Finite Group Databasebeta

← #31796771

Subgroups — #31796771

Order 17536.

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

normal subgroups only

This group has 376 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
1C2×D137⋊M16C211yesmaximal, characteristic, complemented
2C2×#17984153C212yesmaximal, complemented
3C4×C22⋊#10587739C211yesmaximal, characteristic, complemented
4#22578273C212yesmaximal, complemented
5#22577916C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7D137⋊M16—24no—
8C2×#14555519—21no—
9C2×#14555518—21no—
10C2×C22⋊#10587739—21no—
11C2×C4×#10587739C2212yes—
12#17984153—24no—
13(C2×#10587739)⋊C4—21no—
14C2×#14555511C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984073—21no—
17#17984064C411yescharacteristic, complemented
18#17984063C411yescharacteristic, complemented
19#17984062C411yescharacteristic, complemented
20#17984058—21no—
21#17984050—21no—
22#17984048—21no—
23#17984046C2211yescharacteristic
24C22⋊Q8×C137C411yescharacteristic, complemented
25C22×#10587739—22no—
26#14555520—22no—
27#14555520D414yescomplemented
28#14555519—24no—
29#14555518—24no—
30C2×#11764047—22no—
31C22⋊#10587739—24no—
32C4×#10587739—22no—
33C4×#10587739—22no—
34C23×D137D411yescharacteristic
35#14555511—22no—
36#14555511—22no—
37C2×D548—21no—
38C2×D548—21no—
39C2×D548C2×C411yescharacteristic
40C2×C22⋊D137—21no—
41C22×Dic137D411yescharacteristic
42C2×C4×D137—22no—
43C2×C4×D137—22no—
44C2×C4×D137—21no—
45C2×C4×D137C2×C411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137D411yescharacteristic
48C2×C4×D137C2311yescharacteristic
49C4⋊Dic137C2×C411yescharacteristic
50#14555489—21no—
51#14555487—21no—
52#14555487—21no—
53#14555487—21no—
54#14555487—21no—
55C4×Dic137—21no—
56C2×Q8×C137C2×C411yescharacteristic
57C22×C4×C137C2×C411yescharacteristic
58C4⋊C4×C137—21no—
59C4⋊C4×C137C2×C411yescharacteristic
60C22⋊C4×C137—21no—
61C2×#10587739—24no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64#11764047—24no—
65C22×D137—22no—
66C22×D137—22no—
67C22×D137—21no—
68C22×D137—41no—
69C22×D137C22⋊C411yescharacteristic
70C22×D137C2×D411yescharacteristic
71D548—22no—
72D548—42no—
73D548—42no—
74C22⋊D137—42no—
75C2×Dic137—21no—
76C2×Dic137—21no—
77C2×Dic137—21no—
78C2×Dic137C22⋊C411yescharacteristic
79C2×Dic137C2×D411yescharacteristic
80C4×D137—22no—
81C4×D137—22no—
82C4×D137—21no—
83C4×D137—21no—
84C4×D137—41no—
85C4×D137—41no—
86C4×D137C2×D412yes—
87C4×D137#4112yes—
88C23×C137C22⋊C411yescharacteristic
89Q8×C137—22no—
90C2×C4×C137—22no—
91C2×C4×C137—21no—
92C2×C4×C137—21no—
93C2×C4×C137C22⋊C411yescharacteristic
94C2×C4×C137C22×C411yescharacteristic
95C22×C137—21no—
96C22×C137—21no—
97C22×C137—21no—
98C22×C137C2×C22⋊C411yescharacteristic
99C2×D137—21no—
100C2×D137—21no—
101C2×D137—21no—
102C2×D137—21no—
103C2×D137—44no—
104C2×D137—41no—
105C2×D137—41no—
106C2×D137—41no—
107C2×D137#10412yescomplemented
108C2×D137#12112yes—
109#10587739—44no—
110C4×C137—21no—
111C4×C137—41no—
112C4×C137—41no—
113C4×C137C4×D412yes—
114Dic137—21no—
115Dic137—41no—
116Dic137#12011yescharacteristic
117Dic137#12311yescharacteristic
118C2×C137—41no—
