Finite Group Databasebeta

← #31797266

Subgroups — #31797266

Order 17536.

512 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 162 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#22578352C218yesmaximal, complemented
2#22578346C212yesmaximal
3#22578345C212yesmaximal
4C4×C22⋊#10587739C212yesmaximal
5C2×C4⋊C4×D137C211yesmaximal, characteristic
6C2×C4⋊#10587739C2214yes—
7C2×C22⋊#10587739C2214yes—
8C2×C4×#10587739C2214yes—
9(C2×#10587739)⋊C4C2214yes—
10(C2×#10587739)⋊C4C2214yes—
11(C2×#10587739)⋊C4C2218yes—
12C22×C4×D137C2211yescharacteristic
13C22×C4×D137C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15C2×C4⋊Dic137C411yescharacteristic, complemented
16C2×#14555489C411yescharacteristic, complemented
17C2×#14555489C411yescharacteristic, complemented
18C4⋊C4×D137C414yescomplemented
19C4⋊C4×D137C2214yes—
20C2×C4⋊C4×C137C411yescharacteristic, complemented
21C22×#10587739C2318yes—
22C4⋊#10587739—44no—
23C22⋊#10587739—28no—
24C4×#10587739—44no—
25C23×D137C2311yescharacteristic
26C22×Dic137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2×C411yescharacteristic
29C2×C4×D137—24no—
30C2×C4×D137—42no—
31C2×C4×D137—42no—
32C2×C4×D137C2×C412yes—
33C2×C4×D137C2×C412yes—
34C2×C4×D137C2×C412yes—
35C2×C4×D137C2312yes—
36C2×C4×D137C2312yes—
37C2×C4×D137C2312yes—
38C4⋊Dic137C2×C414yes—
39#14555489C2×C414yes—
40#14555489C2×C414yes—
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137C2×C411yescharacteristic
43C22×C4×C137C2×C411yescharacteristic
44C4⋊C4×C137C2×C414yes—
45C2×#10587739—28no—
46C2×#10587739—28no—
47C2×#10587739—28no—
48C22×D137—22no—
49C22×D137—22no—
50C22×D137C22×C411yescharacteristic
51C22×D137#4114yes—
52C22×D137C2411yescharacteristic
53C2×Dic137—22no—
54C2×Dic137—41no—
55C2×Dic137—41no—
56C2×Dic137C22×C412yes—
57C2×Dic137C22×C412yes—
58C2×Dic137C22×C412yes—
59C4×D137—24no—
60C4×D137—24no—
61C4×D137—24no—
62C4×D137—24no—
63C4×D137—42no—
64C4×D137—42no—
65C23×C137C22×C411yescharacteristic
66C2×C4×C137—22no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137C22×C412yes—
70C2×C4×C137C22×C412yes—
71C2×C4×C137C22×C412yes—
72C22×C137—21no—
73C22×C137—21no—
74C22×C137#11712yes—
75C22×C137C23×C411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—28no—
79C2×D137#11712yes—
80C2×D137C2×#4112yes—
81C2×D137H(5,𝔽2)12yes—
82C2×D1372−1+412yes—
83#10587739—48no—
84C4×C137—22no—
85C4×C137—22no—
86C4×C137—22no—
87Dic137—22no—
88Dic137—22no—
89Dic137—22no—
90C2×C137—22no—
91C2×C137C2×#11711yescharacteristic
92C2×C137#51811yescharacteristic
93C2×C137#51911yescharacteristic
94D137—22no—
95D137#53914yes—
96C137#231311yescharacteristic, complemented, Sylow
97#2313—1371nomaximal, Sylow
98C22×C4⋊C4—1371no—
99#387—1378no—
100#381—1372no—
101#380—1372no—
102C4×C22⋊C4—1372no—
103C23×C4—1371no—
104C23×C4—1371no—
105C23×C4—1371no—
106C2×C4⋊C4—1374no—
107C2×C4⋊C4—1371no—
108C2×C4⋊C4—1371no—
109C2×C4⋊C4—1371no—
110C2×C4⋊C4—1374no—
111C2×C4⋊C4—1374no—
112C2×C4⋊C4#1058773911yescharacteristic, complemented
113C2×C22⋊C4—1374no—
114C2×C42—1374no—
115#95—1374no—
116#95—1374no—
117#95—1378no—
118C24—1371no—
119C22×C4—2744no—
120C22×C4—5482no—
121C22×C4—5482no—
122C22×C4—1378no—
123C22×C4—1371no—
124C22×C4—1372no—
125C22×C4—1372no—
126C22×C4—1372no—
127C22×C4—1372no—
128C22×C4—1371no—
129C22×C4—1371no—
130C22×C4—1372no—
131C22×C4—1372no—
132C22×C4C2×#1058773911yescharacteristic
133C22×C4C2×#1058773911yescharacteristic
134C22×C4C2×#1058773911yescharacteristic
135C4⋊C4—5484no—
136C4⋊C4—1374no—
137C4⋊C4—1374no—
138C4⋊C4—1374no—
139C4⋊C4C2×#1058773914yes—
140C22⋊C4—2748no—
141C42—5484no—
142C23—2742no—
143C23—2742no—
144C23—1374no—
145C23—1371no—
146C23—1371no—
147C23C22×#1058773911yescharacteristic
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—2744no—
152C2×C4—2744no—
153C2×C4—2744no—
154C2×C4—2744no—
155C2×C4—2742no—
156C2×C4—2748no—
157C2×C4—2748no—
158C2×C4—2748no—
159C2×C4—5482no—
160C2×C4—5482no—
161C2×C4—5481no—
162C2×C4—5481no—
163C2×C4—1372no—
164C2×C4—1372no—
165C2×C4—1372no—
166C2×C4C22×#1058773912yes—
167C2×C4C22×#1058773912yes—
168C2×C4C22×#1058773912yes—
169C22—21no—
170C22—21no—
171C22—2748no—
172C22—2741no—
173C22—2741no—
174C22—1372no—
175C22—1372no—
176C22—1372no—
177C22—1372no—
178C22#1798416712yes—
179C22C23×#1058773911yescharacteristic
180C4—22no—
181C4—22no—
182C4—22no—
183C4—2742no—
184C4—2742no—
185C4—2742no—
186C4—5488no—
187C2—22no—
188C2—2742no—
189C2—1374no—
190C2C2×#1798416711yescharacteristic
191C2#2257847011yescharacteristic
192C2#2257847111yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
FittingC2×C4⋊C4×C137C4
prime radicalC23×D137C23
cosocleC22×D137C24
hypercenterC2×C4⋊C4#10587739
hypocenterC137#2313
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578352C2semidirect
15. C2×C4⋊Dic137C4semidirect
16. C2×#14555489C4semidirect
17. C2×#14555489C4semidirect
18. C4⋊C4×D137C4semidirect
20. C2×C4⋊C4×C137C4semidirect
96. C137#2313semidirect
112. C2×C4⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783522
#225783462
#225783452
C4×C22⋊#105877392
C2×C4⋊C4×D1372
#2313137