Finite Group Databasebeta

← #31796310

Subgroups — #31796310

Order 17536.

208 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578423C211yesmaximal, characteristic
2C2×#17984155C211yesmaximal, characteristic
3#22578262C212yesmaximal, complemented
4#22578262C212yesmaximal, complemented
5#22577956C211yesmaximal, characteristic
6C22×#11764047—21no—
7#17984155—21no—
8#17984155C2212yes—
9#17984155C2212yes—
10C22⋊#11764047—21no—
11#17984143—21no—
12#17984087C2211yescharacteristic
13#17984087C2211yescharacteristic
14#17984079C411yescharacteristic
15#17984079C411yescharacteristic
16#17984065—21no—
17#17984065—21no—
18C4×Dic274—21no—
19C4×Dic274—21no—
20#17984047C411yescharacteristic
21C2×C4×Dic137C2211yescharacteristic
22#117×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C2×Dic274—21no—
27C2×Dic274—21no—
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C411yescharacteristic
33C4⋊Dic137C2×C411yescharacteristic
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37#14555489—21no—
38C22⋊Dic137—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C22×C4×C137C2×C411yescharacteristic
45C4⋊C4×C137C2×C411yescharacteristic
46C4⋊C4×C137C2×C411yescharacteristic
47C22⋊C4×C137—21no—
48C42×C137—21no—
49#11764047—42no—
50#11764047—42no—
51Dic274—42no—
52Dic274—41no—
53Dic274—41no—
54C2×Dic137—22no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C23×C137C22⋊C411yescharacteristic
64C2×C4×C137—22no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22×C411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137#10112yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C4×C137—21no—
74C4×C137—21no—
75C4×C137—41no—
76C4×C137—41no—
77Dic137—21no—
78Dic137—21no—
79Dic137—41no—
80Dic137—41no—
81Dic137#13512yes—
82Dic137AGL(1,ℤ/8)11yescharacteristic
83Dic137#13711yescharacteristic
84C2×C137—22no—
85C2×C137C2×#10111yescharacteristic
86C2×C137#41611yescharacteristic
87C2×C137#41711yescharacteristic
88C137#155211yescharacteristic, complemented, Sylow
89#1552—1371nomaximal, Sylow
90#533—1371no—
91#432—1371no—
92C2×C4⋊C8—1371no—
93#332—1372no—
94#332—1372no—
95C22×C8—2741no—
96C4⋊Q8—1371no—
97C4⋊Q8—1371no—
98#125—1371no—
99#125—1371no—
100C22⋊Q8—2741no—
101C22⋊Q8—2741no—
102C4×Q8—2741no—
103C4×Q8—2741no—
104#117—1371no—
105#117#1058773911yescharacteristic
106C2×C42—1371no—
107C4⋊C8—2741no—
108C4⋊C8—1372no—
109C4⋊C8—1372no—
110C22⋊C8—2741no—
111#97—2741no—
112C2×Q8—2741no—
113C2×Q8—2741no—
114C22×C4—1371no—
115C22×C4—1371no—
116C22×C4C2×#1058773911yescharacteristic
117C2×C8—2742no—
118C2×C8—2742no—
119C2×C8—5482no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—1371no—
127C4⋊C4—1371no—
128C4⋊C4C2×#1058773911yescharacteristic
129C4⋊C4C2×#1058773911yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—2741no—
132C42—21no—
133C42—2741no—
134C42—1371no—
135C42—1371no—
136C42—1371no—
137C42—1371no—
138C23C22⋊#1058773911yescharacteristic
139Q8—5482no—
140Q8—5481no—
141Q8—5481no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—2741no—
146C2×C4—2742no—
147C2×C4—2741no—
148C2×C4—2741no—
149C2×C4—2741no—
150C2×C4—1371no—
151C2×C4—1371no—
152C2×C4—1371no—
153C2×C4—1371no—
154C2×C4C22⋊#1058773911yescharacteristic
155C2×C4C22×#1058773911yescharacteristic
156C8—5482no—
157C8—5482no—
158C22—21no—
159C22—21no—
160C22#1798414912yes—
161C22C2×C22⋊#1058773911yescharacteristic
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—2741no—
167C4—2741no—
168C4—5481no—
169C4—5481no—
170C4—1372no—
171C4—1371no—
172C4—1371no—
173C2—22no—
174C2C2×#1798414911yescharacteristic
175C2#2257839311yescharacteristic
176C2#2257839411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×C137C22⋊C4
cosocleC4×Dic137C23
hypercenter#117#10587739
hypocenterC137#1552
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #22578262C2semidirect
4. #22578262C2semidirect
88. C137#1552semidirect

Maximal subgroups

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

subgroupindex
#225784232
C2×#179841552
#225782622
#225779562
#1552137