Finite Group Databasebeta

← #31796775

Subgroups — #31796775

Order 17536.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×#17984167C211yesmaximal, characteristic
2C2×#17984151C212yesmaximal, complemented
3#22578371C211yesmaximal, characteristic
4#22578276C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7#17984167—24no—
8C2×C4⋊#10587739C2212yes—
9C2×C22⋊#10587739—21no—
10C2×C4×#10587739—21no—
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—22no—
26#14555518—22no—
27#14555518—22no—
28C2×#11764047—22no—
29C4⋊#10587739—22no—
30C4⋊#10587739D414yes—
31C22⋊#10587739—24no—
32C4×#10587739—24no—
33C23×D137D411yescharacteristic
34C2×Dic274—21no—
35C2×Dic274C2×C411yescharacteristic
36Q8×D137—22no—
37Q8×D137—22no—
38C22×Dic137D411yescharacteristic
39C2×C4×D137—22no—
40C2×C4×D137—22no—
41C2×C4×D137—21no—
42C2×C4×D137—21no—
43C2×C4×D137C2×C411yescharacteristic
44C2×C4×D137D411yescharacteristic
45C2×C4×D137D411yescharacteristic
46C2×C4×D137C2311yescharacteristic
47C4⋊Dic137—21no—
48C4⋊Dic137C2×C411yescharacteristic
49#14555489—21no—
50#14555489—21no—
51#14555489—21no—
52C22⋊Dic137—21no—
53#14555487—21no—
54#14555487—21no—
55C2×Q8×C137C2×C411yescharacteristic
56C22×C4×C137C2×C411yescharacteristic
57C4⋊C4×C137—21no—
58C4⋊C4×C137C2×C411yescharacteristic
59C22⋊C4×C137—21no—
60C2×#10587739—24no—
61C2×#10587739—42no—
62C2×#10587739—42no—
63#11764047—42no—
64C22×D137—21no—
65C22×D137—21no—
66C22×D137—22no—
67C22×D137—22no—
68C22×D137C22⋊C411yescharacteristic
69C22×D137C2×D411yescharacteristic
70Dic274—22no—
71Dic274—42no—
72C2×Dic137—22no—
73C2×Dic137—21no—
74C2×Dic137—21no—
75C2×Dic137C22⋊C411yescharacteristic
76C2×Dic137C2×D411yescharacteristic
77C4×D137—21no—
78C4×D137—21no—
79C4×D137—22no—
80C4×D137—22no—
81C4×D137—41no—
82C4×D137—41no—
83C4×D137—42no—
84C4×D137C2×D412yes—
85C4×D137#4112yes—
86C23×C137C22⋊C411yescharacteristic
87Q8×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—21no—
90C2×C4×C137—21no—
91C2×C4×C137C22⋊C411yescharacteristic
92C2×C4×C137C22×C411yescharacteristic
93C22×C137—21no—
94C22×C137—21no—
95C22×C137—21no—
96C22×C137C2×C22⋊C411yescharacteristic
97C2×D137—21no—
98C2×D137—21no—
99C2×D137—21no—
100C2×D137—42no—
101C2×D137—42no—
102C2×D137#12011yescharacteristic
103C2×D137#12311yescharacteristic
104C2×D137#13512yes—
105C2×D137#13712yes—
106#10587739—44no—
107C4×C137—21no—
108C4×C137—41no—
109C4×C137—41no—
110C4×C137C4×D412yes—
111Dic137—21no—
112Dic137—41no—
113Dic137—41no—
114Dic137#12112yes—
115C2×C137—41no—
116C2×C137#38611yescharacteristic
117C2×C137#41611yescharacteristic
118C2×C137#41911yescharacteristic
119D137—41no—
120D137#45212yes—
121D137#48512yes—
122C137#187311yescharacteristic, complemented, Sylow
123#1873—1371nomaximal, Sylow
124C2×C22⋊Q8—1371no—
125C2×#117—1371no—
126C2×#103—1372no—
127C2×C22⋊C8—1371no—
128#340—1372no—
129C22×Q8—1371no—
130C23×C4—1371no—
131C22×C8—1372no—
132C22⋊Q8—2741no—
133C22⋊Q8—2741no—
134C22⋊Q8—1371no—
135C22⋊Q8—1371no—
136C22⋊Q8—1371no—
137C22⋊Q8#1058773911yescharacteristic, complemented
138#117—2744no—
139C2×C4⋊C4—2741no—
140C2×C4⋊C4—1372no—
141C2×C4⋊C4—1371no—
142C2×C22⋊C4—2741no—
143C2×C22⋊C4—2741no—
144C2×C42—2741no—
145#103—2744no—
146C22⋊C8—2741no—
147C22⋊C8—2741no—
148C24—1371no—
149C2×Q8—2741no—
150C2×Q8—2742no—
151C2×Q8—2742no—
152C2×Q8—1371no—
153C2×Q8C2×#1058773911yescharacteristic
154C22×C4—2742no—
155C22×C4—2742no—
156C22×C4—2741no—
157C22×C4—2741no—
158C22×C4—2742no—
159C22×C4—1371no—
160C22×C4—1371no—
161C22×C4—1371no—
162C22×C4—1371no—
163C22×C4—1371no—
164C22×C4C2×#1058773911yescharacteristic
165C2×C8—2742no—
166C2×C8—2742no—
167C2×C8—2742no—
168C4⋊C4—21no—
169C4⋊C4—2741no—
170C4⋊C4—2741no—
171C4⋊C4—2741no—
172C4⋊C4—2742no—
173C4⋊C4—2741no—
174C4⋊C4—1374no—
175C4⋊C4—1371no—
176C4⋊C4C2×#1058773911yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—2741no—
179C22⋊C4—2741no—
180C22⋊C4—2744no—
181C22⋊C4—2741no—
182C42—2744no—
183C23—2741no—
184C23—2741no—
185C23—2742no—
186C23—2742no—
187C23—1371no—
188C23—1371no—
189C23C22⋊#1058773911yescharacteristic
190Q8—22no—
191Q8—2742no—
192Q8—5482no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—2742no—
199C2×C4—2742no—
200C2×C4—2742no—
201C2×C4—2744no—
202C2×C4—2741no—
203C2×C4—2741no—
204C2×C4—5481no—
205C2×C4—5481no—
206C2×C4—5482no—
207C2×C4—5482no—
208C2×C4—5482no—
209C2×C4—1371no—
210C2×C4—1371no—
211C2×C4—1372no—
212C2×C4—1372no—
213C2×C4C22⋊#1058773911yescharacteristic
214C2×C4C22×#1058773911yescharacteristic
215C8—5482no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—2741no—
220C22—2741no—
221C22—2741no—
222C22—5482no—
223C22—5482no—
224C22—1371no—
225C22—1371no—
226C22—1372no—
227C22—1372no—
228C22C2×C22⋊#1058773911yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—2741no—
233C4—5484no—
234C4—5481no—
235C4—5481no—
236C4—1372no—
237C4D4×#1058773912yes—
238C2—41no—
239C2—5481no—
240C2—1372no—
241C2—1372no—
242C2#2257835111yescharacteristic
243C2#2257839311yescharacteristic
244C2#2257839611yescharacteristic
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#1873
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
122. C137#1873semidirect
137. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841672
C2×#179841512
#225783712
#225782762
C22⋊Q8×D1372
#1873137