Finite Group Databasebeta

← #31796802

Subgroups — #31796802

Order 17536.

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

normal subgroups only

This group has 380 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×#14555518C211yesmaximal, characteristic
2C2×#17984151C212yesmaximal, complemented
3#22578345C211yesmaximal, characteristic
4#22578276C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C2×#14555518—24no—
7C2×#14555518—21no—
8C2×#14555518C2212yes—
9C22×#11764047—21no—
10C2×C4⋊#10587739C2212yes—
11C2×C22⋊#10587739—21no—
12#17984151—24no—
13(C2×#10587739)⋊C4—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—22no—
26#14555518—24no—
27#14555518—22no—
28#14555518D414yes—
29C2×#11764047—24no—
30C2×#11764047—22no—
31C4⋊#10587739—22no—
32C4⋊#10587739—22no—
33C22⋊#10587739—42no—
34C23×D137D411yescharacteristic
35C2×Dic274—21no—
36C2×Dic274C2×C411yescharacteristic
37Q8×D137—22no—
38Q8×D137—22no—
39C22×Dic137D411yescharacteristic
40C2×C4×D137—22no—
41C2×C4×D137—22no—
42C2×C4×D137—21no—
43C2×C4×D137—21no—
44C2×C4×D137C2×C411yescharacteristic
45C2×C4×D137D411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137C2311yescharacteristic
48C4⋊Dic137—21no—
49C4⋊Dic137C2×C411yescharacteristic
50#14555489—21no—
51#14555489—21no—
52#14555489—21no—
53C22⋊Dic137—21no—
54#14555487—21no—
55#14555487—21no—
56C2×Q8×C137C2×C411yescharacteristic
57C22×C4×C137C2×C411yescharacteristic
58C4⋊C4×C137—21no—
59C4⋊C4×C137C2×C411yescharacteristic
60C22⋊C4×C137—21no—
61C2×#10587739—42no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64#11764047—24no—
65C22×D137—22no—
66C22×D137—22no—
67C22×D137C22⋊C411yescharacteristic
68C22×D137SD1612yes—
69C22×D137Q1612yes—
70C22×D137C2×D411yescharacteristic
71Dic274—22no—
72Dic274—41no—
73Dic274—41no—
74C2×Dic137—21no—
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—42no—
85C4×D137—42no—
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#10312yescomplemented
98C22×C137C2×C22⋊C411yescharacteristic
99C2×D137—21no—
100C2×D137—21no—
101C2×D137—24no—
102C2×D137—24no—
103C2×D137#10312yescomplemented
104C2×D137#12112yes—
105C2×D137C2×SD1611yescharacteristic
106C2×D137C2×Q1611yescharacteristic
107C2×D137#13512yes—
108#10587739—82no—
109C4×C137—21no—
110C4×C137—41no—
111C4×C137—41no—
112C4×C137C4×D412yes—
113Dic137—21no—
114Dic137—41no—
115Dic137—41no—
116Dic137#12011yescharacteristic
117Dic137#12311yescharacteristic
118C2×C137—22no—
119C2×C137#38611yescharacteristic
120C2×C137C2×#10311yescharacteristic
121C2×C137#41611yescharacteristic
122D137—22no—
123D137#46512yes—
124D137#46712yes—
125C137#189211yescharacteristic, complemented, Sylow
126#1892—1371nomaximal, Sylow
127C23×C8—1371no—
128C2×C22⋊Q8—1371no—
129C2×#103—1372no—
130#380—1371no—
131#340—1372no—
132C22×Q8—1371no—
133C23×C4—1371no—
134C22×C8—2741no—
135C22×C8—2741no—
136C22×C8—2744no—
137C22×C8—1372no—
138C22⋊Q8—2741no—
139C22⋊Q8—2741no—
140C22⋊Q8—1371no—
141C22⋊Q8—1371no—
142C22⋊Q8—1371no—
143C22⋊Q8#1058773911yescharacteristic, complemented
144C2×C4⋊C4—2741no—
145C2×C4⋊C4—1372no—
146C2×C4⋊C4—1371no—
147C2×C22⋊C4—2741no—
148C2×C22⋊C4—2741no—
149#103—2744no—
150#95—2741no—
151C24—1371no—
152C2×Q8—2741no—
153C2×Q8—2742no—
154C2×Q8—2742no—
155C2×Q8—1371no—
156C2×Q8C2×#1058773911yescharacteristic
157C22×C4—2742no—
158C22×C4—2742no—
159C22×C4—2741no—
160C22×C4—2741no—
161C22×C4—2742no—
162C22×C4—1371no—
163C22×C4—1371no—
164C22×C4—1371no—
165C22×C4—1371no—
166C22×C4—1371no—
167C22×C4C2×#1058773911yescharacteristic
168C2×C8—2744no—
169C2×C8—2744no—
170C2×C8—2742no—
171C2×C8—2742no—
172C2×C8—1374no—
173C4⋊C4—21no—
174C4⋊C4—2741no—
175C4⋊C4—2741no—
176C4⋊C4—2741no—
177C4⋊C4—2741no—
178C4⋊C4—2742no—
179C4⋊C4—2742no—
180C4⋊C4—1371no—
181C4⋊C4C2×#1058773911yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—2741no—
184C22⋊C4—2741no—
185C22⋊C4—2741no—
186C22⋊C4—5482no—
187C23—2742no—
188C23—2742no—
189C23—1372no—
190C23—1372no—
191C23—1371no—
192C23—1371no—
193C23C22⋊#1058773911yescharacteristic
194Q8—22no—
195Q8—2742no—
196Q8—5481no—
197Q8—5481no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—2742no—
202C2×C4—2741no—
203C2×C4—2741no—
204C2×C4—2741no—
205C2×C4—2741no—
206C2×C4—2742no—
207C2×C4—2741no—
208C2×C4—2741no—
209C2×C4—5482no—
210C2×C4—5482no—
211C2×C4—5482no—
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—2744no—
224C22—2744no—
225C22—2741no—
226C22—2741no—
227C22—1372no—
228C22—1372no—
229C22—1371no—
230C22—1371no—
231C22—1372no—
232C22#1798415112yescomplemented
233C22C2×C22⋊#1058773911yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—2741no—
238C4—5481no—
239C4—5481no—
240C4—10962no—
241C4—1371no—
242C4—1371no—
243C4D4×#1058773912yes—
244C2—22no—
245C2—2742no—
246C2—1372no—
247C2—1372no—
248C2#2257835111yescharacteristic
249C2C2×#1798415111yescharacteristic
250C2#2257839311yescharacteristic
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#1892
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
97. C22×C137#103semidirect
103. C2×D137#103semidirect
125. C137#1892semidirect
143. C22⋊Q8#10587739semidirect
232. C22#17984151semidirect

Maximal subgroups

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

subgroupindex
C22×#145555182
C2×#179841512
#225783452
#225782762
C22⋊Q8×D1372
#1892137