Finite Group Databasebeta

← #31796774

Subgroups — #31796774

Order 17536.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#10587739C211yesmaximal, characteristic
2C2×#17984151C212yesmaximal, complemented
3#22578371C211yesmaximal, characteristic
4#22578276C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C23×#10587739—21no—
7C2×#14555518C2212yes—
8C2×C4⋊#10587739—24no—
9C2×C4⋊#10587739—21no—
10C2×C4⋊#10587739C2212yes—
11#17984151—24no—
12#17984145—21no—
13C22⋊#11764047—21no—
14C2×Q8×D137C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984068C411yescharacteristic, complemented
17#17984067C411yescharacteristic, complemented
18#17984066—21no—
19#17984065—21no—
20#17984064C411yescharacteristic, complemented
21C4⋊C4×D137—21no—
22C4⋊C4×D137C2211yescharacteristic
23C22⋊C4×D137—21no—
24C22⋊Q8×C137C411yescharacteristic, complemented
25C22×#10587739—24no—
26C22×#10587739—22no—
27C22×#10587739—42no—
28C22×#10587739—42no—
29#14555518—22no—
30#14555518—22no—
31C2×#11764047—22no—
32C4⋊#10587739—24no—
33C4⋊#10587739—22no—
34C4⋊#10587739D414yes—
35C23×D137D411yescharacteristic
36C2×Dic274—21no—
37C2×Dic274C2×C411yescharacteristic, complemented
38Q8×D137—22no—
39Q8×D137—22no—
40C22×Dic137D411yescharacteristic
41C2×C4×D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137—21no—
44C2×C4×D137—21no—
45C2×C4×D137C2×C411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137D411yescharacteristic
48C2×C4×D137C2311yescharacteristic
49C4⋊Dic137—21no—
50C4⋊Dic137C2×C411yescharacteristic, complemented
51#14555489—21no—
52#14555489—21no—
53#14555489—21no—
54C22⋊Dic137—21no—
55#14555487—21no—
56#14555487—21no—
57C2×Q8×C137C2×C411yescharacteristic, complemented
58C22×C4×C137C2×C411yescharacteristic
59C4⋊C4×C137—21no—
60C4⋊C4×C137C2×C411yescharacteristic, complemented
61C22⋊C4×C137—21no—
62C2×#10587739—24no—
63C2×#10587739—48no—
64C2×#10587739—42no—
65C2×#10587739—42no—
66#11764047—42no—
67C22×D137—21no—
68C22×D137—21no—
69C22×D137—21no—
70C22×D137—21no—
71C22×D137C22⋊C411yescharacteristic
72C22×D137SD1612yes—
73C22×D137Q1612yes—
74C22×D137C2×D411yescharacteristic
75Dic274—22no—
76Dic274—42no—
77C2×Dic137—22no—
78C2×Dic137—21no—
79C2×Dic137—21no—
80C2×Dic137C22⋊C411yescharacteristic
81C2×Dic137C2×D411yescharacteristic
82C4×D137—21no—
83C4×D137—21no—
84C4×D137—22no—
85C4×D137—22no—
86C4×D137—41no—
87C4×D137—41no—
88C4×D137—41no—
89C4×D137—41no—
90C4×D137C2×D412yes—
91C4×D137#4112yes—
92C23×C137C22⋊C411yescharacteristic
93Q8×C137—22no—
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—21no—
97C2×C4×C137C22⋊C411yescharacteristic
98C2×C4×C137C22×C411yescharacteristic
99C22×C137—21no—
100C22×C137—21no—
101C22×C137#10312yescomplemented
102C22×C137C2×C22⋊C411yescharacteristic
103C2×D137—21no—
104C2×D137—21no—
105C2×D137—22no—
106C2×D137—22no—
107C2×D137—22no—
108C2×D137—22no—
109C2×D137#10312yescomplemented
110C2×D137#12011yescharacteristic
111C2×D137#12311yescharacteristic
112C2×D137C2×SD1611yescharacteristic
113C2×D137C2×Q1611yescharacteristic
114C2×D137AGL(1,ℤ/8)11yescharacteristic
115C2×D137#13711yescharacteristic
116#10587739—44no—
117C4×C137—21no—
118C4×C137—41no—
119C4×C137—41no—
120C4×C137C4×D412yes—
121Dic137—21no—
122Dic137—41no—
123Dic137—41no—
124Dic137#12112yes—
125C2×C137—22no—
126C2×C137#38611yescharacteristic
127C2×C137C2×#10311yescharacteristic
128C2×C137#41711yescharacteristic
129D137—22no—
130D137#45011yescharacteristic
