Finite Group Databasebeta

← #31796955

Subgroups — #31796955

Order 17536.

324 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 88 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#22578404C211yesmaximal, characteristic
2C2×#17984151C214yesmaximal
3C4×C4⋊#10587739C211yesmaximal, characteristic
4C4⋊Q8×D137C211yesmaximal, characteristic
5C2×#14555518C2212yes—
6C2×C4⋊#10587739C2212yes—
7C2×C4×#10587739—21no—
8#17984155—21no—
9C4⋊#11764047—21no—
10#17984151—28no—
11(C2×#10587739)⋊C4—21no—
12C2×Q8×D137C2212yes—
13#17984088C411yescharacteristic, complemented
14#17984087—21no—
15#17984087—21no—
16#17984086C412yescomplemented
17C4⋊C4×D137—21no—
18C4⋊C4×D137—21no—
19C42×D137C2211yescharacteristic
20C4⋊Q8×C137C411yescharacteristic, complemented
21C22×#10587739—22no—
22#14555518—24no—
23C2×#11764047—22no—
24C4⋊#10587739—22no—
25C4⋊#10587739D414yes—
26C4×#10587739—42no—
27C2×Dic274—22no—
28C2×Dic274C2×C412yes—
29Q8×D137—24no—
30Q8×D137—24no—
31C2×C4×D137—22no—
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137C2311yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37#14555489—22no—
38#14555489—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—42no—
50C22×D137C2×D411yescharacteristic
51Dic274—24no—
52Dic274—42no—
53Dic274—42no—
54C2×Dic137—22no—
55C2×Dic137C22⋊C411yescharacteristic
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C2×D411yescharacteristic
58C4×D137—22no—
59C4×D137—44no—
60C4×D137SD1612yes—
61C4×D137Q1612yes—
62C4×D137C2×D412yes—
63C4×D137#4112yes—
64Q8×C137—24no—
65C2×C4×C137—22no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22×C411yescharacteristic
69C22×C137C2×C22⋊C411yescharacteristic
70C2×D137#12112yes—
71C2×D137C2×SD1611yescharacteristic
72C2×D137C2×Q1611yescharacteristic
73C2×D137#13712yes—
74#10587739—44no—
75C4×C137—21no—
76C4×C137—42no—
77C4×C137#10312yescomplemented
78C4×C137C4×D412yes—
79Dic137—21no—
80Dic137—42no—
81Dic137#10312yescomplemented
82Dic137#12411yescharacteristic
83Dic137#12711yescharacteristic
84C2×C137#39011yescharacteristic
85C2×C137C2×#10311yescharacteristic
86C2×C137#41911yescharacteristic
87D137#46112yes—
88D137#46212yes—
89C137#198111yescharacteristic, complemented, Sylow
90#1981—1371nomaximal, Sylow
91C2×C4⋊Q8—1371no—
92C2×C4⋊C8—1371no—
93C2×#103—1374no—
94C4×C4⋊C4—1371no—
95C22×Q8—1372no—
96C22×C8—1372no—
97C4⋊Q8—2741no—
98C4⋊Q8—2741no—
99C4⋊Q8—1372no—
100C4⋊Q8—1371no—
101C4⋊Q8#1058773911yescharacteristic, complemented
102C2×C4⋊C4—2741no—
103C2×C4⋊C4—2741no—
104C2×C4⋊C4—1372no—
105C2×C42—2741no—
106C2×C42—1371no—
107C4⋊C8—2741no—
108C4⋊C8—2741no—
109#103—2748no—
110#95—2741no—
111C2×Q8—2742no—
112C2×Q8—2744no—
113C2×Q8—2744no—
114C2×Q8—1372no—
115C2×Q8C2×#1058773912yes—
116C22×C4—2742no—
117C22×C4—2742no—
118C22×C4—1371no—
119C22×C4—1371no—
120C22×C4—1371no—
121C2×C8—2744no—
122C2×C8—2742no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—2742no—
126C4⋊C4—2742no—
127C4⋊C4—2742no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—1374no—
131C42—5482no—
132C42—1371no—
133C42—1371no—
134C42—1371no—
135C42C2×#1058773911yescharacteristic
136C23—1371no—
137Q8—24no—
138Q8—2744no—
139Q8—5482no—
140Q8—5482no—
141C2×C4—22no—
142C2×C4—2742no—
143C2×C4—2744no—
144C2×C4—2742no—
145C2×C4—5484no—
146C2×C4—5482no—
147C2×C4—5482no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4—1372no—
152C2×C4—1372no—
153C2×C4—1372no—
154C2×C4—1372no—
155C2×C4C22⋊#1058773911yescharacteristic
156C2×C4C22⋊#1058773911yescharacteristic
157C2×C4C22×#1058773911yescharacteristic
158C8—5482no—
159C22—1372no—
160C22—1371no—
161C22—1371no—
162C22—1372no—
163C22C2×C22⋊#1058773911yescharacteristic
164C4—21no—
165C4—42no—
166C4—2741no—
167C4—5484no—
168C4—5482no—
169C4—1371no—
170C4—1371no—
171C4—1372no—
172C4#1798415112yescomplemented
173C4D4×#1058773912yes—
174C2—1372no—
175C2—1372no—
176C2#2257835511yescharacteristic
177C2C2×#1798415111yescharacteristic
178C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1981
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
13. #17984088C4semidirect
16. #17984086C4semidirect
20. C4⋊Q8×C137C4semidirect
77. C4×C137#103semidirect
81. Dic137#103semidirect
89. C137#1981semidirect
101. C4⋊Q8#10587739semidirect
172. C4#17984151semidirect

Maximal subgroups

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

subgroupindex
#225784042
C2×#179841512
C4×C4⋊#105877392
C4⋊Q8×D1372
#1981137