Finite Group Databasebeta

← #31796952

Subgroups — #31796952

Order 17536.

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

normal subgroups only

This group has 336 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
1C2×#17984153C214yesmaximal, complemented
2C4×#14555520C211yesmaximal, characteristic, complemented
3C4×C4⋊#10587739C211yesmaximal, characteristic, complemented
4#22577954C211yesmaximal, characteristic
5C2×#14555520C2212yes—
6C2×C4⋊#10587739—21no—
7C2×C4×#10587739C2212yes—
8#17984153—28no—
9#17984143—21no—
10C4×#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12C2×#14555511C2212yes—
13#17984085C411yescharacteristic, complemented
14#17984077C412yescomplemented
15#17984063—21no—
16#17984063—21no—
17#17984050—21no—
18#17984050—21no—
19C42×D137C2211yescharacteristic
20C4⋊Q8×C137C411yescharacteristic, complemented
21C22×#10587739—22no—
22#14555520—22no—
23#14555520D414yescomplemented
24C2×#11764047—22no—
25C4⋊#10587739—24no—
26C4×#10587739—24no—
27#14555511—24no—
28#14555511—24no—
29C2×D548—22no—
30C2×D548—21no—
31C2×D548—21no—
32C2×D548C2×C412yes—
33C2×C4×D137—22no—
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137C2311yescharacteristic
37#14555487—22no—
38#14555487—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×Q8×C137C2×C412yes—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C42×C137C2×C411yescharacteristic
46C2×#10587739—24no—
47C2×#10587739—42no—
48C2×#10587739—42no—
49#11764047—24no—
50C22×D137—22no—
51C22×D137C2×D411yescharacteristic
52D548—24no—
53D548—42no—
54D548—42no—
55D548—42no—
56D548—42no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C4×D137—22no—
61C4×D137—41no—
62C4×D137—44no—
63C4×D137C2×D412yes—
64C4×D137#4112yes—
65Q8×C137—24no—
66C2×C4×C137—22no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22×C411yescharacteristic
70C22×C137C2×C22⋊C411yescharacteristic
71C2×D137—21no—
72C2×D137—21no—
73C2×D137—42no—
74C2×D137—42no—
75C2×D137—42no—
76C2×D137#12411yescharacteristic
77C2×D137#12711yescharacteristic
78#10587739—44no—
79C4×C137—21no—
80C4×C137—42no—
81C4×C137#10412yescomplemented
82C4×C137C4×D412yes—
83Dic137—21no—
84Dic137—21no—
85Dic137#12112yes—
86C2×C137#39011yescharacteristic
87C2×C137C2×#10411yescharacteristic
88C2×C137#42111yescharacteristic
89D137—21no—
90D137—21no—
91D137—82no—
92C137#198011yescharacteristic, complemented, Sylow
93#1980—1371nomaximal, Sylow
94#532—1371no—
95C2×#104—1374no—
96C4×M16—1371no—
97C4×C4⋊C4—1371no—
98C2×#41—1372no—
99C2×M16—1372no—
100C4⋊Q8#1058773911yescharacteristic, complemented
101#127—1371no—
102#124—1372no—
103#121—2741no—
104#121—2741no—
105C4×D4—2741no—
106C4×D4—2741no—
107C2×C4⋊C4—2741no—
108C2×C42—1372no—
109C2×C42—1371no—
110#104—2748no—
111#97—2741no—
112C4×C8—2741no—
113#95—2741no—
114#41—2744no—
115#41—2744no—
116C2×Q8C2×#1058773912yes—
117C2×D4—2742no—
118C2×D4—2741no—
119C2×D4—2741no—
120C2×D4—1372no—
121C22×C4—2742no—
122C22×C4—2742no—
123C22×C4—1371no—
124C22×C4—1371no—
125C22×C4—1371no—
126M16—2742no—
127M16—1374no—
128C2×C8—2742no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2744no—
132C22⋊C4—2742no—
133C22⋊C4—2742no—
134C42—2744no—
135C42—1371no—
136C42—1371no—
137C42—1371no—
138C42C2×#1058773911yescharacteristic
139C23—2742no—
140C23—1371no—
141Q8—24no—
142D4—2744no—
143D4—5482no—
144D4—5482no—
145D4—5482no—
146D4—5482no—
147C2×C4—22no—
148C2×C4—2742no—
149C2×C4—2744no—
150C2×C4—5481no—
151C2×C4—5484no—
152C2×C4—5482no—
153C2×C4—5482no—
154C2×C4—1371no—
155C2×C4—1371no—
156C2×C4—1371no—
157C2×C4—1372no—
158C2×C4—1372no—
159C2×C4C22⋊#1058773911yescharacteristic
160C2×C4C22⋊#1058773911yescharacteristic
161C2×C4C22×#1058773911yescharacteristic
162C8—2744no—
163C22—2741no—
164C22—2741no—
165C22—5482no—
166C22—5482no—
167C22—5482no—
168C22—1371no—
169C22—1371no—
170C22C2×C22⋊#1058773911yescharacteristic
171C4—21no—
172C4—42no—
173C4—2741no—
174C4—2741no—
175C4—5484no—
176C4—1372no—
177C4#1798415312yescomplemented
178C4D4×#1058773912yes—
179C2—2741no—
180C2—2741no—
181C2—10962no—
182C2#2257835511yescharacteristic
183C2C2×#1798415311yescharacteristic
184C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1980
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17984153C2semidirect
2. C4×#14555520C2semidirect
3. C4×C4⋊#10587739C2semidirect
13. #17984085C4semidirect
14. #17984077C4semidirect
20. C4⋊Q8×C137C4semidirect
23. #14555520D4semidirect
81. C4×C137#104semidirect
92. C137#1980semidirect
100. C4⋊Q8#10587739semidirect
177. C4#17984153semidirect

Maximal subgroups

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

subgroupindex
C2×#179841532
C4×#145555202
C4×C4⋊#105877392
#225779542
#1980137