Finite Group Databasebeta

← #31796978

Subgroups — #31796978

Order 17536.

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

normal subgroups only

This group has 324 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#22578345C211yesmaximal, characteristic
2C4×C22⋊#10587739C211yesmaximal, characteristic
3#22578281C214yesmaximal
4C2×#17984064C211yesmaximal, characteristic
5C2×C4⋊#10587739—21no—
6C2×C22⋊#10587739C2212yes—
7C2×C22⋊#10587739C2212yes—
8C2×C4×#10587739—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4—21no—
11C22×C4×D137C2211yescharacteristic
12#17984064—21no—
13#17984064—21no—
14#17984064—21no—
15#17984064—21no—
16C2×C4⋊Dic137C411yescharacteristic, complemented
17C2×#14555489C412yescomplemented
18C2×#14555487C2212yes—
19C22×Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C22×#10587739—22no—
22C4⋊#10587739—42no—
23C22⋊#10587739—22no—
24C22⋊#10587739—24no—
25C22⋊#10587739Q814yescomplemented
26C4×#10587739—42no—
27C23×D137C2311yescharacteristic
28C22×Dic137C2×C412yes—
29C22×Dic137C2×C411yescharacteristic
30C2×C4×D137—42no—
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37#14555489—22no—
38#14555489—22no—
39#14555487—22no—
40#14555487—22no—
41C2×Q8×C137—21no—
42C2×Q8×C137—21no—
43C2×Q8×C137—42no—
44C22×C4×C137C2×C412yes—
45C22×C4×C137C2×C411yescharacteristic
46C2×#10587739—24no—
47C2×#10587739—42no—
48C2×#10587739—42no—
49C2×#10587739—42no—
50C2×#10587739—42no—
51C2×#10587739—42no—
52C22×D137—41no—
53C22×D137—41no—
54C22×D137C2×D411yescharacteristic
55C22×D137C2×D411yescharacteristic
56C22×D137C2×Q812yes—
57C22×D137#4112yes—
58C2×Dic137—22no—
59C2×Dic137—41no—
60C2×Dic137—42no—
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C4×D137—22no—
64C4×D137—22no—
65C4×D137—41no—
66C4×D137—41no—
67C23×C137C22×C411yescharacteristic
68Q8×C137—42no—
69Q8×C137—42no—
70C2×C4×C137—22no—
71C2×C4×C137—22no—
72C2×C4×C137—42no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22⋊C411yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137C2×C22⋊C411yescharacteristic
78C22×C137C4×Q812yes—
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—42no—
84C2×D137—42no—
85C2×D137C22⋊Q812yes—
86C2×D137#12411yescharacteristic
87C2×D137C4⋊Q811yescharacteristic
88#10587739—44no—
89#10587739—82no—
90C4×C137—21no—
91C4×C137—42no—
92C4×C137#9912yes—
93Dic137—21no—
94Dic137—21no—
95Dic137—42no—
96C2×C137—22no—
97C2×C137#39111yescharacteristic
98C2×C137C2×#9911yescharacteristic
99C2×C137#41011yescharacteristic
100D137—22no—
101D137—21no—
102D137—21no—
103C137#199611yescharacteristic, complemented, Sylow
104#1996—1371nomaximal, Sylow
105C2×C22⋊Q8—1371no—
106#380—1371no—
107C4×C22⋊C4—1371no—
108#342—1374no—
109C22×Q8#1058773911yescharacteristic, complemented
110C23×C4—1371no—
111C22⋊Q8—2741no—
112C22⋊Q8—2741no—
113C22⋊Q8—2741no—
114C22⋊Q8—2741no—
115C2×C4⋊C4—2741no—
116C2×C4⋊C4—1372no—
117C2×C4⋊C4—1371no—
118C2×C22⋊C4—1372no—
119C2×C22⋊C4—1372no—
120C2×C22⋊C4—1372no—
121C2×C42—2741no—
122#95—2741no—
123#95—2741no—
124C24—1371no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—2742no—
129C22×C4—2742no—
130C22×C4—5482no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1372no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4C2×#1058773912yes—
138C22×C4C2×#1058773911yescharacteristic
139C4⋊C4—2742no—
140C4⋊C4—2742no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—5482no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C22⋊C4—2742no—
147C22⋊C4—2744no—
148C22⋊C4—1374no—
149C42—5482no—
150C23—5481no—
151C23—5481no—
152C23—1371no—
153C23—1371no—
154C23—1372no—
155C23—1372no—
156C23C22×#1058773911yescharacteristic
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—2742no—
163C2×C4—2742no—
164C2×C4—2744no—
165C2×C4—2742no—
166C2×C4—5481no—
167C2×C4—5481no—
168C2×C4—5481no—
169C2×C4—5482no—
170C2×C4—5482no—
171C2×C4—5482no—
172C2×C4—5482no—
173C2×C4—5482no—
174C2×C4—5482no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4C22⋊#1058773911yescharacteristic
178C2×C4C22⋊#1058773911yescharacteristic
179C22—21no—
180C22—21no—
181C22—2741no—
182C22—2741no—
183C22—2741no—
184C22—2741no—
185C22—5482no—
186C22—5482no—
187C22—1372no—
188C22—1371no—
189C22—1371no—
190C22C2×C22⋊#1058773911yescharacteristic
191C22Q8×#1058773912yes—
192C4—21no—
193C4—42no—
194C4—2741no—
195C4—2741no—
196C4—5484no—
197C4—5482no—
198C4—10962no—
199C4#1798414612yes—
200C2—22no—
201C2—2742no—
202C2—2741no—
203C2—2741no—
204C2(C2×#10587739)⋊Q811yescharacteristic
205C2C2×#1798414611yescharacteristic
206C2#2257838411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC22×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical, cosocleC23×D137C23
hypercenterC22×Q8#10587739
hypocenterC137#1996
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊Dic137C4semidirect
17. C2×#14555489C4semidirect
19. C22×Q8×C137C4semidirect
25. C22⋊#10587739Q8semidirect
103. C137#1996semidirect
109. C22×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783452
C4×C22⋊#105877392
#225782812
C2×#179840642
#1996137