Finite Group Databasebeta

← #31790366

Subgroups — #31790366

Order 17536.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C22×C8×D137C211yesmaximal, characteristic, complemented
2#22577910C212yesmaximal, complemented
3#22577641C212yesmaximal, complemented
4#22577636C212yesmaximal, complemented
5#22577629C214yesmaximal, complemented
6#22577441C211yesmaximal, characteristic, complemented
7#22577399C212yesmaximal, complemented
8C2×#135×C137C211yesmaximal, characteristic, complemented
9C22×C4×D137C2211yescharacteristic
10C22×#11764040C2211yescharacteristic
11C2×C8×D137—24no—
12C2×C8×D137C2212yes—
13C2×C8×D137C2212yes—
14#17984069—22no—
15#17984064C2214yes—
16#17984059C2214yes—
17#17984056—22no—
18C4×C22⋊D137—24no—
19#17984047C2212yes—
20#17984022C2212yescomplemented
21#17984020C2212yescomplemented
22#17984012C2214yescomplemented
23#17984009C2214yescomplemented
24C2×#41×C137C2212yes—
25#135×C137—24no—
26C2×Q16×C137C2211yescharacteristic, complemented
27C2×SD16×C137C2212yescomplemented
28C2×D8×C137C2211yescharacteristic, complemented
29C22×C8×C137C2211yescharacteristic
30C23×D137D411yescharacteristic
31C2×C22⋊D137—24no—
32C22×Dic137D411yescharacteristic
33C2×C4×D137—22no—
34C2×C4×D137—22no—
35C2×C4×D137D412yes—
36C2×C4×D137C2312yes—
37C2×#11764040—22no—
38C2×#11764040C2312yes—
39C8×D137—24no—
40C8×D137—24no—
41C4⋊Dic137C2314yes—
42#14555489—24no—
43C22⋊Dic137—24no—
44C22⋊Dic137—22no—
45#14555487—24no—
46C4×Dic137—22no—
47#41×C137—24no—
48#41×C137—24no—
49C2×Q8×C137C2312yes—
50C2×D4×C137—22no—
51C2×D4×C137C2312yes—
52C22×C4×C137—22no—
53C22×C4×C137C2311yescharacteristic
54Q16×C137—22no—
55SD16×C137—24no—
56D8×C137—22no—
57C2×C8×C137D414yescomplemented
58C2×C8×C137C2311yescharacteristic
59C2×C8×C137C2311yescharacteristic
60C22×D137—22no—
61C22×D137—24no—
62C22×D137C2×D412yes—
63C22⋊D137—48no—
64C2×Dic137—22no—
65C2×Dic137—24no—
66C2×Dic137C2×D412yes—
67C4×D137—24no—
68C4×D137—22no—
69C4×D137—22no—
70#11764040—22no—
71C23×C137—22no—
72C23×C137C2×D411yescharacteristic
73Q8×C137—22no—
74Q8×C137—22no—
75D4×C137—22no—
76D4×C137—22no—
77D4×C137—44no—
78C2×C4×C137—22no—
79C2×C4×C137—44no—
80C2×C4×C137C2×D411yescharacteristic
81C2×C4×C137C2×D412yes—
82C2×C4×C137#4112yes—
83C2×C4×C137C2411yescharacteristic
84C8×C137C2×D412yes—
85C8×C137C2×D412yes—
86C22×C137—22no—
87C22×C137—42no—
88C22×C137—44no—
89C22×C137#12112yes—
90C22×C137C22×D411yescharacteristic
91C2×D137—28no—
92C2×D137—22no—
93C2×D137—22no—
94C2×D137#13514yescomplemented
95C4×C137—42no—
96C4×C137#12112yes—
97C4×C137C22×D411yescharacteristic
98C4×C137C2×#4111yescharacteristic
99Dic137—22no—
100Dic137—44no—
101C2×C137—22no—
102C2×C137—82no—
103C2×C137C2×#12111yescharacteristic
104C2×C137C2×#13512yes—
105D137—24no—
106C137#305011yescharacteristic, complemented, Sylow
107#3050—1371nomaximal, Sylow
108C2×#135D13711yescharacteristic, complemented
109C23×C8—1371no—
110#525—1372no—
111#467—1372no—
