Finite Group Databasebeta

← #31796670

Subgroups — #31796670

Order 17536.

272 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×#17984159C211yesmaximal, characteristic, complemented
2C2×#17984158C211yesmaximal, characteristic, complemented
3#22578374C212yesmaximal
4#22578279C212yesmaximal, complemented
5C2×M16×D137C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7C2×#14555520C2212yes—
8#17984159—22no—
9#17984158—22no—
10#17984145—22no—
11C22⋊#11764047—22no—
12C22×C4×D137C2211yescharacteristic
13C2×#14555494C411yescharacteristic
14C2×#14555493C411yescharacteristic
15C2×#14555493C411yescharacteristic
16M16×D137—22no—
17M16×D137—22no—
18C2×C8×D137C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20C2×M16×C137C411yescharacteristic
21#14555520—22no—
22#14555520—22no—
23#14555520—22no—
24#14555520—22no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27C23×D137Q811yescharacteristic
28C22×Dic137D411yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137Q811yescharacteristic
36C2×C4×D137C2311yescharacteristic
37#14555494—22no—
38#14555493—22no—
39#14555493—22no—
40C2×#11764040C2×C411yescharacteristic
41C2×#11764040C2×C411yescharacteristic
42C8×D137—21no—
43C8×D137—21no—
44C8×D137D412yes—
45C8×D137D412yes—
46C22×C4×C137C2×C411yescharacteristic
47M16×C137—22no—
48C2×C8×C137C2×C411yescharacteristic
49C2×C8×C137C2×C411yescharacteristic
50#11764047—42no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137—21no—
54C22×D137—42no—
55C22×D137C4⋊C411yescharacteristic
56C22×D137C2×Q811yescharacteristic
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C4⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—21no—
64C4×D137—21no—
65C4×D137—21no—
66C4×D137—21no—
67C4×D137C2×D411yescharacteristic
68C4×D137C2×D411yescharacteristic
69C4×D137#4111yescharacteristic
70C4×D137#4111yescharacteristic
71#11764040—21no—
72#11764040C22⋊C412yes—
73C23×C137C4⋊C411yescharacteristic
74C2×C4×C137—21no—
75C2×C4×C137—21no—
76C2×C4×C137C4⋊C411yescharacteristic
77C2×C4×C137C22×C411yescharacteristic
78C8×C137—21no—
79C8×C137C22⋊C412yes—
80C22×C137—21no—
81C22×C137—22no—
82C22×C137C2×C4⋊C411yescharacteristic
83C2×D137—22no—
84C2×D137—21no—
85C2×D137—22no—
86C2×D137—42no—
87C2×D137—42no—
88C2×D137C22⋊Q811yescharacteristic
89C2×D137C22⋊Q811yescharacteristic
90C4×C137—21no—
91C4×C137C2×C22⋊C411yescharacteristic
92C4×C137#11711yescharacteristic
93Dic137—21no—
94Dic137#12111yescharacteristic
95Dic137#12111yescharacteristic
96C2×C137—41no—
97C2×C137#38011yescharacteristic
98C2×C137#43012yes—
99D137—21no—
100D137—21no—
101D137—41no—
102C137#181111yescharacteristic, complemented, Sylow
103#1811—1371nomaximal, Sylow
104C22×M16—1371no—
105C2×#108—1371no—
106C2×#108—1371no—
107#407—1372no—
108#341—1372no—
109C23×C4—1371no—
110C2×M16—2742no—
111C2×M16—2742no—
112C2×M16—1372no—
113C2×M16—1372no—
114C2×M16—1371no—
115C2×M16—1371no—
116C2×M16—1371no—
117C2×M16#1058773911yescharacteristic
118C22×C8—1371no—
119C22×C8—1371no—
120#108—2742no—
121#108—2742no—
122C22⋊C8—2742no—
123C22⋊C8—2742no—
124C24—1371no—
125C22×C4—2741no—
126C22×C4—2741no—
127C22×C4—2741no—
128C22×C4—2741no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4C2×#1058773911yescharacteristic
135M16—22no—
136M16—2742no—
137M16—2742no—
138M16—2742no—
139M16—2742no—
140M16—2742no—
141M16—2742no—
142M16—2742no—
143C2×C8—2741no—
144C2×C8—2741no—
145C2×C8—2742no—
146C2×C8—2742no—
147C2×C8—1371no—
148C2×C8—1371no—
149C2×C8—1372no—
150C2×C8—1372no—
151C2×C8C2×#1058773911yescharacteristic
152C2×C8C2×#1058773911yescharacteristic
153C23—2741no—
154C23—2741no—
155C23—5482no—
156C23—1371no—
157C23—1371no—
158C23C4⋊#1058773911yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—2741no—
164C2×C4—2741no—
165C2×C4—2741no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—2741no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4C4⋊#1058773911yescharacteristic
176C2×C4C22×#1058773911yescharacteristic
177C8—21no—
178C8—2741no—
179C8—5482no—
180C8—5482no—
181C8—1372no—
182C8C22⋊#1058773912yes—
183C22—21no—
184C22—22no—
185C22—2742no—
186C22—2741no—
187C22—2742no—
188C22—5482no—
189C22—5482no—
190C22—1371no—
191C22—1371no—
192C22C2×C4⋊#1058773911yescharacteristic
193C4—21no—
194C4—2741no—
195C4—1371no—
196C4—1371no—
197C4C2×C22⋊#1058773911yescharacteristic
198C4#1798416711yescharacteristic
199C2—41no—
200C2—2741no—
201C2—2741no—
202C2—5481no—
203C2#2257834511yescharacteristic
204C2#2257841812yes—
subgroupG/H
centerC22C2×C4⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×M16×C137C4
socleC22×C137C2×C4⋊C4
prime radicalC23×D137Q8
cosocleC2×C4×D137C23
hypercenterC2×M16#10587739
hypocenterC137#1811
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#17984159C2semidirect
2. C2×#17984158C2semidirect
4. #22578279C2semidirect
102. C137#1811semidirect

Maximal subgroups

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

subgroupindex
C2×#179841592
C2×#179841582
#225783742
#225782792
C2×M16×D1372
#1811137