Finite Group Databasebeta

← #31796307

Subgroups — #31796307

Order 17536.

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

normal subgroups only

This group has 216 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#22578406C211yesmaximal, characteristic
2C2×#17984155C211yesmaximal, characteristic
3#22578263C212yesmaximal, complemented
4#22578262C212yesmaximal, complemented
5#22577955C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8#17984155—21no—
9#17984155C2212yes—
10#17984155C2212yes—
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
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
27C2×Dic274—21no—
28C2×Dic274C2×C411yescharacteristic
29C22×Dic137D411yescharacteristic
30C22×Dic137D411yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C411yescharacteristic
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37#14555489—21no—
38C22⋊Dic137—21no—
39C4×Dic137—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×Q8×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50#11764047—42no—
51#11764047—42no—
52Dic274—41no—
53Dic274—41no—
54Dic274—41no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137SD1612yes—
62C2×Dic137Q1612yes—
63C2×Dic137C2×D411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C23×C137C22⋊C411yescharacteristic
66Q8×C137—41no—
67C2×C4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137—41no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22×C411yescharacteristic
72C22×C137—21no—
73C22×C137—21no—
74C22×C137#10312yes—
75C22×C137C2×C22⋊C411yescharacteristic
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—41no—
79C4×C137—41no—
80Dic137—21no—
81Dic137—41no—
82Dic137—41no—
83Dic137#10312yes—
84Dic137C2×SD1611yescharacteristic
85Dic137C2×Q1611yescharacteristic
86Dic137AGL(1,ℤ/8)11yescharacteristic
87Dic137#13711yescharacteristic
88C2×C137—22no—
89C2×C137#41311yescharacteristic
90C2×C137C2×#10311yescharacteristic
91C2×C137#41711yescharacteristic
92C137#155011yescharacteristic, complemented, Sylow
93#1550—1371nomaximal, Sylow
94#533—1371no—
95#423—1371no—
96C2×C4⋊C8—1371no—
97#333—1372no—
98#332—1372no—
99C2×M16—2741no—
100C22×C8—2741no—
101C4⋊Q8—1371no—
102C4⋊Q8—1371no—
103#125—2741no—
104C22⋊Q8—2741no—
105C22⋊Q8—1371no—
106C22⋊Q8#1058773911yescharacteristic
107C4×Q8—2741no—
108C4×Q8—1371no—
109C4×Q8—1371no—
110#117—2741no—
111C2×C42—1371no—
112C4⋊C8—2741no—
113C4⋊C8—2741no—
114C4⋊C8—1372no—
115C4⋊C8—1372no—
116C2×Q8—2741no—
117C2×Q8—1371no—
118C2×Q8C2×#1058773911yescharacteristic
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4C2×#1058773911yescharacteristic
122M16—5482no—
123C2×C8—2742no—
124C2×C8—2742no—
125C2×C8—5482no—
126C4⋊C4—21no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—1371no—
134C4⋊C4C2×#1058773911yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—2741no—
137C42—2741no—
138C42—2741no—
139C42—1371no—
140C42—1371no—
141C42—1371no—
142C42—1371no—
143C23C22⋊#1058773911yescharacteristic
144Q8—41no—
145Q8—5481no—
146Q8—5481no—
147Q8—5481no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—2741no—
152C2×C4—2741no—
153C2×C4—2741no—
154C2×C4—2741no—
155C2×C4—1371no—
156C2×C4—1372no—
157C2×C4—1372no—
158C2×C4—1371no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4C22⋊#1058773911yescharacteristic
162C2×C4C22×#1058773911yescharacteristic
163C8—5482no—
164C8—5482no—
165C22—21no—
166C22—21no—
167C22#1798415112yes—
168C22C2×C22⋊#1058773911yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—2741no—
174C4—5481no—
175C4—5481no—
176C4—1371no—
177C4—1372no—
178C4—1371no—
179C4—1371no—
180C4—1371no—
181C2—22no—
182C2#2257839011yescharacteristic
183C2C2×#1798415111yescharacteristic
184C2#2257839411yescharacteristic
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#1550
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
92. C137#1550semidirect

Maximal subgroups

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

subgroupindex
#225784062
C2×#179841552
#225782632
#225782622
#225779552
#1550137