Finite Group Databasebeta

← #31796813

Subgroups — #31796813

Order 17536.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#17984167C211yesmaximal, characteristic, complemented
2C2×#17984147C212yesmaximal
3#22578346C211yesmaximal, characteristic, complemented
4#22578281C212yesmaximal, complemented
5C2×#17984073C211yesmaximal, characteristic
6#17984167—24no—
7C2×C4⋊#10587739—21no—
8C2×C22⋊#10587739C2212yes—
9C2×C22⋊#10587739C2212yes—
10C2×C4×#10587739—21no—
11#17984147—24no—
12(C2×#10587739)⋊C4—21no—
13(C2×#10587739)⋊C4—21no—
14C22×D548C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984073—21no—
17#17984073—21no—
18#17984073—21no—
19#17984073—21no—
20C2×#14555489C411yescharacteristic, complemented
21C2×#14555487C411yescharacteristic, complemented
22C2×#14555487C411yescharacteristic, complemented
23C2×#14555487C2211yescharacteristic
24C2×C4⋊C4×C137C411yescharacteristic, complemented
25C22×#10587739—22no—
26C22×#10587739—22no—
27C4⋊#10587739—24no—
28C22⋊#10587739—22no—
29C22⋊#10587739—22no—
30C22⋊#10587739—22no—
31C22⋊#10587739D414yes—
32C4×#10587739—24no—
33C23×D137C2×C411yescharacteristic
34C23×D137C2311yescharacteristic
35C2×D548—21no—
36C2×D548—21no—
37C2×D548—22no—
38C2×D548—22no—
39C22×Dic137C2×C411yescharacteristic
40C22×Dic137C2×C411yescharacteristic
41C2×C4×D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137D411yescharacteristic
44C2×C4×D137D411yescharacteristic
45C2×C4×D137D411yescharacteristic
46C2×C4×D137D411yescharacteristic
47#14555489—21no—
48#14555489—21no—
49#14555487—21no—
50#14555487—21no—
51#14555487—21no—
52#14555487—21no—
53#14555487—21no—
54#14555487—21no—
55C22×C4×C137C2×C411yescharacteristic
56C22×C4×C137C2×C411yescharacteristic
57C22×C4×C137C2×C411yescharacteristic
58C4⋊C4×C137—21no—
59C4⋊C4×C137—21no—
60C2×#10587739—24no—
61C2×#10587739—42no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64C2×#10587739—42no—
65C2×#10587739—42no—
66C22×D137—21no—
67C22×D137—22no—
68C22×D137—41no—
69C22×D137—41no—
70C22×D137—41no—
71C22×D137—41no—
72C22×D137C2×D411yescharacteristic
73C22×D137C2×D411yescharacteristic
74C22×D137C2×D412yes—
75C22×D137#4112yes—
76D548—41no—
77D548—42no—
78D548—41no—
79C2×Dic137—22no—
80C2×Dic137—21no—
81C2×Dic137—41no—
82C2×Dic137C22⋊C411yescharacteristic
83C2×Dic137C22⋊C411yescharacteristic
84C4×D137—21no—
85C4×D137—21no—
86C4×D137—22no—
87C4×D137—22no—
88C4×D137—21no—
89C4×D137—21no—
90C23×C137C22×C411yescharacteristic
91C2×C4×C137—22no—
92C2×C4×C137—21no—
93C2×C4×C137—21no—
94C2×C4×C137—22no—
95C2×C4×C137—41no—
96C2×C4×C137C22⋊C411yescharacteristic
97C2×C4×C137C22⋊C411yescharacteristic
98C22×C137—21no—
99C22×C137—21no—
100C22×C137C2×C22⋊C411yescharacteristic
101C22×C137C4×D412yes—
102C2×D137—21no—
103C2×D137—21no—
104C2×D137—21no—
105C2×D137—21no—
106C2×D137—42no—
107C2×D137—42no—
108C2×D137—41no—
109C2×D137—41no—
110C2×D137—41no—
111C2×D137—82no—
112C2×D137#12011yescharacteristic
113C2×D137#12112yes—
114C2×D137#12311yescharacteristic
115#10587739—44no—
116#10587739—82no—
117C4×C137—21no—
118C4×C137—21no—
119C4×C137—41no—
120C4×C137—41no—
121Dic137—21no—
122Dic137—21no—
123Dic137—41no—
124C2×C137—22no—
