Finite Group Databasebeta

← #31784124

Subgroups — #31784124

Order 17536.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14555510C211yesmaximal, characteristic
2C2×#17984071C212yesmaximal, complemented
3#22577323C212yesmaximal, complemented
4#22577320C212yesmaximal, complemented
5#22577282C212yesmaximal, complemented
6#22577274C214yesmaximal, complemented
7#22577266C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C137C211yesmaximal, characteristic, complemented
9C2×#14555510—28no—
10C2×#14555510—24no—
11C22×Dic274C2211yescharacteristic
12C22×D548C2211yescharacteristic
13C22×C22⋊D137C2212yes—
14C22×C4×D137C2212yes—
15#17984071—28no—
16C2×#14555489C2214yes—
17C2×C22⋊Dic137C2212yes—
18C2×#14555487C2214yes—
19C2×#14555487C2214yes—
20#17983995C2212yes—
21#17983995C2212yes—
22#17983995C2214yes—
23C23×C4×C137C2211yescharacteristic
24C23×C4×C137C2211yescharacteristic
25C23×C4×C137C2211yescharacteristic
26C2×C4⋊C4×C137—24no—
27C2×C4⋊C4×C137C2212yes—
28C2×C4⋊C4×C137C2212yes—
29C23×D137C2312yes—
30#14555510—232no—
31C2×Dic274—24no—
32C2×Dic274—22no—
33C2×D548—24no—
34C2×D548—22no—
35C2×C22⋊D137—24no—
36C2×C22⋊D137—24no—
37C2×C22⋊D137D418yes—
38C22×Dic137C2314yes—
39C22×Dic137C2312yes—
40C2×C4×D137—24no—
41C2×C4×D137—24no—
42C2×C4×D137—24no—
43#14555489—28no—
44C22⋊Dic137—24no—
45#14555487—28no—
46#14555487—28no—
47C24×C137C2311yescharacteristic
48C22×C4×C137—24no—
49C22×C4×C137—22no—
50C22×C4×C137—24no—
51C22×C4×C137—24no—
52C22×C4×C137—22no—
53C22×C4×C137D414yes—
54C22×C4×C137C2312yes—
55C22×C4×C137C2311yescharacteristic
56C22×C4×C137C2311yescharacteristic
57C22×C4×C137C2312yes—
58C4⋊C4×C137—24no—
59C4⋊C4×C137—24no—
60C22×D137—22no—
61C22×D137—22no—
62C22×D137—24no—
63C22×D137—24no—
64C22×D137C2×D414yes—
65Dic274—28no—
66D548—28no—
67C22⋊D137—216no—
68C2×Dic137—22no—
69C2×Dic137—22no—
70C2×Dic137—24no—
71C2×Dic137—24no—
72C2×Dic137—24no—
73C2×Dic137C2×D414yes—
74C4×D137—216no—
75C23×C137—22no—
76C23×C137—22no—
77C23×C137C2×D414yes—
78C23×C137C2×D412yes—
79C23×C137C2411yescharacteristic
80C2×C4×C137—28no—
81C2×C4×C137—28no—
82C2×C4×C137—22no—
83C2×C4×C137—22no—
84C2×C4×C137—22no—
85C2×C4×C137—22no—
86C2×C4×C137—22no—
87C2×C4×C137—24no—
88C2×C4×C137—44no—
89C2×C4×C137C2×D412yes—
90C2×C4×C137C2×D412yes—
91C2×C4×C137#4114yescomplemented
92C22×C137—22no—
93C22×C137—22no—
94C22×C137—28no—
95C22×C137#12014yes—
96C22×C137C22×D412yes—
97C22×C137C22×D411yescharacteristic
98C22×C137C2×#4112yes—
99C22×C1372−1+412yes—
100C2×D137—24no—
101C2×D137—28no—
102C2×D137—42no—
103C2×D137—42no—
104C2×D137—44no—
105C4×C137—22no—
106C4×C137—22no—
107C4×C137—24no—
108C4×C137—42no—
109Dic137—24no—
110Dic137—44no—
111C2×C137—24no—
112C2×C137C2×#12011yescharacteristic
113C2×C137#52511yescharacteristic
114C2×C137#53611yescharacteristic
115C2×C137#54714yes—
116D137—44no—
117C137#240611yescharacteristic, complemented, Sylow
