Finite Group Databasebeta

← #31796291

Subgroups — #31796291

Order 17536.

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

normal subgroups only

This group has 212 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#22578420C211yesmaximal, characteristic
2C2×#17984155C211yesmaximal, characteristic
3#22578263C212yesmaximal, complemented
4#22578262C212yesmaximal, complemented
5#22577955C211yesmaximal, characteristic
6C22×#11764047—21no—
7#17984155—21no—
8#17984155C2212yes—
9#17984155C2212yes—
10C22⋊#11764047—21no—
11C4×#11764047—21no—
12#17984087C2211yescharacteristic
13#17984086C2211yescharacteristic
14#17984078—21no—
15#17984067C411yescharacteristic
16#17984065—21no—
17Q8×Dic137C411yescharacteristic
18#17984053—21no—
19#17984053C411yescharacteristic
20#17984044—21no—
21C2×C4×Dic137C2211yescharacteristic
22C22⋊Q8×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C2×Dic274—21no—
27C2×Dic274C2×C411yescharacteristic
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C411yescharacteristic
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37C22⋊Dic137—21no—
38C4×Dic137—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×Q8×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C4⋊C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49#11764047—42no—
50#11764047—42no—
51Dic274—42no—
52Dic274—41no—
53C2×Dic137—22no—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137SD1612yes—
59C2×Dic137Q1612yes—
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C23×C137C22⋊C411yescharacteristic
63Q8×C137—41no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137—41no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22×C411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137#10312yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C4×C137—21no—
74C4×C137—21no—
75C4×C137—41no—
76C4×C137—41no—
77Dic137—21no—
78Dic137—41no—
79Dic137—41no—
80Dic137#10312yes—
81Dic137C2×SD1611yescharacteristic
82Dic137C2×Q1611yescharacteristic
83Dic137#13512yes—
84C2×C137—22no—
85C2×C137#41311yescharacteristic
86C2×C137C2×#10311yescharacteristic
87C2×C137#41611yescharacteristic
88C137#154211yescharacteristic, complemented, Sylow
89#1542—1371nomaximal, Sylow
90#533—1371no—
91#431—1371no—
92C2×C4⋊C8—1371no—
93#333—1372no—
94#332—1372no—
95C22×C8—2741no—
96C4⋊Q8—1371no—
97C4⋊Q8—1371no—
98#125—2741no—
99C22⋊Q8—2741no—
100C22⋊Q8—1371no—
101C22⋊Q8#1058773911yescharacteristic
102C4×Q8—2741no—
103C4×Q8—1371no—
104C4×Q8—1371no—
105#117—2741no—
106C2×C42—1371no—
107C4⋊C8—2741no—
108C4⋊C8—1372no—
109C4⋊C8—1372no—
110C22⋊C8—2741no—
111C4×C8—2741no—
112C2×Q8—2741no—
113C2×Q8—1371no—
114C2×Q8C2×#1058773911yescharacteristic
115C22×C4—1371no—
116C22×C4—1371no—
117C22×C4C2×#1058773911yescharacteristic
118C2×C8—2742no—
119C2×C8—2742no—
120C2×C8—5482no—
121C4⋊C4—21no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—1371no—
129C4⋊C4C2×#1058773911yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—2741no—
132C42—2741no—
133C42—2741no—
134C42—1371no—
135C42—1371no—
136C42—1371no—
137C42—1371no—
138C23C22⋊#1058773911yescharacteristic
139Q8—41no—
140Q8—5481no—
141Q8—5482no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—2742no—
146C2×C4—2741no—
147C2×C4—2741no—
148C2×C4—1371no—
149C2×C4—1372no—
150C2×C4—1372no—
151C2×C4—1371no—
152C2×C4—1371no—
153C2×C4—1371no—
154C2×C4C22⋊#1058773911yescharacteristic
155C2×C4C22×#1058773911yescharacteristic
156C8—5482no—
157C8—5482no—
158C22—21no—
159C22—21no—
160C22#1798415112yes—
161C22C2×C22⋊#1058773911yescharacteristic
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—2741no—
167C4—5481no—
168C4—5481no—
169C4—1371no—
170C4—1372no—
171C4—1371no—
172C4—1372no—
173C2—22no—
174C2#2257839011yescharacteristic
175C2C2×#1798415111yescharacteristic
176C2#2257839311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×C137C22⋊C4
cosocleC4×Dic137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1542
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22578263C2semidirect
4. #22578262C2semidirect
88. C137#1542semidirect

Maximal subgroups

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

subgroupindex
#225784202
C2×#179841552
#225782632
#225782622
#225779552
#1542137