Finite Group Databasebeta

← #31796736

Subgroups — #31796736

Order 17536.

324 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 92 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×#14555519C211yesmaximal, characteristic
2C2×#17984155C211yesmaximal, characteristic, complemented
3C2×C4⋊#11764047C211yesmaximal, characteristic, complemented
4C2×C22⋊#11764047C211yesmaximal, characteristic
5#22578269C211yesmaximal, characteristic, complemented
6#22578269C211yesmaximal, characteristic, complemented
7C22⋊C4×Dic137C211yesmaximal, characteristic
8C2×#14555519—22no—
9C2×#14555519—22no—
10C2×#14555519—22no—
11C22×#11764047C2211yescharacteristic
12C22×#11764047C2211yescharacteristic
13C22×#11764047C2211yescharacteristic
14C22×#11764047C2211yescharacteristic
15#17984155—22no—
16C4⋊#11764047—22no—
17C22⋊#11764047—22no—
18C23×Dic137C2211yescharacteristic
19C2×C22⋊Dic137C411yescharacteristic
20C2×C4×Dic137C2211yescharacteristic
21C2×C4×Dic137C2211yescharacteristic
22#17983995C411yescharacteristic
23#17983995C411yescharacteristic
24C2×C22⋊C4×C137C411yescharacteristic
25#14555519—28no—
26C2×#11764047—21no—
27C2×#11764047—21no—
28C2×#11764047—21no—
29C2×#11764047—21no—
30C2×#11764047—21no—
31C2×#11764047—21no—
32C2×#11764047—21no—
33C2×#11764047—21no—
34C2×#11764047—21no—
35C2×#11764047—21no—
36C2×#11764047D412yes—
37C2×#11764047D412yes—
38C22×Dic137—21no—
39C22×Dic137—21no—
40C22×Dic137—21no—
41C22×Dic137—21no—
42C22×Dic137C2×C411yescharacteristic
43C22×Dic137C2×C411yescharacteristic
44C22×Dic137C2×C411yescharacteristic
45C22×Dic137D412yes—
46C22×Dic137Q812yes—
47C22×Dic137C2311yescharacteristic
48C22⋊Dic137—22no—
49C4×Dic137—21no—
50C4×Dic137—21no—
51C4×Dic137—21no—
52C4×Dic137—21no—
53C24×C137C2×C411yescharacteristic
54C22×C4×C137C2×C411yescharacteristic
55C22×C4×C137C2×C411yescharacteristic
56C22⋊C4×C137—22no—
57#11764047—22no—
58#11764047—22no—
59#11764047—41no—
60#11764047—41no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67C2×Dic137—22no—
68C2×Dic137—22no—
69C2×Dic137—21no—
70C2×Dic137—21no—
71C2×Dic137—21no—
72C2×Dic137C4⋊C412yes—
73C2×Dic137M1612yescomplemented
74C2×Dic137C2×D411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C2×Dic137C2×D411yescharacteristic
77C2×Dic137C2×Q811yescharacteristic
78C2×Dic137#4111yescharacteristic
79C2×Dic137#4111yescharacteristic
80C23×C137—21no—
81C23×C137—21no—
82C23×C137—21no—
83C23×C137—21no—
84C23×C137—21no—
85C23×C137—21no—
86C23×C137C4⋊C412yes—
87C23×C137C22×C411yescharacteristic
88C2×C4×C137—21no—
89C2×C4×C137—21no—
90C2×C4×C137—21no—
91C2×C4×C137—21no—
92C2×C4×C137—21no—
93C2×C4×C137M1612yescomplemented
94C22×C137—22no—
95C22×C137—22no—
96C22×C137—22no—
97C22×C137—41no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137—41no—
101C22×C137C2×C4⋊C411yescharacteristic
102C22×C137C4×D411yescharacteristic
103C22×C137C4×D411yescharacteristic
104C22×C137C2×M1611yescharacteristic
105C22×C137C2×M1611yescharacteristic
106C22×C137#13111yescharacteristic
107C22×C137#13111yescharacteristic
108C4×C137—22no—
109C4×C137—41no—
110Dic137—22no—
111Dic137—22no—
112Dic137—41no—
113Dic137#12011yescharacteristic
114Dic137C22⋊Q811yescharacteristic
115Dic137C22⋊Q811yescharacteristic
116Dic137#12311yescharacteristic
