Finite Group Databasebeta

← #31796692

Subgroups — #31796692

Order 17536.

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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#22578276C214yesmaximal, complemented
4C2×C4⋊C4×D137C211yesmaximal, characteristic
5C2×#14555518C2212yes—
6C2×C4⋊#10587739C2212yes—
7C2×C22⋊#10587739—21no—
8#17984145—21no—
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11C22×C4×D137—21no—
12C22×C4×D137C2211yescharacteristic
13C2×C4⋊Dic137C411yescharacteristic, complemented
14C2×#14555489—21no—
15C4⋊C4×D137—21no—
16C4⋊C4×D137—21no—
17C4⋊C4×D137C412yescomplemented
18C4⋊C4×D137C2212yes—
19C2×C4⋊C4×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555518—24no—
22C2×#11764047—22no—
23C4⋊#10587739—24no—
24C22⋊#10587739—42no—
25C23×D137D411yescharacteristic
26C22×Dic137—21no—
27C22×Dic137D411yescharacteristic
28C2×C4×D137—24no—
29C2×C4×D137—22no—
30C2×C4×D137—22no—
31C2×C4×D137—42no—
32C2×C4×D137—42no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137C2311yescharacteristic
37C4⋊Dic137—21no—
38C4⋊Dic137C2×C412yes—
39#14555489—22no—
40#14555489—22no—
41C22×C4×C137—21no—
42C22×C4×C137C2×C411yescharacteristic
43C4⋊C4×C137—21no—
44C4⋊C4×C137C2×C412yes—
45C2×#10587739—42no—
46C2×#10587739—42no—
47C2×#10587739—42no—
48#11764047—42no—
49C22×D137—22no—
50C22×D137—22no—
51C22×D137C22⋊C411yescharacteristic
52C22×D137D812yes—
53C22×D137SD1612yes—
54C22×D137C2×D411yescharacteristic
55C2×Dic137—22no—
56C2×Dic137—22no—
57C2×Dic137—42no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C4×D137—22no—
61C4×D137—22no—
62C4×D137—22no—
63C4×D137—44no—
64C4×D137—42no—
65C4×D137—42no—
66C4×D137#4114yes—
67C23×C137C22⋊C411yescharacteristic
68C2×C4×C137—22no—
69C2×C4×C137—22no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22×C411yescharacteristic
73C22×C137—21no—
74C22×C137—21no—
75C22×C137#10212yes—
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—24no—
80C2×D137—24no—
81C2×D137#10212yes—
82C2×D137#12312yes—
83C2×D137C2×D811yescharacteristic
84C2×D137C2×SD1611yescharacteristic
85C2×D137#13712yes—
86#10587739—82no—
87C4×C137—21no—
88C4×C137—42no—
89C4×C137#11712yes—
90Dic137—21no—
91Dic137—42no—
92Dic137#12312yes—
93C2×C137—22no—
94C2×C137#38111yescharacteristic
95C2×C137C2×#10211yescharacteristic
96C2×C137#41911yescharacteristic
97D137—22no—
98D137#48012yes—
99D137#48312yes—
100C137#182311yescharacteristic, complemented, Sylow
101#1823—1371nomaximal, Sylow
102C22×C4⋊C4—1371no—
103C2×C22⋊C8—1371no—
104#380—1371no—
105#340—1374no—
106C23×C4—2741no—
107C23×C4—1371no—
108C22×C8—1372no—
109C2×C4⋊C4—2741no—
110C2×C4⋊C4—2741no—
111C2×C4⋊C4—2741no—
112C2×C4⋊C4—1372no—
113C2×C4⋊C4—1371no—
114C2×C4⋊C4—1372no—
115C2×C4⋊C4—1372no—
116C2×C4⋊C4#1058773911yescharacteristic, complemented
117C2×C22⋊C4—2741no—
118C22⋊C8—2741no—
119C22⋊C8—2741no—
120#95—2741no—
121C24—1371no—
122C22×C4—21no—
123C22×C4—2744no—
124C22×C4—2742no—
125C22×C4—2742no—
126C22×C4—2741no—
127C22×C4—2742no—
128C22×C4—5482no—
129C22×C4—5482no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4C2×#1058773911yescharacteristic
136C2×C8—2742no—
137C2×C8—2744no—
138C4⋊C4—21no—
139C4⋊C4—2742no—
140C4⋊C4—2742no—
141C4⋊C4—2741no—
142C4⋊C4—2744no—
143C4⋊C4—1372no—
144C4⋊C4C2×#1058773912yes—
145C22⋊C4—5482no—
146C23—2742no—
147C23—2742no—
148C23—1372no—
149C23—1372no—
150C23—1371no—
151C23—1371no—
152C23C22⋊#1058773911yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—2742no—
157C2×C4—2742no—
158C2×C4—2742no—
159C2×C4—2742no—
160C2×C4—2742no—
161C2×C4—5484no—
162C2×C4—5482no—
163C2×C4—5482no—
164C2×C4—5482no—
165C2×C4—5482no—
166C2×C4—5482no—
167C2×C4—5482no—
168C2×C4—1371no—
169C2×C4—1371no—
170C2×C4—1374no—
171C2×C4C22⋊#1058773911yescharacteristic
172C2×C4C22×#1058773911yescharacteristic
173C8—5482no—
174C22—21no—
175C22—21no—
176C22—2744no—
177C22—2744no—
178C22—2741no—
179C22—2741no—
180C22—1372no—
181C22—1372no—
182C22—1371no—
183C22—1371no—
184C22—1372no—
185C22#1798415012yes—
186C22C2×C22⋊#1058773911yescharacteristic
187C4—21no—
188C4—42no—
189C4—2741no—
190C4—5482no—
191C4—10962no—
192C4—1372no—
193C4#1798416712yes—
194C2—22no—
195C2—2742no—
196C2—1372no—
197C2—1372no—
198C2#2257834611yescharacteristic
199C2C2×#1798415011yescharacteristic
200C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1823
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22578276C2semidirect
13. C2×C4⋊Dic137C4semidirect
17. C4⋊C4×D137C4semidirect
19. C2×C4⋊C4×C137C4semidirect
100. C137#1823semidirect
116. C2×C4⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783712
#225783452
#225782762
C2×C4⋊C4×D1372
#1823137