Finite Group Databasebeta

← #31796795

Subgroups — #31796795

Order 17536.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#22578471C211yesmaximal, characteristic
2#22578396C212yesmaximal
3D137⋊#101C211yesmaximal, characteristic
4#22578274C212yesmaximal
5C4⋊Q8×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7Q8×#10587739—24no—
8#17984167—22no—
9#17984167C2212yes—
10#17984151—24no—
11#17984149—22no—
12C2×Q8×D137C2211yescharacteristic
13C2×Q8×D137C2211yescharacteristic
14#17984088C411yescharacteristic, complemented
15#17984087—22no—
16#17984086C411yescharacteristic, complemented
17#17984086C411yescharacteristic, complemented
18C4⋊C4×D137—22no—
19C42×D137C2211yescharacteristic
20C4⋊Q8×C137C411yescharacteristic, complemented
21#14555520—22no—
22#14555519—22no—
23#14555518—22no—
24C4⋊#10587739—24no—
25C4⋊#10587739D414yes—
26C22⋊#10587739—22no—
27C4×#10587739—24no—
28C4×#10587739—22no—
29C2×Dic274—21no—
30C2×Dic274C2×C411yescharacteristic
31C2×Dic274C2×C411yescharacteristic
32C2×Dic274D412yes—
33Q8×D137—22no—
34Q8×D137—22no—
35Q8×D137—22no—
36Q8×D137—22no—
37C2×C4×D137—21no—
38C2×C4×D137—21no—
39C2×C4×D137D412yes—
40C2×C4×D137C2311yescharacteristic
41C4⋊Dic137—22no—
42#14555489—22no—
43#14555489—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C2×Q8×C137C2×C411yescharacteristic
48C4⋊C4×C137—22no—
49C42×C137C2×C411yescharacteristic
50C2×#10587739—24no—
51#11764047—42no—
52C22×D137C2×D411yescharacteristic
53Dic274—22no—
54Dic274—24no—
55Dic274—22no—
56Dic274—42no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C412yes—
60C2×Dic137C2×D411yescharacteristic
61C4×D137—24no—
62C4×D137—22no—
63C4×D137—22no—
64C4×D137—41no—
65C4×D137—41no—
66C4×D137C2×D412yes—
67C4×D137#4112yes—
68Q8×C137—22no—
69Q8×C137—22no—
70C2×C4×C137—21no—
71C2×C4×C137—21no—
72C2×C4×C137C22⋊C412yes—
73C2×C4×C137C22×C411yescharacteristic
74C22×C137C2×C22⋊C411yescharacteristic
75C2×D137—22no—
76C2×D137#12011yescharacteristic
77C2×D137#12311yescharacteristic
78#10587739—44no—
79C4×C137—22no—
80C4×C137—22no—
81C4×C137—41no—
82C4×C137C4×D412yes—
83Dic137—22no—
84Dic137—22no—
85Dic137—41no—
86Dic137#12112yes—
87C2×C137—21no—
88C2×C137#38611yescharacteristic
89D137—21no—
90D137#45612yes—
91C137#188611yescharacteristic, complemented, Sylow
92#1886—1371nomaximal, Sylow
93C2×C4⋊Q8—1371no—
94#519—1371no—
95#419—1372no—
96C2×#101—1371no—
97#339—1372no—
98C22×Q8—1371no—
99C22×Q8—1371no—
100C2×M16—1372no—
101C4⋊Q8—2742no—
102C4⋊Q8—1371no—
103C4⋊Q8—1371no—
104C4⋊Q8—1371no—
105C4⋊Q8#1058773911yescharacteristic, complemented
106C4×Q8—2744no—
107#117—2742no—
108#117—1372no—
109C2×C4⋊C4—2742no—
110C2×C42—1371no—
111#103—2744no—
112#101—2742no—
113C2×Q8—2741no—
114C2×Q8—2742no—
115C2×Q8—2742no—
116C2×Q8—2742no—
117C2×Q8—2742no—
118C2×Q8—1371no—
119C2×Q8—1372no—
120C2×Q8—1371no—
121C2×Q8C2×#1058773911yescharacteristic
122C2×Q8C2×#1058773911yescharacteristic
123C22×C4—2741no—
124C22×C4—2741no—
125C22×C4—1372no—
126C22×C4—1371no—
127M16—2742no—
128M16—2742no—
129C2×C8—2742no—
130C4⋊C4—22no—
131C4⋊C4—2742no—
132C4⋊C4—2742no—
133C4⋊C4—2744no—
134C4⋊C4—2742no—
135C4⋊C4—1374no—
136C22⋊C4—2742no—
137C42—2741no—
138C42—2744no—
139C42—2742no—
140C42—1371no—
141C42C2×#1058773911yescharacteristic
142C23—1371no—
143Q8—22no—
144Q8—22no—
145Q8—2744no—
146Q8—2742no—
147Q8—2742no—
148Q8—5482no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—2744no—
152C2×C4—2742no—
153C2×C4—2742no—
154C2×C4—2741no—
155C2×C4—2744no—
156C2×C4—2741no—
157C2×C4—5481no—
158C2×C4—5481no—
159C2×C4—1372no—
160C2×C4—1371no—
161C2×C4—1372no—
162C2×C4—1372no—
163C2×C4C22⋊#1058773912yes—
164C2×C4C22×#1058773911yescharacteristic
165C8—5482no—
166C22—2742no—
167C22—1371no—
168C22—1371no—
169C22C2×C22⋊#1058773911yescharacteristic
170C4—22no—
171C4—22no—
172C4—41no—
173C4—2742no—
174C4—2742no—
175C4—5484no—
176C4—5481no—
177C4—1372no—
178C4D4×#1058773912yes—
179C2—21no—
180C2—2741no—
181C2—1372no—
182C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137#386
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1886
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #17984088C4semidirect
16. #17984086C4semidirect
17. #17984086C4semidirect
20. C4⋊Q8×C137C4semidirect
91. C137#1886semidirect
105. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784712
#225783962
D137⋊#1012
#225782742
C4⋊Q8×D1372
#1886137