131D137#45111yescharacteristic
132D137#48311yescharacteristic
133D137#48411yescharacteristic
134C137#187211yescharacteristic, complemented, Sylow
135#1872—1371nomaximal, Sylow
136C2×C22⋊Q8—1371no—
137C22×C4⋊C4—1371no—
138C2×#103—1372no—
139C2×C22⋊C8—1371no—
140#340—1372no—
141C22×Q8—1371no—
142C23×C4—2741no—
143C23×C4—1371no—
144C22×C8—1372no—
145C22⋊Q8—2741no—
146C22⋊Q8—2741no—
147C22⋊Q8—1371no—
148C22⋊Q8—1371no—
149C22⋊Q8—1371no—
150C22⋊Q8#1058773911yescharacteristic, complemented
151C2×C4⋊C4—2741no—
152C2×C4⋊C4—2741no—
153C2×C4⋊C4—2744no—
154C2×C4⋊C4—1372no—
155C2×C4⋊C4—1371no—
156C2×C22⋊C4—2741no—
157#103—2744no—
158C22⋊C8—2741no—
159C22⋊C8—2741no—
160C24—1371no—
161C2×Q8—2741no—
162C2×Q8—2742no—
163C2×Q8—2742no—
164C2×Q8—1371no—
165C2×Q8C2×#1058773911yescharacteristic, complemented
166C22×C4—2742no—
167C22×C4—2742no—
168C22×C4—2741no—
169C22×C4—2741no—
170C22×C4—2744no—
171C22×C4—2742no—
172C22×C4—5482no—
173C22×C4—5482no—
174C22×C4—1371no—
175C22×C4—1371no—
176C22×C4—1371no—
177C22×C4—1371no—
178C22×C4—1371no—
179C22×C4C2×#1058773911yescharacteristic
180C2×C8—2742no—
181C2×C8—2742no—
182C2×C8—2742no—
183C4⋊C4—21no—
184C4⋊C4—2741no—
185C4⋊C4—2741no—
186C4⋊C4—2741no—
187C4⋊C4—2744no—
188C4⋊C4—2742no—
189C4⋊C4—2741no—
190C4⋊C4—1374no—
191C4⋊C4—1371no—
192C4⋊C4C2×#1058773911yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—2741no—
195C22⋊C4—2741no—
196C22⋊C4—2741no—
197C23—2741no—
198C23—2741no—
199C23—2741no—
200C23—2741no—
201C23—1372no—
202C23—1372no—
203C23—1371no—
204C23—1371no—
205C23C22⋊#1058773911yescharacteristic
206Q8—22no—
207Q8—2742no—
208Q8—5482no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—2741no—
213C2×C4—2741no—
214C2×C4—2742no—
215C2×C4—2742no—
216C2×C4—2742no—
217C2×C4—2744no—
218C2×C4—2741no—
219C2×C4—2741no—
220C2×C4—5481no—
221C2×C4—5481no—
222C2×C4—5481no—
223C2×C4—5481no—
224C2×C4—5488no—
225C2×C4—5482no—
226C2×C4—5482no—
227C2×C4—1371no—
228C2×C4—1371no—
229C2×C4—1372no—
230C2×C4—1372no—
231C2×C4C22⋊#1058773911yescharacteristic
232C2×C4C22×#1058773911yescharacteristic
233C8—5482no—
234C22—21no—
235C22—21no—
236C22—2742no—
237C22—2742no—
238C22—2742no—
239C22—2742no—
240C22—2741no—
241C22—2741no—
242C22—1372no—
243C22—1371no—
244C22—1371no—
245C22—1371no—
246C22—1371no—
247C22—1371no—
248C22—1371no—
249C22#1798415112yescomplemented
250C22C2×C22⋊#1058773911yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—2741no—
255C4—5484no—
256C4—5481no—
257C4—5481no—
258C4—1372no—
259C4D4×#1058773912yes—
260C2—22no—
261C2—2742no—
262C2—1371no—
263C2—1371no—
264C2—1371no—
265C2—1371no—
266C2#2257835111yescharacteristic
267C2C2×#1798415111yescharacteristic
268C2#2257839411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1872
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
2. C2×#17984151C2semidirect
4. #22578276C2semidirect
16. #17984068C4semidirect
17. #17984067C4semidirect
20. #17984064C4semidirect
24. C22⋊Q8×C137C4semidirect
37. C2×Dic274C2×C4semidirect
50. C4⋊Dic137C2×C4semidirect
57. C2×Q8×C137C2×C4semidirect
60. C4⋊C4×C137C2×C4semidirect
101. C22×C137#103semidirect
109. C2×D137#103semidirect
134. C137#1872semidirect
150. C22⋊Q8#10587739semidirect
165. C2×Q8C2×#10587739semidirect
192. C4⋊C4C2×#10587739semidirect
249. C22#17984151semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#105877392
C2×#179841512
#225783712
#225782762
C22⋊Q8×D1372
#1872137