Finite Group Databasebeta

← #31796816

Subgroups — #31796816

Order 17536.

338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 74 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×#17984149C212yesmaximal, complemented
3#22578373C211yesmaximal, characteristic
4#22578283C212yesmaximal, complemented
5C2×#17984065C211yesmaximal, characteristic
6C2×#14555519—24no—
7C2×#14555519—21no—
8C2×#14555519C2212yes—
9C2×#14555519C2212yes—
10C22×#11764047—21no—
11#17984149—24no—
12C22⋊#11764047—21no—
13C22⋊#11764047—21no—
14C22×Dic274C2211yescharacteristic
15C23×Dic137C2211yescharacteristic
16#17984065—21no—
17#17984065—21no—
18#17984065—21no—
19#17984065—21no—
20C2×C4⋊Dic137C411yescharacteristic
21C2×#14555489C411yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×C22⋊Dic137C411yescharacteristic
24C2×C22⋊C4×C137C411yescharacteristic
25#14555519—24no—
26#14555519—22no—
27#14555519—22no—
28#14555519—22no—
29#14555519D414yes—
30C2×#11764047—24no—
31C2×#11764047—22no—
32C2×#11764047—22no—
33C2×Dic274—21no—
34C2×Dic274—21no—
35C2×Dic274—22no—
36C2×Dic274—22no—
37C22×Dic137—22no—
38C22×Dic137—22no—
39C22×Dic137C2×C411yescharacteristic
40C22×Dic137C2×C411yescharacteristic
41C22×Dic137C2×C411yescharacteristic
42C22×Dic137D411yescharacteristic
43C22×Dic137D411yescharacteristic
44C22×Dic137D411yescharacteristic
45C22×Dic137D411yescharacteristic
46C22×Dic137C2311yescharacteristic
47C4⋊Dic137—21no—
48C4⋊Dic137—21no—
49#14555489—21no—
50#14555489—21no—
51#14555489—21no—
52#14555489—21no—
53C22⋊Dic137—21no—
54C22⋊Dic137—21no—
55C24×C137C2×C411yescharacteristic
56C22×C4×C137C2×C411yescharacteristic
57C22×C4×C137C2×C411yescharacteristic
58C22⋊C4×C137—21no—
59C22⋊C4×C137—21no—
60#11764047—24no—
61#11764047—42no—
62Dic274—41no—
63Dic274—42no—
64Dic274—41no—
65C2×Dic137—22no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137—22no—
69C2×Dic137—22no—
70C2×Dic137—21no—
71C2×Dic137—21no—
72C2×Dic137—21no—
73C2×Dic137—21no—
74C2×Dic137—22no—
75C2×Dic137—41no—
76C2×Dic137C22⋊C411yescharacteristic
77C2×Dic137C22⋊C411yescharacteristic
78C2×Dic137C2×D412yes—
79C2×Dic137C2×D411yescharacteristic
80C2×Dic137C2×D411yescharacteristic
81C2×Dic137#4112yes—
82C23×C137—22no—
83C23×C137—22no—
84C23×C137—41no—
85C23×C137C22⋊C411yescharacteristic
86C23×C137C22⋊C411yescharacteristic
87C23×C137C22×C411yescharacteristic
88C2×C4×C137—21no—
89C2×C4×C137—21no—
90C2×C4×C137—22no—
91C2×C4×C137—41no—
92C22×C137—21no—
93C22×C137—21no—
94C22×C137—21no—
95C22×C137—21no—
96C22×C137—21no—
97C22×C137—21no—
98C22×C137—21no—
99C22×C137—44no—
100C22×C137#10112yescomplemented
101C22×C137C2×C22⋊C411yescharacteristic
102C22×C137C4×D412yes—
103C4×C137—41no—
104C4×C137—41no—
105Dic137—21no—
106Dic137—21no—
107Dic137—41no—
108Dic137—41no—
109Dic137#12011yescharacteristic
110Dic137#12112yes—
111Dic137#12311yescharacteristic
112C2×C137—22no—
113C2×C137—22no—
114C2×C137—41no—
115C2×C137#38611yescharacteristic
116C2×C137C2×#10111yescharacteristic
117C2×C137#41311yescharacteristic
118C137#190111yescharacteristic, complemented, Sylow
119#1901—1371nomaximal, Sylow
120C22×M16—1371no—
121C2×C22⋊Q8—1371no—
122C2×#101—1372no—
123#407—1371no—
124#343—1372no—
125C22×Q8—1371no—
126C23×C4—1371no—
127C2×M16—2741no—
128C2×M16—2744no—
129C2×M16—1372no—
130C2×M16—1372no—
131C22×C8—2741no—
132C22⋊Q8—2741no—
133C22⋊Q8—2741no—
134C22⋊Q8—2741no—
135C22⋊Q8—2741no—
136C2×C4⋊C4—1371no—
137C2×C4⋊C4—1371no—
138C2×C4⋊C4—1371no—
139C2×C22⋊C4—1371no—
140C2×C22⋊C4#1058773911yescharacteristic
141#101—2744no—
142C22⋊C8—2741no—
143C22⋊C8—2741no—
144C24C2×#1058773911yescharacteristic
145C2×Q8—2742no—
146C2×Q8—2741no—
147C2×Q8—2741no—
148C2×Q8—2742no—
149C22×C4—2742no—
150C22×C4—2742no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4—1371no—
156C22×C4—1371no—
157C22×C4—1371no—
158C22×C4—1371no—
159C22×C4C2×#1058773911yescharacteristic
160C22×C4C2×#1058773911yescharacteristic
161M16—2744no—
162M16—2742no—
163M16—2742no—
164M16—2742no—
165M16—1374no—
166C2×C8—2744no—
167C2×C8—2742no—
168C2×C8—2742no—
169C4⋊C4—2741no—
170C4⋊C4—2741no—
171C4⋊C4—2741no—
172C4⋊C4—2741no—
173C4⋊C4—2741no—
174C4⋊C4—2741no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—2741no—
178C22⋊C4—2741no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#1058773911yescharacteristic
183C23C22⋊#1058773911yescharacteristic
184C23C22×#1058773911yescharacteristic
185Q8—5481no—
186Q8—5482no—
187Q8—5481no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—2742no—
193C2×C4—2741no—
194C2×C4—2741no—
195C2×C4—2742no—
196C2×C4—2742no—
197C2×C4—2741no—
198C2×C4—2741no—
199C2×C4—2741no—
200C2×C4—2741no—
201C2×C4—2742no—
202C2×C4—5481no—
203C2×C4—1371no—
204C2×C4—1371no—
205C2×C4—1371no—
206C2×C4—1371no—
207C2×C4—1372no—
208C2×C4—1372no—
209C8—2744no—
210C8—5482no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1798414912yescomplemented
220C22C2×C22⋊#1058773911yescharacteristic
221C22D4×#1058773912yes—
222C4—41no—
223C4—41no—
224C4—2741no—
225C4—2741no—
226C4—5481no—
227C4—5481no—
228C4—1371no—
229C4—1372no—
230C4—1371no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#2257835111yescharacteristic
235C2C2×#1798414911yescharacteristic
236C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1901
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17984149C2semidirect
4. #22578283C2semidirect
100. C22×C137#101semidirect
118. C137#1901semidirect
219. C22#17984149semidirect

Maximal subgroups

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

subgroupindex
C22×#145555192
C2×#179841492
#225783732
#225782832
C2×#179840652
#1901137