Finite Group Databasebeta

← #31796934

Subgroups — #31796934

Order 17536.

348 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 82 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×#17984156C211yesmaximal, characteristic, complemented
2C2×#17984151C211yesmaximal, characteristic, complemented
3C2×#17984150C211yesmaximal, characteristic
4C2×C8×#10587739C211yesmaximal, characteristic, complemented
5(C2×#10587739)⋊Q8C211yesmaximal, characteristic, complemented
6#22578355C211yesmaximal, characteristic
7C2×SD16×D137C211yesmaximal, characteristic
8C2×#14555518C2211yescharacteristic
9C2×C4⋊#10587739C2211yescharacteristic
10C2×C4⋊#10587739C2211yescharacteristic
11C2×C22⋊#10587739—22no—
12C2×C4×#10587739C2211yescharacteristic
13#17984156—22no—
14#17984151—22no—
15#17984150—22no—
16C8×#10587739—22no—
17(C2×#10587739)⋊C4—22no—
18C2×Q8×D137C2211yescharacteristic
19C2×D4×D137C2211yescharacteristic
20C2×#14555499C411yescharacteristic, complemented
21C2×#14555498C411yescharacteristic, complemented
22C2×#14555497C411yescharacteristic, complemented
23SD16×D137—22no—
24SD16×D137—22no—
25C2×C8×D137C2211yescharacteristic
26C2×SD16×C137C411yescharacteristic, complemented
27C22×#10587739—21no—
28C22×#10587739—21no—
29C22×#10587739D412yes—
30#14555518—21no—
31#14555518D412yes—
32C2×#11764047—21no—
33C4⋊#10587739—21no—
34C4⋊#10587739—21no—
35C4⋊#10587739—21no—
36C4⋊#10587739—21no—
37C22⋊#10587739—44no—
38C4×#10587739—21no—
39C4×#10587739—21no—
40C23×D137—21no—
41C2×Dic274—21no—
42C2×Dic274C2×C411yescharacteristic
43Q8×D137—21no—
44Q8×D137—21no—
45Q8×D137—21no—
46Q8×D137—21no—
47C2×D548C2×C411yescharacteristic
48C2×C22⋊D137—21no—
49D4×D137—21no—
50D4×D137—21no—
51D4×D137—21no—
52D4×D137—21no—
53C2×C4×D137—21no—
54C2×C4×D137C2311yescharacteristic
55#14555499—22no—
56#14555498—22no—
57#14555497—22no—
58C2×#11764040C2×C411yescharacteristic
59C8×D137D412yes—
60C8×D137D412yes—
61C2×Q8×C137C2×C411yescharacteristic
62C2×D4×C137C2×C411yescharacteristic
63SD16×C137—22no—
64C2×C8×C137C2×C411yescharacteristic
65C2×#10587739—22no—
66C2×#10587739—22no—
67C2×#10587739—41no—
68C2×#10587739—41no—
69C2×#10587739—42no—
70C2×#10587739—42no—
71C2×#10587739SD1614yescomplemented
72#11764047—22no—
73C22×D137—21no—
74C22×D137—41no—
75C22×D137—41no—
76C22×D137—42no—
77C22×D137—42no—
78C22×D137C2×D411yescharacteristic
79Dic274—21no—
80Dic274—21no—
81Dic274—41no—
82Dic274—41no—
83D548—21no—
84D548—21no—
85C22⋊D137—41no—
86C22⋊D137—41no—
87C2×Dic137—21no—
88C2×Dic137#4111yescharacteristic
89C4×D137—41no—
90C4×D137—41no—
91C4×D137C2×D411yescharacteristic
92C4×D137C2×D411yescharacteristic
93C4×D137C2×D411yescharacteristic
94C4×D137#4111yescharacteristic
95#11764040C22⋊C412yescomplemented
96C23×C137—21no—
97Q8×C137—21no—
98Q8×C137—21no—
99D4×C137—21no—
100D4×C137—21no—
101C2×C4×C137—21no—
102C2×C4×C137C22×C411yescharacteristic
103C8×C137C22⋊C412yescomplemented
104C22×C137—41no—
105C22×C137—42no—
106C22×C137C4×D411yescharacteristic
107C2×D137—41no—
108C2×D137—42no—
109C2×D137—82no—
110C2×D137—81no—
111C2×D137—81no—
112C2×D137#12111yescharacteristic
113C2×D137#12711yescharacteristic
114C2×D137C2×SD1612yes—
115C2×D137#13512yes—
116#10587739—24no—
117#10587739—81no—
118#10587739—81no—
119C4×C137—41no—
120C4×C137C2×C22⋊C411yescharacteristic
121C4×C137C4×D411yescharacteristic
122Dic137—41no—
123Dic137#12111yescharacteristic
124Dic137#12411yescharacteristic
125C2×C137—81no—
