Finite Group Databasebeta

← #31796694

Subgroups — #31796694

Order 17536.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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#22578371C211yesmaximal, characteristic
2#22578345C211yesmaximal, characteristic
3#22578276C212yesmaximal, complemented
4#22578276C212yesmaximal, complemented
5#117×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7C2×C4⋊#10587739C2212yes—
8C2×C22⋊#10587739—21no—
9#17984145—21no—
10C22⋊#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12C22×C4×D137C2211yescharacteristic
13#17984047C411yescharacteristic, complemented
14#17984046C411yescharacteristic, complemented
15#17984046C411yescharacteristic, complemented
16#17984045—21no—
17#17984044—21no—
18C4⋊C4×D137C2211yescharacteristic
19C4⋊C4×D137C2211yescharacteristic
20C22⋊C4×D137—21no—
21C42×D137—21no—
22#117×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555518—24no—
25C2×#11764047—22no—
26C4⋊#10587739—24no—
27C22⋊#10587739—42no—
28C23×D137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C2×C4×D137—24no—
31C2×C4×D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137C2311yescharacteristic
37C4⋊Dic137C2×C411yescharacteristic
38C4⋊Dic137C2×C411yescharacteristic
39#14555489—21no—
40#14555489—21no—
41C22⋊Dic137—21no—
42#14555487—21no—
43#14555487—21no—
44C4×Dic137—21no—
45C4×Dic137—21no—
46C4×Dic137—21no—
47C22×C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C4⋊C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51C42×C137—21no—
52C2×#10587739—42no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55#11764047—42no—
56C22×D137—22no—
57C22×D137—22no—
58C22×D137—21no—
59C22×D137—21no—
60C22×D137C22⋊C411yescharacteristic
61C22×D137C2×D411yescharacteristic
62C2×Dic137—22no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C4×D137—22no—
68C4×D137—22no—
69C4×D137—22no—
70C4×D137—42no—
71C4×D137—41no—
72C4×D137—41no—
73C4×D137#4114yes—
74C23×C137C22⋊C411yescharacteristic
75C2×C4×C137—22no—
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137C22⋊C411yescharacteristic
79C2×C4×C137C22×C411yescharacteristic
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—21no—
83C22×C137C2×C22⋊C411yescharacteristic
84C2×D137—21no—
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—42no—
88C2×D137—42no—
89C2×D137#12312yes—
90C2×D137#13512yes—
91C2×D137AGL(1,ℤ/8)11yescharacteristic
92C2×D137#13711yescharacteristic
93#10587739—82no—
94C4×C137—21no—
95C4×C137—41no—
96C4×C137—41no—
97C4×C137#11712yes—
98Dic137—21no—
99Dic137—41no—
100Dic137—41no—
101Dic137#12312yes—
102C2×C137—41no—
103C2×C137#38111yescharacteristic
104C2×C137#41611yescharacteristic
105C2×C137#41711yescharacteristic
106D137—41no—
107D137#48212yes—
108D137#48512yes—
109C137#182511yescharacteristic, complemented, Sylow
110#1825—1371nomaximal, Sylow
111C2×#117—1371no—
112C2×C22⋊C8—1371no—
113#380—1371no—
114#340—1372no—
115#340—1372no—
116C23×C4—1371no—
117C22×C8—1372no—
118#117—2741no—
119#117—2741no—
120#117—1371no—
121#117—1371no—
122#117—1371no—
123#117#1058773911yescharacteristic, complemented
124C2×C4⋊C4—1372no—
125C2×C4⋊C4—1371no—
126C2×C4⋊C4—1371no—
127C2×C22⋊C4—2741no—
128C2×C22⋊C4—2741no—
129C2×C42—2741no—
130C22⋊C8—2741no—
131C22⋊C8—2741no—
132#95—2741no—
133C24—1371no—
134C22×C4—2744no—
135C22×C4—2741no—
136C22×C4—2741no—
137C22×C4—2742no—
138C22×C4—1371no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4—1371no—
143C22×C4C2×#1058773911yescharacteristic
144C2×C8—2742no—
145C2×C8—2744no—
146C4⋊C4—2741no—
147C4⋊C4—2741no—
148C4⋊C4—2744no—
149C4⋊C4—1371no—
150C4⋊C4—1371no—
151C4⋊C4C2×#1058773911yescharacteristic
152C4⋊C4C2×#1058773911yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C22⋊C4—2741no—
157C22⋊C4—5482no—
158C42—21no—
159C42—2741no—
160C42—2741no—
161C42—2741no—
162C23—2741no—
163C23—2742no—
164C23—2742no—
165C23—2741no—
166C23—1371no—
167C23—1371no—
168C23C22⋊#1058773911yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—2742no—
173C2×C4—2742no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—2741no—
177C2×C4—2741no—
178C2×C4—5482no—
179C2×C4—5481no—
180C2×C4—5481no—
181C2×C4—5482no—
182C2×C4—5482no—
183C2×C4—5482no—
184C2×C4—1371no—
185C2×C4—1371no—
186C2×C4—1374no—
187C2×C4C22⋊#1058773911yescharacteristic
188C2×C4C22×#1058773911yescharacteristic
189C8—5482no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—2741no—
194C22—2741no—
195C22—2741no—
196C22—5482no—
197C22—5482no—
198C22—1372no—
199C22—1372no—
200C22—1371no—
201C22—1371no—
202C22C2×C22⋊#1058773911yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—2741no—
207C4—5481no—
208C4—5481no—
209C4—10962no—
210C4—1372no—
211C4#1798416712yes—
212C2—41no—
213C2—5481no—
214C2—1372no—
215C2—1372no—
216C2#2257834611yescharacteristic
217C2#2257839311yescharacteristic
218C2#2257839411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenter#117#10587739
hypocenterC137#1825
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #22578276C2semidirect
4. #22578276C2semidirect
13. #17984047C4semidirect
14. #17984046C4semidirect
15. #17984046C4semidirect
22. #117×C137C4semidirect
109. C137#1825semidirect
123. #117#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783712
#225783452
#225782762
#117×D1372
#1825137