Finite Group Databasebeta

← #31796657

Subgroups — #31796657

Order 17536.

372 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 94 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#22578404C211yesmaximal, characteristic, complemented
2C2×#17984150C212yesmaximal
3#22578349C211yesmaximal, characteristic, complemented
4#22578276C212yesmaximal, complemented
5C4×D4×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7C2×C4⋊#10587739—21no—
8C2×C4⋊#10587739—21no—
9C2×C4⋊#10587739C2212yes—
10#17984155—21no—
11C4⋊#11764047—21no—
12#17984150—24no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137—21no—
15C4×D548C411yescharacteristic, complemented
16D4×Dic137C411yescharacteristic, complemented
17#17984050C411yescharacteristic, complemented
18C4×C22⋊D137—21no—
19#17984048—21no—
20C4⋊C4×D137C2211yescharacteristic
21C22⋊C4×D137—21no—
22C42×D137C2211yescharacteristic
23C4×D4×C137C411yescharacteristic, complemented
24C22×#10587739—22no—
25#14555518—22no—
26#14555518—22no—
27C2×#11764047—22no—
28C4⋊#10587739—22no—
29C4⋊#10587739—22no—
30C4⋊#10587739—42no—
31C4⋊#10587739—42no—
32C23×D137—21no—
33C2×D548C2×C411yescharacteristic
34C2×C22⋊D137—21no—
35D4×D137—21no—
36D4×D137—21no—
37D4×D137D412yes—
38D4×D137Q812yes—
39C22×Dic137—21no—
40C2×C4×D137—21no—
41C2×C4×D137—21no—
42C2×C4×D137—21no—
43C2×C4×D137—42no—
44C2×C4×D137—42no—
45C2×C4×D137D411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137C2311yescharacteristic
48C4⋊Dic137C2×C411yescharacteristic
49#14555489—21no—
50C22⋊Dic137—21no—
51#14555487—21no—
52#14555487—21no—
53C4×Dic137C2×C411yescharacteristic
54C4×Dic137D411yescharacteristic
55C4×Dic137D411yescharacteristic
56C2×D4×C137C2×C411yescharacteristic
57C22×C4×C137—21no—
58C4⋊C4×C137C2×C411yescharacteristic
59C22⋊C4×C137—21no—
60C42×C137C2×C411yescharacteristic
61C2×#10587739—42no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64#11764047—42no—
65C22×D137—21no—
66C22×D137—22no—
67C22×D137—22no—
68C22×D137—41no—
69C22×D137—41no—
70C22×D137—41no—
71C22×D137—41no—
72C22×D137C2×D411yescharacteristic
73D548—21no—
74D548C4⋊C412yescomplemented
75C22⋊D137—22no—
76C22⋊D137—22no—
77C2×Dic137—21no—
78C2×Dic137—42no—
79C2×Dic137C22⋊C411yescharacteristic
80C2×Dic137C22⋊C411yescharacteristic
81C2×Dic137C2×D411yescharacteristic
82C4×D137—21no—
83C4×D137—21no—
84C4×D137—41no—
85C4×D137—41no—
86C4×D137—42no—
87C4×D137—42no—
88C4×D137D812yes—
89C4×D137SD1612yes—
90C4×D137C2×D411yescharacteristic
91C4×D137C2×Q811yescharacteristic
92C4×D137#4111yescharacteristic
93C4×D137#4111yescharacteristic
94C23×C137—21no—
95D4×C137—21no—
96D4×C137C4⋊C412yescomplemented
97C2×C4×C137—21no—
98C2×C4×C137—42no—
99C2×C4×C137C22⋊C411yescharacteristic
100C2×C4×C137C22⋊C411yescharacteristic
101C2×C4×C137C22×C411yescharacteristic
102C22×C137—22no—
103C22×C137—41no—
104C22×C137—41no—
105C22×C137C2×C22⋊C411yescharacteristic
106C2×D137—22no—
107C2×D137—41no—
108C2×D137—41no—
109C2×D137—42no—
110C2×D137—42no—
111C2×D137—42no—
112C2×D137—42no—
113C2×D137#12111yescharacteristic
114C2×D137C22⋊Q811yescharacteristic
115C2×D137C2×D811yescharacteristic
116C2×D137C2×SD1611yescharacteristic
117C2×D137AGL(1,ℤ/8)11yescharacteristic
118C2×D137#13711yescharacteristic
119#10587739—82no—
120C4×C137—21no—
121C4×C137—41no—
122C4×C137#10212yescomplemented
123C4×C137C2×C4⋊C411yescharacteristic
124C4×C137#11711yescharacteristic
125Dic137—21no—
126Dic137—41no—
127Dic137#10212yescomplemented
128Dic137#12111yescharacteristic
129Dic137C22⋊Q811yescharacteristic
130C2×C137—42no—
131C2×C137#38011yescharacteristic
132C2×C137C2×#10211yescharacteristic