126C2×C137#39011yescharacteristic
127C2×C137C4×SD1612yes—
128D137—81no—
129D137#46512yes—
130D137#49211yescharacteristic
131D137#49511yescharacteristic
132C137#196811yescharacteristic, complemented, Sylow
133#1968—1371nomaximal, Sylow
134C22×SD16—1371no—
135C2×#106—1371no—
136C2×#103—1371no—
137C2×#102—1371no—
138C2×C4×C8—1371no—
139#391—1371no—
140#390—1371no—
141C22×Q8—1371no—
142C22×D4—1371no—
143C2×SD16—2742no—
144C2×SD16—2742no—
145C2×SD16—1371no—
146C2×SD16—1371no—
147C2×SD16—1371no—
148C2×SD16#1058773911yescharacteristic, complemented
149C22×C8—1371no—
150C22×C8—1371no—
151C2×C4⋊C4—1371no—
152C2×C4⋊C4—1371no—
153C2×C22⋊C4—2742no—
154C2×C42—1371no—
155#106—2742no—
156#103—2742no—
157#102—2742no—
158C4×C8—2742no—
159#95—2742no—
160C24—2741no—
161C2×Q8—2741no—
162C2×Q8—2741no—
163C2×Q8—2741no—
164C2×Q8—2741no—
165C2×Q8—2741no—
166C2×Q8—1371no—
167C2×Q8C2×#1058773911yescharacteristic
168C2×D4—2741no—
169C2×D4—2741no—
170C2×D4—2741no—
171C2×D4—2741no—
172C2×D4—2741no—
173C2×D4—1371no—
174C2×D4C2×#1058773911yescharacteristic
175C22×C4—2741no—
176C22×C4—2741no—
177C22×C4—2741no—
178C22×C4—1372no—
179C22×C4—1371no—
180SD16—22no—
181SD16—2742no—
182SD16—2742no—
183SD16—2742no—
184C2×C8—2741no—
185C2×C8—2741no—
186C2×C8—1372no—
187C2×C8—1371no—
188C2×C8—1372no—
189C2×C8—1372no—
190C2×C8C2×#1058773911yescharacteristic
191C4⋊C4—2741no—
192C4⋊C4—2741no—
193C4⋊C4—2741no—
194C4⋊C4—2741no—
195C22⋊C4—5484no—
196C42—2741no—
197C42—2741no—
198C23—21no—
199C23—2741no—
200C23—5482no—
201C23—5481no—
202C23—5481no—
203C23—5482no—
204C23—1371no—
205Q8—21no—
206Q8—21no—
207Q8—2741no—
208Q8—2741no—
209Q8—5481no—
210Q8—5481no—
211D4—21no—
212D4—21no—
213D4—2741no—
214D4—2741no—
215D4—5481no—
216D4—5481no—
217C2×C4—21no—
218C2×C4—2742no—
219C2×C4—2742no—
220C2×C4—2741no—
221C2×C4—5481no—
222C2×C4—5481no—
223C2×C4—5482no—
224C2×C4—5481no—
225C2×C4—5481no—
226C2×C4—5482no—
227C2×C4—1374no—
228C2×C4—1371no—
229C2×C4—1371no—
230C2×C4—1371no—
231C2×C4—1371no—
232C2×C4—1371no—
233C2×C4C22×#1058773911yescharacteristic
234C8—2742no—
235C8—1372no—
236C8C22⋊#1058773912yescomplemented
237C22—42no—
238C22—41no—
239C22—5482no—
240C22—5481no—
241C22—10962no—
242C22—10961no—
243C22—10961no—
244C22—1372no—
245C22—1372no—
246C22—1371no—
247C22—1371no—
248C22D4×#1058773911yescharacteristic
249C4—41no—
250C4—2744no—
251C4—5481no—
252C4—10961no—
253C4—10961no—
254C4—1371no—
255C4—1371no—
256C4C2×C22⋊#1058773911yescharacteristic
257C4D4×#1058773911yescharacteristic
258C2—81no—
259C2—10961no—
260C2—1372no—
261C2—1371no—
262C2—1371no—
263C2#2257835511yescharacteristic
264C2SD16×#1058773912yes—
subgroupG/H
centerC22D4×#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×SD16×C137C4
socleC22×C137C4×D4
prime radicalC2×D4×D137C22
cosocleC2×C4×D137C23
hypercenterC2×SD16#10587739
hypocenterC137#1968
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#17984156C2semidirect
2. C2×#17984151C2semidirect
4. C2×C8×#10587739C2semidirect
5. (C2×#10587739)⋊Q8C2semidirect
20. C2×#14555499C4semidirect
21. C2×#14555498C4semidirect
22. C2×#14555497C4semidirect
26. C2×SD16×C137C4semidirect
71. C2×#10587739SD16semidirect
95. #11764040C22⋊C4semidirect
103. C8×C137C22⋊C4semidirect
132. C137#1968semidirect
148. C2×SD16#10587739semidirect
236. C8C22⋊#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841562
C2×#179841512
C2×#179841502
C2×C8×#105877392
(C2×#10587739)⋊Q82
#225783552
C2×SD16×D1372
#1968137