112#466—1372no—
113#465—1374no—
114#427—1371no—
115#416—1372no—
116C2×#41C2×D13712yes—
117C23×C4—1371no—
118#135—24no—
119C2×Q16C2×D13711yescharacteristic, complemented
120C2×SD16C2×D13712yescomplemented
121C2×D8C2×D13711yescharacteristic, complemented
122C22×C8—2744no—
123C22×C8—1371no—
124C22×C8—1372no—
125C22×C8—1372no—
126C22×C8C2×D13711yescharacteristic
127#123—2742no—
128C22⋊Q8—1374no—
129#121—1374no—
130#120—2742no—
131C4×D4—2744no—
132#117—1372no—
133#107—1372no—
134#106—1372no—
135#103—1374no—
136#102—1374no—
137C24—1371no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D13712yes—
141C2×D4—22no—
142C2×D4—2744no—
143C2×D4C22×D13712yes—
144C22×C4—22no—
145C22×C4—2742no—
146C22×C4—2742no—
147C22×C4—1371no—
148C22×C4—1372no—
149C22×C4—1372no—
150C22×C4C22×D13711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—2742no—
155C2×C8—2744no—
156C2×C8—2744no—
157C2×C8—1372no—
158C2×C8C22⋊D13714yescomplemented
159C2×C8C22×D13711yescharacteristic
160C2×C8C22×D13711yescharacteristic
161C4⋊C4—2744no—
162C4⋊C4—1374no—
163C22⋊C4—2744no—
164C22⋊C4—2742no—
165C22⋊C4—2744no—
166C42—2742no—
167C23—22no—
168C23—2744no—
169C23—2742no—
170C23—1372no—
171C23D4×D13711yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—5488no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—2742no—
181C2×C4—2744no—
182C2×C4—2742no—
183C2×C4—2742no—
184C2×C4—2744no—
185C2×C4—1372no—
186C2×C4D4×D13711yescharacteristic
187C2×C4C2×C22⋊D13712yes—
188C2×C4#1455550912yes—
189C2×C4C23×D13711yescharacteristic
190C8—2742no—
191C8C2×C22⋊D13712yes—
192C8C2×C22⋊D13712yes—
193C22—22no—
194C22—44no—
195C22—42no—
196C22—2748no—
197C22—2742no—
198C22—2742no—
199C22—1374no—
200C22#1798405912yes—
201C22C2×D4×D13711yescharacteristic
202C4—42no—
203C4—2742no—
204C4—5484no—
205C4#1798405912yes—
206C4C22×C22⋊D13711yescharacteristic
207C4C2×#1455550911yescharacteristic
208C2—22no—
209C2—82no—
210C2—2744no—
211C2C2×#1798405911yescharacteristic
212C2#135×D13712yes—
subgroupG/H
centerC2×C4D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#135×C137C2
socleC22×C137C22×D4
hypercenterC2×#135D137
hypocenterC137#3050
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×C8×D137C2semidirect
2. #22577910C2semidirect
3. #22577641C2semidirect
4. #22577636C2semidirect
5. #22577629C2semidirect
6. #22577441C2semidirect
7. #22577399C2semidirect
8. C2×#135×C137C2semidirect
20. #17984022C22semidirect
21. #17984020C22semidirect
22. #17984012C22semidirect
23. #17984009C22semidirect
26. C2×Q16×C137C22semidirect
27. C2×SD16×C137C22semidirect
28. C2×D8×C137C22semidirect
57. C2×C8×C137D4semidirect
94. C2×D137#135semidirect
106. C137#3050semidirect
108. C2×#135D137semidirect
119. C2×Q16C2×D137semidirect
120. C2×SD16C2×D137semidirect
121. C2×D8C2×D137semidirect
158. C2×C8C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D1372
#225779102
#225776412
#225776362
#225776292
#225774412
#225773992
C2×#135×C1372
#3050137