125C2×C137#38611yescharacteristic
126C2×C137#41011yescharacteristic
127C2×C137#41011yescharacteristic
128D137—21no—
129D137—22no—
130D137—21no—
131D137—81no—
132C137#189911yescharacteristic, complemented, Sylow
133#1899—1371nomaximal, Sylow
134C2×#123—1371no—
135C2×#117—1371no—
136C2×#99—1372no—
137#381—1371no—
138#342—1372no—
139C22×D4—1371no—
140C23×C4—1371no—
141#123—2741no—
142#123—2741no—
143#123—2741no—
144#123—2741no—
145#117—2744no—
146C2×C4⋊C4—2741no—
147C2×C4⋊C4—1371no—
148C2×C4⋊C4#1058773911yescharacteristic, complemented
149C2×C22⋊C4—1372no—
150C2×C22⋊C4—1371no—
151C2×C22⋊C4—1371no—
152C2×C22⋊C4—1371no—
153C2×C22⋊C4—1372no—
154C2×C42—2741no—
155#99—2744no—
156#95—2741no—
157#95—2741no—
158C24—1371no—
159C24—1371no—
160C2×D4—2742no—
161C2×D4—2741no—
162C2×D4—2741no—
163C2×D4—2742no—
164C22×C4—2742no—
165C22×C4—2742no—
166C22×C4—2742no—
167C22×C4—2742no—
168C22×C4—1371no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4—1371no—
172C22×C4—1371no—
173C22×C4—1371no—
174C22×C4C2×#1058773911yescharacteristic
175C22×C4C2×#1058773911yescharacteristic
176C22×C4C2×#1058773911yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—2741no—
180C4⋊C4—2741no—
181C4⋊C4—2744no—
182C22⋊C4—2741no—
183C22⋊C4—2741no—
184C22⋊C4—2741no—
185C22⋊C4—2741no—
186C22⋊C4—2742no—
187C22⋊C4—2741no—
188C22⋊C4—2741no—
189C22⋊C4—2742no—
190C22⋊C4—2742no—
191C22⋊C4—1374no—
192C42—2744no—
193C23—2741no—
194C23—2742no—
195C23—5481no—
196C23—5481no—
197C23—5481no—
198C23—5481no—
199C23—1371no—
200C23—1371no—
201C23—1372no—
202C23—1372no—
203C23C22×#1058773911yescharacteristic
204D4—5481no—
205D4—5482no—
206D4—5481no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—2741no—
213C2×C4—2741no—
214C2×C4—2742no—
215C2×C4—2742no—
216C2×C4—2741no—
217C2×C4—2741no—
218C2×C4—2742no—
219C2×C4—2744no—
220C2×C4—2741no—
221C2×C4—5482no—
222C2×C4—5482no—
223C2×C4—5481no—
224C2×C4—5482no—
225C2×C4—5482no—
226C2×C4—5482no—
227C2×C4—1371no—
228C2×C4—1371no—
229C2×C4C22⋊#1058773911yescharacteristic
230C2×C4C22⋊#1058773911yescharacteristic
231C22—21no—
232C22—21no—
233C22—2741no—
234C22—2741no—
235C22—2741no—
236C22—2741no—
237C22—5482no—
238C22—5482no—
239C22—5481no—
240C22—5481no—
241C22—5481no—
242C22—10962no—
243C22—1371no—
244C22—1372no—
245C22—1371no—
246C22C2×C22⋊#1058773911yescharacteristic
247C22D4×#1058773912yes—
248C4—21no—
249C4—21no—
250C4—41no—
251C4—41no—
252C4—2741no—
253C4—2741no—
254C4—5484no—
255C4—5481no—
256C4—10962no—
257C2—22no—
258C2—2741no—
259C2—2742no—
260C2—2741no—
261C2—10961no—
262C2#2257835111yescharacteristic
263C2#2257838211yescharacteristic
264C2#2257838411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C22
cosocleC23×D137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1899
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
1. C2×#17984167C2semidirect
3. #22578346C2semidirect
4. #22578281C2semidirect
20. C2×#14555489C4semidirect
21. C2×#14555487C4semidirect
22. C2×#14555487C4semidirect
24. C2×C4⋊C4×C137C4semidirect
132. C137#1899semidirect
148. C2×C4⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841672
C2×#179841472
#225783462
#225782812
C2×#179840732
#1899137