirect
6. #22577513C2semidirect
7. C4×#14555495C2semidirect
8. #22577464C2semidirect
9. #518×C137C2semidirect
12. #17984085C22semidirect
13. #17984080C22semidirect
17. C4×D548C22semidirect
18. #17984021C22semidirect
19. #17984019C22semidirect
20. C4⋊#11764040C22semidirect
23. #17984008C22semidirect
24. #17984008C22semidirect
26. #17983999C22semidirect
27. C4×#11764040C22semidirect
108. C137#3122semidirect
111. #518D137semidirect

Maximal subgroups

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

subgroupindex
#225780082
#225777272
#225777032
#225775982
#225775372
#225775132
C4×#145554952
#225774642
#518×C1372
#3122137