Finite Group Databasebeta

← #31796697

Subgroups — #31796697

Order 17536.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C4×C22⋊#10587739C211yesmaximal, characteristic
3#22578273C212yesmaximal, complemented
4#22578273C212yesmaximal, complemented
5#117×D137C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7C2×C22⋊#10587739—21no—
8C2×C4×#10587739C2212yes—
9#17984145—21no—
10C22⋊#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12C22×C4×D137C2211yescharacteristic
13#17984047C411yescharacteristic, complemented
14#17984046C2211yescharacteristic
15#17984046C2211yescharacteristic
16#17984045—21no—
17#17984044—21no—
18C4⋊C4×D137C411yescharacteristic, complemented
19C4⋊C4×D137C411yescharacteristic, complemented
20C22⋊C4×D137—21no—
21C42×D137—21no—
22#117×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555520—24no—
25C2×#11764047—22no—
26C22⋊#10587739—24no—
27C4×#10587739—24no—
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—24no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55#11764047—42no—
56C22×D137—22no—
57C22×D137—22no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137C2×D411yescharacteristic
61C2×Dic137—22no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C4×D137—22no—
67C4×D137—22no—
68C4×D137—22no—
69C4×D137—42no—
70C4×D137—42no—
71C4×D137#4114yes—
72C23×C137C22⋊C411yescharacteristic
73C2×C4×C137—22no—
74C2×C4×C137—21no—
75C2×C4×C137—21no—
76C2×C4×C137C22⋊C411yescharacteristic
77C2×C4×C137C22×C411yescharacteristic
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—21no—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—21no—
85C2×D137—21no—
86C2×D137—44no—
87C2×D137#10412yes—
88C2×D137#12312yes—
89#10587739—44no—
90C4×C137—21no—
91C4×C137—41no—
92C4×C137—41no—
93C4×C137#11712yes—
94Dic137—21no—
95Dic137—41no—
96Dic137—41no—
97Dic137#12312yes—
98C2×C137—41no—
99C2×C137#38111yescharacteristic
100C2×C137C2×#10411yescharacteristic
101C2×C137#42111yescharacteristic
102D137—22no—
103D137—22no—
104C137#182611yescharacteristic, complemented, Sylow
105#1826—1371nomaximal, Sylow
106C2×#117—1371no—
107#407—1371no—
108C4×C22⋊C4—1371no—
109#339—1372no—
110#339—1372no—
111C23×C4—1371no—
112C2×M16—1372no—
113#117—2741no—
114#117—2741no—
115#117—1371no—
116#117—1371no—
117#117—1371no—
118#117#1058773911yescharacteristic, complemented
119C2×C4⋊C4—1371no—
120C2×C4⋊C4—1371no—
121C2×C22⋊C4—2741no—
122C2×C22⋊C4—2741no—
123C2×C42—2741no—
124C2×C42—1372no—
125C22⋊C8—2741no—
126C22⋊C8—2741no—
127#95—2741no—
128C24—1371no—
129C22×C4—2744no—
130C22×C4—2741no—
131C22×C4—2741no—
132C22×C4—2742no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4C2×#1058773911yescharacteristic
139M16—2744no—
140C2×C8—2742no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—1371no—
144C4⋊C4—1371no—
145C4⋊C4C2×#1058773911yescharacteristic
146C4⋊C4C2×#1058773911yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—2741no—
149C22⋊C4—2741no—
150C22⋊C4—2744no—
151C22⋊C4—2741no—
152C42—21no—
153C42—2741no—
154C42—2741no—
155C42—2741no—
156C42—2744no—
157C23—2742no—
158C23—2742no—
159C23—5481no—
160C23—1371no—
161C23—1371no—
162C23C22⋊#1058773911yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—2742no—
167C2×C4—2742no—
168C2×C4—2742no—
169C2×C4—2742no—
170C2×C4—2744no—
171C2×C4—2741no—
172C2×C4—2741no—
173C2×C4—5482no—
174C2×C4—5482no—
175C2×C4—5482no—
176C2×C4—5482no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4—1374no—
180C2×C4C22⋊#1058773911yescharacteristic
181C2×C4C22×#1058773911yescharacteristic
182C8—5482no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—2741no—
187C22—2741no—
188C22—2741no—
189C22—2741no—
190C22—5484no—
191C22—1372no—
192C22—1372no—
193C22C2×C22⋊#1058773911yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—2741no—
198C4—5484no—
199C4—5481no—
200C4—5481no—
201C4—1372no—
202C4#1798416712yes—
203C2—41no—
204C2—2742no—
205C2—2742no—
206C2#2257834611yescharacteristic
207C2D137⋊#10411yescharacteristic
208C2#2257840011yescharacteristic
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#1826
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. #22578273C2semidirect
4. #22578273C2semidirect
13. #17984047C4semidirect
18. C4⋊C4×D137C4semidirect
19. C4⋊C4×D137C4semidirect
22. #117×C137C4semidirect
104. C137#1826semidirect
118. #117#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783742
C4×C22⋊#105877392
#225782732
#117×D1372
#1826137