118#2406—1371nomaximal, Sylow
119C22×#41—1371no—
120C2×#123—1372no—
121C22×C4⋊C4D13711yescharacteristic, complemented
122#395—1372no—
123#394—1372no—
124#386—1372no—
125#385—1374no—
126#381—1371no—
127C2×#41—2748no—
128C2×#41—2744no—
129C22×Q8—1371no—
130C22×D4—1372no—
131C22×D4—1371no—
132C23×C4—1372no—
133C23×C4C2×D13711yescharacteristic
134C23×C4C2×D13711yescharacteristic
135C23×C4C2×D13711yescharacteristic
136#123—2748no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1374no—
139C2×C4⋊C4C2×D13712yes—
140C2×C4⋊C4C2×D13712yes—
141C2×C22⋊C4—1372no—
142C2×C22⋊C4—1374no—
143C2×C22⋊C4—1374no—
144#95—1372no—
145#95—1372no—
146#95—1374no—
147C24—1372no—
148C24C22×D13711yescharacteristic
149#41—27432no—
150C2×Q8—2744no—
151C2×Q8—2742no—
152C2×D4—2744no—
153C2×D4—2744no—
154C2×D4—2744no—
155C2×D4—2742no—
156C2×D4—1378no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—2744no—
163C22×C4—2744no—
164C22×C4—2744no—
165C22×C4—1374no—
166C22×C4—1372no—
167C22×C4C22⋊D13714yes—
168C22×C4C22×D13712yes—
169C22×C4C22×D13711yescharacteristic
170C22×C4C22×D13711yescharacteristic
171C22×C4C22×D13712yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—2748no—
175C22⋊C4—2748no—
176C22⋊C4—2744no—
177C22⋊C4—2748no—
178C23—22no—
179C23—22no—
180C23—2742no—
181C23—2742no—
182C23—2744no—
183C23—2744no—
184C23—1374no—
185C23D4×D13714yes—
186C23C2×C22⋊D13712yes—
187C23C23×D13711yescharacteristic
188Q8—2748no—
189D4—2748no—
190D4—27416no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—27416no—
201C2×C4—2742no—
202C2×C4—2742no—
203C2×C4—2744no—
204C2×C4—2744no—
205C2×C4—2744no—
206C2×C4—1374no—
207C2×C4C2×C22⋊D13712yes—
208C2×C4C2×C22⋊D13712yes—
209C2×C4#1455551014yescomplemented
210C22—22no—
211C22—28no—
212C22—22no—
213C22—2748no—
214C22—2744no—
215C22—5482no—
216C22—5484no—
217C22—5482no—
218C22#1798405614yes—
219C22C2×D4×D13712yes—
220C22C22×C22⋊D13711yescharacteristic
221C22C2×#1455551012yes—
222C22#1798413612yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—2744no—
228C4—5484no—
229C2—24no—
230C2—5484no—
231C2C2×#1798405611yescharacteristic
232C2#2257791411yescharacteristic
233C2#2257797711yescharacteristic
234C2#2257805414yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC22×C4⋊C4×C137C2
prime radicalC22×#14555510C2
hypercenterC22×C4⋊C4D137
hypocenterC137#2406
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17984071C2semidirect
3. #22577323C2semidirect
4. #22577320C2semidirect
5. #22577282C2semidirect
6. #22577274C2semidirect
7. #22577266C2semidirect
8. C22×C4⋊C4×C137C2semidirect
91. C2×C4×C137#41semidirect
117. C137#2406semidirect
121. C22×C4⋊C4D137semidirect
209. C2×C4#14555510semidirect

Maximal subgroups

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

subgroupindex
C22×#145555102
C2×#179840712
#225773232
#225773202
#225772822
#225772742
#225772662
C22×C4⋊C4×C1372
#2406137