Finite Group Databasebeta

← #31796980

Subgroups — #31796980

Order 17536.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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#22578345C211yesmaximal, characteristic
2C4×C22⋊#10587739C211yesmaximal, characteristic
3#22578281C212yesmaximal
4#22578281C212yesmaximal
5C2×#17984066C211yesmaximal, characteristic
6C2×C4⋊#10587739—21no—
7C2×C22⋊#10587739C2212yes—
8C2×C22⋊#10587739C2212yes—
9C2×C4×#10587739—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4—21no—
12C22×Dic274C411yescharacteristic, complemented
13C22×C4×D137C2211yescharacteristic
14#17984066—21no—
15#17984066—21no—
16#17984066—21no—
17#17984066—21no—
18C2×#14555489C411yescharacteristic, complemented
19C2×#14555489C411yescharacteristic, complemented
20C2×#14555487C2211yescharacteristic
21C2×#14555487C2211yescharacteristic
22C2×C4⋊C4×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C22×#10587739—22no—
25C4⋊#10587739—42no—
26C22⋊#10587739—22no—
27C22⋊#10587739—24no—
28C22⋊#10587739Q814yescomplemented
29C4×#10587739—42no—
30C23×D137C2311yescharacteristic
31C2×Dic274—21no—
32C2×Dic274—21no—
33C2×Dic274—42no—
34C22×Dic137C2×C411yescharacteristic
35C22×Dic137C2×C411yescharacteristic
36C22×Dic137C2×C411yescharacteristic
37C2×C4×D137—42no—
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C2×C4×D137D411yescharacteristic
41C2×C4×D137D411yescharacteristic
42#14555489—21no—
43#14555489—21no—
44#14555489—21no—
45#14555489—21no—
46#14555487—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50C22×C4×C137C2×C411yescharacteristic
51C22×C4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic
53C4⋊C4×C137—21no—
54C4⋊C4×C137—21no—
55C2×#10587739—24no—
56C2×#10587739—42no—
57C2×#10587739—42no—
58C2×#10587739—42no—
59C2×#10587739—42no—
60C2×#10587739—42no—
61C22×D137—41no—
62C22×D137—41no—
63C22×D137C2×D411yescharacteristic
64C22×D137C2×D411yescharacteristic
65C22×D137C2×Q812yes—
66C22×D137#4112yes—
67Dic274—42no—
68Dic274—42no—
69C2×Dic137—22no—
70C2×Dic137—21no—
71C2×Dic137—21no—
72C2×Dic137—41no—
73C2×Dic137—41no—
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C22⋊C411yescharacteristic
76C4×D137—22no—
77C4×D137—22no—
78C4×D137—41no—
79C4×D137—41no—
80C23×C137C22×C411yescharacteristic
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—41no—
84C2×C4×C137—41no—
85C2×C4×C137—41no—
86C2×C4×C137C22⋊C411yescharacteristic
87C2×C4×C137C22⋊C411yescharacteristic
88C22×C137—21no—
89C22×C137—21no—
90C22×C137C2×C22⋊C411yescharacteristic
91C22×C137C4×Q812yes—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137C22⋊Q812yes—
99C2×D137#12411yescharacteristic
100C2×D137C4⋊Q811yescharacteristic
101#10587739—44no—
102#10587739—82no—
103C4×C137—21no—
104C4×C137—21no—
105C4×C137—41no—
106C4×C137—41no—
107Dic137—21no—
108Dic137—41no—
109Dic137—41no—
110Dic137#9912yes—
111C2×C137—22no—
112C2×C137#39111yescharacteristic
113C2×C137C2×#9911yescharacteristic
114C2×C137#41011yescharacteristic
115D137—22no—
116D137—21no—
117D137—21no—
118C137#199611yescharacteristic, complemented, Sylow
119#1996—1371nomaximal, Sylow
120C2×C22⋊Q8—1371no—
121#380—1371no—
122C4×C22⋊C4—1371no—
123#342—1372no—
124#342—1372no—
125C22×Q8—1371no—
126C23×C4—1371no—
127C22⋊Q8—2741no—
128C22⋊Q8—2741no—
129C22⋊Q8—2741no—
130C22⋊Q8—2741no—
131C2×C4⋊C4—2741no—
132C2×C4⋊C4—1371no—
133C2×C4⋊C4—1371no—
134C2×C4⋊C4#1058773911yescharacteristic, complemented
135C2×C22⋊C4—1372no—
136C2×C22⋊C4—1371no—
137C2×C22⋊C4—1371no—
138C2×C22⋊C4—1372no—
139C2×C42—2741no—
140#95—2741no—
141#95—2741no—
142C24—1371no—
143C2×Q8—2741no—
144C2×Q8—2741no—
145C2×Q8—5482no—
146C22×C4—2742no—
147C22×C4—2742no—
148C22×C4—5482no—
149C22×C4—1371no—
150C22×C4—1371no—
151C22×C4—1371no—
152C22×C4—1371no—
153C22×C4—1371no—
154C22×C4—1371no—
155C22×C4—1371no—
156C22×C4C2×#1058773911yescharacteristic
157C22×C4C2×#1058773911yescharacteristic
158C22×C4C2×#1058773911yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2741no—
162C4⋊C4—2741no—
163C4⋊C4—2741no—
164C4⋊C4—2741no—
165C4⋊C4—5482no—
166C22⋊C4—2741no—
167C22⋊C4—2741no—
168C22⋊C4—2741no—
169C22⋊C4—2741no—
170C22⋊C4—2742no—
171C22⋊C4—2744no—
172C22⋊C4—1374no—
173C42—5482no—
174C23—5481no—
175C23—5481no—
176C23—1371no—
177C23—1371no—
178C23—1372no—
179C23—1372no—
180C23C22×#1058773911yescharacteristic
181Q8—5482no—
182Q8—5482no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—2742no—
189C2×C4—2742no—
190C2×C4—2742no—
191C2×C4—2744no—
192C2×C4—2741no—
193C2×C4—2741no—
194C2×C4—5481no—
195C2×C4—5481no—
196C2×C4—5482no—
197C2×C4—5482no—
198C2×C4—5481no—
199C2×C4—5481no—
200C2×C4—5482no—
201C2×C4—5482no—
202C2×C4—5482no—
203C2×C4—1371no—
204C2×C4—1371no—
205C2×C4C22⋊#1058773911yescharacteristic
206C2×C4C22⋊#1058773911yescharacteristic
207C22—21no—
208C22—21no—
209C22—2741no—
210C22—2741no—
211C22—2741no—
212C22—2741no—
213C22—5482no—
214C22—5482no—
215C22—1372no—
216C22—1371no—
217C22—1371no—
218C22C2×C22⋊#1058773911yescharacteristic
219C22Q8×#1058773912yes—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—2741no—
225C4—5484no—
226C4—5481no—
227C4—5481no—
228C4—10962no—
229C4—1372no—
230C2—22no—
231C2—2742no—
232C2—2741no—
233C2—2741no—
234C2(C2×#10587739)⋊Q811yescharacteristic
235C2D137⋊#9911yescharacteristic
236C2#2257838211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radical, cosocleC23×D137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1996
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C22×Dic274C4semidirect
18. C2×#14555489C4semidirect
19. C2×#14555489C4semidirect
22. C2×C4⋊C4×C137C4semidirect
28. C22⋊#10587739Q8semidirect
118. C137#1996semidirect
134. C2×C4⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783452
C4×C22⋊#105877392
#225782812
C2×#179840662
#1996137