119C2×C137#38611yescharacteristic
120C2×C137C2×#10411yescharacteristic
121C2×C137#42111yescharacteristic
122D137—22no—
123D137—22no—
124D137—81no—
125C137#187111yescharacteristic, complemented, Sylow
126#1871—1371nomaximal, Sylow
127C22×M16—1371no—
128#525—1371no—
129C2×#104—1372no—
130C4×C22⋊C4—1371no—
131#339—1372no—
132C2×#41—1371no—
133C23×C4—1371no—
134C2×M16—2741no—
135C2×M16—2744no—
136C2×M16—1372no—
137C22×C8—2741no—
138#123—2741no—
139C22⋊Q8—1371no—
140C22⋊Q8#1058773911yescharacteristic, complemented
141#121—1371no—
142#121—1371no—
143#120—2741no—
144C4×D4—2741no—
145C4×D4—2741no—
146#117—1371no—
147C2×C22⋊C4—2741no—
148C2×C42—1372no—
149#104—2744no—
150#95—2741no—
151C24—1371no—
152#41—2742no—
153#41—2742no—
154C2×Q8C2×#1058773911yescharacteristic
155C2×D4—2741no—
156C2×D4—2741no—
157C2×D4—2741no—
158C2×D4—1371no—
159C22×C4—2742no—
160C22×C4—2742no—
161C22×C4—2741no—
162C22×C4—2742no—
163C22×C4—1371no—
164C22×C4—1371no—
165C22×C4—1371no—
166C22×C4—1371no—
167C22×C4—1371no—
168C22×C4C2×#1058773911yescharacteristic
169M16—2744no—
170M16—2742no—
171M16—1374no—
172C2×C8—2744no—
173C2×C8—2742no—
174C4⋊C4—21no—
175C4⋊C4—2741no—
176C4⋊C4—1371no—
177C4⋊C4C2×#1058773911yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—2741no—
180C22⋊C4—2741no—
181C22⋊C4—2741no—
182C22⋊C4—2744no—
183C22⋊C4—2741no—
184C42—2741no—
185C42—2742no—
186C42—2742no—
187C23—2742no—
188C23—2742no—
189C23—2741no—
190C23—5481no—
191C23—1371no—
192C23—1371no—
193C23C22⋊#1058773911yescharacteristic
194Q8—22no—
195D4—2742no—
196D4—5482no—
197D4—5482no—
198D4—5482no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—2742no—
203C2×C4—2741no—
204C2×C4—2741no—
205C2×C4—2741no—
206C2×C4—2741no—
207C2×C4—2742no—
208C2×C4—2744no—
209C2×C4—2741no—
210C2×C4—5481no—
211C2×C4—5481no—
212C2×C4—5482no—
213C2×C4—5482no—
214C2×C4—1371no—
215C2×C4—1371no—
216C2×C4—1372no—
217C2×C4—1372no—
218C2×C4C22⋊#1058773911yescharacteristic
219C2×C4C22×#1058773911yescharacteristic
220C8—2744no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—2741no—
225C22—2741no—
226C22—2741no—
227C22—2741no—
228C22—5484no—
229C22—5481no—
230C22—5481no—
231C22—5481no—
232C22—1372no—
233C22—1372no—
234C22C2×C22⋊#1058773911yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—2741no—
239C4—5484no—
240C4—5481no—
241C4—1371no—
242C4—1371no—
243C4D4×#1058773912yes—
244C2—41no—
245C2—2742no—
246C2—2742no—
247C2—10961no—
248C2#2257835111yescharacteristic
249C2D137⋊#10411yescharacteristic
250C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577916C2
cosocleC2×C4×D137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1871
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×D137⋊M16C2semidirect
2. C2×#17984153C2semidirect
3. C4×C22⋊#10587739C2semidirect
4. #22578273C2semidirect
17. #17984064C4semidirect
18. #17984063C4semidirect
19. #17984062C4semidirect
24. C22⋊Q8×C137C4semidirect
27. #14555520D4semidirect
107. C2×D137#104semidirect
125. C137#1871semidirect
140. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×D137⋊M162
C2×#179841532
C4×C22⋊#105877392
#225782732
#225779162
#1871137