133C2×C137#41711yescharacteristic
134D137—42no—
135D137#45911yescharacteristic
136D137#46111yescharacteristic
137D137#47411yescharacteristic
138D137#47611yescharacteristic
139C137#180211yescharacteristic, complemented, Sylow
140#1802—1371nomaximal, Sylow
141C2×C4×D4—1371no—
142C2×C4⋊C8—1371no—
143C2×#102—1372no—
144#384—1371no—
145#340—1372no—
146C22×D4—1371no—
147C23×C4—2741no—
148C22×C8—1372no—
149C4×D4—2741no—
150C4×D4—2741no—
151C4×D4—1371no—
152C4×D4—1371no—
153C4×D4—1371no—
154C4×D4#1058773911yescharacteristic, complemented
155C2×C4⋊C4—2741no—
156C2×C4⋊C4—2741no—
157C2×C4⋊C4—1372no—
158C2×C4⋊C4—1371no—
159C2×C22⋊C4—2741no—
160C2×C42—1371no—
161C4⋊C8—2741no—
162C4⋊C8—2741no—
163#102—2744no—
164C24—2741no—
165C2×D4—2741no—
166C2×D4—2741no—
167C2×D4—2741no—
168C2×D4—1372no—
169C2×D4—1372no—
170C2×D4—1371no—
171C2×D4C2×#1058773911yescharacteristic
172C22×C4—21no—
173C22×C4—2741no—
174C22×C4—2741no—
175C22×C4—2741no—
176C22×C4—2741no—
177C22×C4—2742no—
178C22×C4—5482no—
179C22×C4—5482no—
180C22×C4—1371no—
181C22×C4—1371no—
182C22×C4—1371no—
183C2×C8—2742no—
184C2×C8—2742no—
185C2×C8—2742no—
186C4⋊C4—2741no—
187C4⋊C4—2742no—
188C4⋊C4—2742no—
189C4⋊C4—5482no—
190C4⋊C4—5482no—
191C4⋊C4—1371no—
192C4⋊C4C2×#1058773911yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—2741no—
195C22⋊C4—2741no—
196C22⋊C4—2741no—
197C42—1371no—
198C42—1371no—
199C42—1371no—
200C42C2×#1058773911yescharacteristic
201C23—21no—
202C23—2742no—
203C23—2742no—
204C23—2741no—
205C23—5481no—
206C23—5481no—
207C23—5481no—
208C23—5481no—
209C23—1371no—
210D4—21no—
211D4—2742no—
212D4—2742no—
213D4—2741no—
214D4—1372no—
215D4C4⋊#1058773912yescomplemented
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—2741no—
219C2×C4—2741no—
220C2×C4—2741no—
221C2×C4—5481no—
222C2×C4—5481no—
223C2×C4—5482no—
224C2×C4—5482no—
225C2×C4—5482no—
226C2×C4—5482no—
227C2×C4—5482no—
228C2×C4—5482no—
229C2×C4—1371no—
230C2×C4—1371no—
231C2×C4—1371no—
232C2×C4—1372no—
233C2×C4—1372no—
234C2×C4—1371no—
235C2×C4—1371no—
236C2×C4—1371no—
237C2×C4—1371no—
238C2×C4C22⋊#1058773911yescharacteristic
239C2×C4C22⋊#1058773911yescharacteristic
240C2×C4C22×#1058773911yescharacteristic
241C8—5482no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—2742no—
246C22—5482no—
247C22—5482no—
248C22—5482no—
249C22—5482no—
250C22—5481no—
251C22—5481no—
252C22—1371no—
253C22—1371no—
254C22—1371no—
255C22—1371no—
256C22—1371no—
257C22—1371no—
258C22C2×C22⋊#1058773911yescharacteristic
259C4—21no—
260C4—41no—
261C4—2741no—
262C4—5481no—
263C4—10962no—
264C4—1371no—
265C4—1371no—
266C4—1372no—
267C4#1798415012yescomplemented
268C4C2×C4⋊#1058773911yescharacteristic
269C4#1798416711yescharacteristic
270C2—42no—
271C2—5482no—
272C2—1371no—
273C2—1371no—
274C2—1371no—
275C2—1371no—
276C2#2257834511yescharacteristic
277C2C2×#1798415011yescharacteristic
278C2#2257839411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C22
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1802
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578404C2semidirect
3. #22578349C2semidirect
4. #22578276C2semidirect
15. C4×D548C4semidirect
16. D4×Dic137C4semidirect
17. #17984050C4semidirect
23. C4×D4×C137C4semidirect
74. D548C4⋊C4semidirect
96. D4×C137C4⋊C4semidirect
122. C4×C137#102semidirect
127. Dic137#102semidirect
139. C137#1802semidirect
154. C4×D4#10587739semidirect
215. D4C4⋊#10587739semidirect
267. C4#17984150semidirect

Maximal subgroups

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

subgroupindex
#225784042
C2×#179841502
#225783492
#225782762
C4×D4×D1372
#1802137