Finite Group Databasebeta

← #31796198

Subgroups — #31796198

Order 17536.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578374C211yesmaximal, characteristic
2#22578346C211yesmaximal, characteristic
3#22578257C212yesmaximal, complemented
4#22578256C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7C2×C22⋊#10587739—21no—
8#17984145C2212yes—
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12C2×Q8×D137C411yescharacteristic, complemented
13C22×C4×D137C2211yescharacteristic
14#17984068C2211yescharacteristic
15#17984067C411yescharacteristic, complemented
16#17984066—21no—
17#17984065—21no—
18#17984064C2211yescharacteristic
19C4⋊C4×D137—21no—
20C4⋊C4×D137C411yescharacteristic, complemented
21C22⋊C4×D137—21no—
22C22⋊Q8×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555520—42no—
25C2×#11764047—22no—
26C22⋊#10587739—42no—
27C23×D137D411yescharacteristic
28C2×Dic274—21no—
29C2×Dic274C2×C411yescharacteristic
30Q8×D137—42no—
31C22×Dic137D411yescharacteristic
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—42no—
35C2×C4×D137C2×C411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137C2311yescharacteristic
39C4⋊Dic137—21no—
40C4⋊Dic137C2×C411yescharacteristic
41#14555489—21no—
42#14555489—21no—
43#14555489—21no—
44C22⋊Dic137—21no—
45#14555487—21no—
46#14555487—21no—
47C2×Q8×C137C2×C411yescharacteristic
48C22×C4×C137C2×C411yescharacteristic
49C4⋊C4×C137—21no—
50C4⋊C4×C137C2×C411yescharacteristic
51C22⋊C4×C137—21no—
52C2×#10587739—42no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55#11764047—42no—
56C22×D137—22no—
57C22×D137—22no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137C2×D411yescharacteristic
61Dic274—42no—
62Dic274—41no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—41no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—22no—
69C4×D137—22no—
70C4×D137—41no—
71C4×D137—41no—
72C4×D137—42no—
73C4×D137—42no—
74C23×C137C22⋊C411yescharacteristic
75Q8×C137—41no—
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137—41no—
79C2×C4×C137C22⋊C411yescharacteristic
80C2×C4×C137C22×C411yescharacteristic
81C22×C137—21no—
82C22×C137—21no—
83C22×C137—21no—
84C22×C137C2×C22⋊C411yescharacteristic
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—21no—
89C2×D137—44no—
90C2×D137#9912yes—
91C2×D137#13712yes—
92#10587739—82no—
93C4×C137—21no—
94C4×C137—21no—
95C4×C137—41no—
96C4×C137—41no—
97Dic137—21no—
98Dic137—21no—
99Dic137—41no—
100Dic137—41no—
101C2×C137—41no—
102C2×C137C2×#9911yescharacteristic
103C2×C137#41911yescharacteristic
104C2×C137#42111yescharacteristic
105D137—22no—
106D137—22no—
107C137#151111yescharacteristic, complemented, Sylow
108#1511—1371nomaximal, Sylow
109C2×C22⋊Q8—1371no—
110#407—1371no—
111#381—1371no—
112#328—1372no—
113#328—1372no—
114C22×Q8—1371no—
115C23×C4—1371no—
116C2×M16—2741no—
117C22⋊Q8—2741no—
118C22⋊Q8—2741no—
119C22⋊Q8—1371no—
120C22⋊Q8—1371no—
121C22⋊Q8—1371no—
122C22⋊Q8#1058773911yescharacteristic, complemented
123C2×C4⋊C4—2741no—
124C2×C4⋊C4—1371no—
125C2×C22⋊C4—2741no—
126C2×C22⋊C4—2741no—
127C22⋊C8—2741no—
128C22⋊C8—1372no—
129#95—2741no—
130#95—1372no—
131C24—1371no—
132C2×Q8—2741no—
133C2×Q8—5482no—
134C2×Q8—1371no—
135C2×Q8C2×#1058773911yescharacteristic
136C22×C4—2741no—
137C22×C4—2741no—
138C22×C4—2742no—
139C22×C4—5482no—
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4—1371no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4C2×#1058773911yescharacteristic
146M16—5482no—
147C2×C8—2742no—
148C4⋊C4—21no—
149C4⋊C4—2741no—
150C4⋊C4—2741no—
151C4⋊C4—2741no—
152C4⋊C4—2741no—
153C4⋊C4—1371no—
154C4⋊C4C2×#1058773911yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2741no—
157C22⋊C4—2741no—
158C22⋊C4—2741no—
159C22⋊C4—5482no—
160C23—2742no—
161C23—2742no—
162C23—5481no—
163C23—1371no—
164C23—1371no—
165C23C22⋊#1058773911yescharacteristic
166Q8—41no—
167Q8—5481no—
168Q8—5482no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—2742no—
173C2×C4—2742no—
174C2×C4—2741no—
175C2×C4—2741no—
176C2×C4—5481no—
177C2×C4—5481no—
178C2×C4—5482no—
179C2×C4—5482no—
180C2×C4—5481no—
181C2×C4—5482no—
182C2×C4—5482no—
183C2×C4—5482no—
184C2×C4—1371no—
185C2×C4—1371no—
186C2×C4C22⋊#1058773911yescharacteristic
187C2×C4C22×#1058773911yescharacteristic
188C8—5482no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—2741no—
193C22—2741no—
194C22—2741no—
195C22—2741no—
196C22—5484no—
197C22—1372no—
198C22—1372no—
199C22C2×C22⋊#1058773911yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—2741no—
205C4—2741no—
206C4—5481no—
207C4—5481no—
208C4—10962no—
209C2—41no—
210C2—2742no—
211C2—2742no—
212C2D137⋊#9911yescharacteristic
213C2#2257839611yescharacteristic
214C2#2257840011yescharacteristic
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#1511
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22578257C2semidirect
4. #22578256C2semidirect
12. C2×Q8×D137C4semidirect
15. #17984067C4semidirect
20. C4⋊C4×D137C4semidirect
22. C22⋊Q8×C137C4semidirect
107. C137#1511semidirect
122. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225783462
#225782572
#225782562
C22⋊Q8×D1372
#1511137