117C2×C137—42no—
118C2×C137#38511yescharacteristic
119C2×C137#42311yescharacteristic
120C2×C137#42411yescharacteristic
121C2×C137#43511yescharacteristic
122C2×C137#43511yescharacteristic
123C2×C137#43611yescharacteristic
124C2×C137#43611yescharacteristic
125C137#185111yescharacteristic, complemented, Sylow
126#1851—1371nomaximal, Sylow
127C22×M16—1371no—
128C2×C4⋊C8—1371no—
129C2×C4⋊C8—1371no—
130C2×C22⋊C8—1371no—
131C4×C22⋊C4—1371no—
132#336—1371no—
133#336—1371no—
134C23×C4—1371no—
135C2×M16—2742no—
136C2×M16—2742no—
137C2×M16—2742no—
138C22×C8—1371no—
139C22×C8—1371no—
140C22×C8—1371no—
141C22×C8—1371no—
142C2×C22⋊C4—1371no—
143C2×C22⋊C4#1058773911yescharacteristic
144C2×C42—1371no—
145C2×C42—1371no—
146C4⋊C8—2742no—
147C4⋊C8—2742no—
148C22⋊C8—2742no—
149#95—1371no—
150#95—1371no—
151C24C2×#1058773911yescharacteristic
152C22×C4—2741no—
153C22×C4—2741no—
154C22×C4—2741no—
155C22×C4—2741no—
156C22×C4—1371no—
157C22×C4—1372no—
158C22×C4—1372no—
159C22×C4—1371no—
160C22×C4—1371no—
161C22×C4—1371no—
162C22×C4C2×#1058773911yescharacteristic
163C22×C4C2×#1058773911yescharacteristic
164M16—2748no—
165C2×C8—2741no—
166C2×C8—2741no—
167C2×C8—2741no—
168C2×C8—2741no—
169C2×C8—2741no—
170C2×C8—2741no—
171C2×C8—2741no—
172C2×C8—2741no—
173C2×C8—2741no—
174C2×C8—2741no—
175C2×C8—1372no—
176C2×C8—1372no—
177C22⋊C4—22no—
178C22⋊C4—2742no—
179C42—2741no—
180C42—2741no—
181C42—2741no—
182C42—2741no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23C4⋊#1058773912yes—
190C23C22×#1058773911yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—2741no—
199C2×C4—2741no—
200C2×C4—2742no—
201C2×C4—2742no—
202C2×C4—2742no—
203C2×C4—2742no—
204C2×C4—2741no—
205C2×C4—2741no—
206C2×C4—2741no—
207C2×C4—1372no—
208C2×C4—1372no—
209C2×C4—1371no—
210C2×C4—1371no—
211C2×C4—1371no—
212C2×C4—1371no—
213C2×C4—1371no—
214C2×C4—1371no—
215C2×C4#1455551912yescomplemented
216C8—2742no—
217C8—2742no—
218C8—5481no—
219C8—5481no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22C2×C4⋊#1058773911yescharacteristic
228C22D4×#1058773911yescharacteristic
229C22D4×#1058773911yescharacteristic
230C22C2×#1455551911yescharacteristic
231C22D137⋊M1611yescharacteristic
232C22#1798417511yescharacteristic
233C22#1798417511yescharacteristic
234C4—22no—
235C4—41no—
236C4—2742no—
237C4—2742no—
238C4—5481no—
239C4—1371no—
240C4—1371no—
241C4—1371no—
242C4—1371no—
243C2—42no—
244C2#2257835011yescharacteristic
245C2#2257840611yescharacteristic
246C2#2257840911yescharacteristic
247C2#2257843411yescharacteristic
248C2#2257843511yescharacteristic
249C2#2257843611yescharacteristic
250C2#2257843811yescharacteristic
subgroupG/H
center, FrattiniC23C22×#10587739
commutator, socleC23×C137C22×C4
FittingC2×C22⋊C4×C137C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1851
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17984155C2semidirect
3. C2×C4⋊#11764047C2semidirect
5. #22578269C2semidirect
6. #22578269C2semidirect
73. C2×Dic137M16semidirect
93. C2×C4×C137M16semidirect
125. C137#1851semidirect
215. C2×C4#14555519semidirect

Maximal subgroups

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

subgroupindex
C22×#145555192
C2×#179841552
C2×C4⋊#117640472
C2×C22⋊#117640472
#225782692
C22⋊C4×Dic1372
#1851137