Finite Group Databasebeta

← #31796199

Subgroups — #31796199

Order 17536.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578375C211yesmaximal, characteristic
2#22578346C211yesmaximal, characteristic
3#22578257C214yesmaximal, complemented
4C2×#17984064C211yesmaximal, characteristic
5C2×#14555519—21no—
6C2×C22⋊#10587739—21no—
7#17984145—21no—
8#17984145C2212yes—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11C22×C4×D137C2211yescharacteristic
12#17984064—21no—
13#17984064—21no—
14#17984064C412yescomplemented
15#17984064C2212yes—
16C2×C4⋊Dic137C411yescharacteristic, complemented
17C2×#14555489—21no—
18C2×#14555487—21no—
19C22×Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555519—42no—
22C2×#11764047—22no—
23C22⋊#10587739—42no—
24C23×D137D411yescharacteristic
25C22×Dic137—21no—
26C22×Dic137D411yescharacteristic
27C2×C4×D137—42no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137C2311yescharacteristic
32C4⋊Dic137—21no—
33C4⋊Dic137C2×C412yes—
34#14555489—22no—
35#14555489—22no—
36#14555487—22no—
37#14555487—22no—
38C2×Q8×C137—21no—
39C2×Q8×C137—42no—
40C2×Q8×C137C2×C412yes—
41C22×C4×C137—21no—
42C22×C4×C137C2×C411yescharacteristic
43C2×#10587739—42no—
44C2×#10587739—42no—
45C2×#10587739—42no—
46#11764047—42no—
47C22×D137—22no—
48C22×D137—22no—
49C22×D137—41no—
50C22×D137C22⋊C411yescharacteristic
51C22×D137C2×D411yescharacteristic
52C2×Dic137—22no—
53C2×Dic137—22no—
54C2×Dic137—42no—
55C2×Dic137C22⋊C411yescharacteristic
56C2×Dic137C2×D411yescharacteristic
57C4×D137—41no—
58C4×D137—41no—
59C4×D137—41no—
60C4×D137—41no—
61C23×C137C22⋊C411yescharacteristic
62Q8×C137—42no—
63Q8×C137—42no—
64C2×C4×C137—22no—
65C2×C4×C137—22no—
66C2×C4×C137—42no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22×C411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137#9912yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C2×D137—21no—
74C2×D137—21no—
75C2×D137—21no—
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—44no—
79C2×D137#13712yes—
80#10587739—82no—
81C4×C137—21no—
82C4×C137—21no—
83C4×C137—42no—
84Dic137—21no—
85Dic137—21no—
86Dic137—42no—
87C2×C137—22no—
88C2×C137C2×#9911yescharacteristic
89C2×C137#41911yescharacteristic
90C2×C137#42111yescharacteristic
91D137—22no—
92D137—41no—
93C137#151111yescharacteristic, complemented, Sylow
94#1511—1371nomaximal, Sylow
95C2×C22⋊Q8—1371no—
96#407—1371no—
97#381—1371no—
98#328—1374no—
99C22×Q8#1058773911yescharacteristic, complemented
100C23×C4—1371no—
101C2×M16—2741no—
102C22⋊Q8—2741no—
103C22⋊Q8—2741no—
104C22⋊Q8—1372no—
105C22⋊Q8—1372no—
106C2×C4⋊C4—2741no—
107C2×C4⋊C4—1371no—
108C2×C22⋊C4—2741no—
109C2×C22⋊C4—2741no—
110C22⋊C8—2741no—
111C22⋊C8—1372no—
112#95—2741no—
113#95—1372no—
114C24—1371no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×#1058773912yes—
118C22×C4—21no—
119C22×C4—2741no—
120C22×C4—2742no—
121C22×C4—5482no—
122C22×C4—1371no—
123C22×C4—1371no—
124C22×C4—1371no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4C2×#1058773911yescharacteristic
128M16—5482no—
129C2×C8—2742no—
130C4⋊C4—2742no—
131C4⋊C4—2742no—
132C4⋊C4—2741no—
133C4⋊C4—1372no—
134C22⋊C4—2742no—
135C22⋊C4—2742no—
136C22⋊C4—5482no—
137C23—2742no—
138C23—2742no—
139C23—5481no—
140C23—1371no—
141C23—1371no—
142C23C22⋊#1058773911yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—2742no—
149C2×C4—2742no—
150C2×C4—5481no—
151C2×C4—5481no—
152C2×C4—5481no—
153C2×C4—5481no—
154C2×C4—5482no—
155C2×C4—5482no—
156C2×C4—5482no—
157C2×C4—5482no—
158C2×C4—1371no—
159C2×C4—1371no—
160C2×C4C22⋊#1058773911yescharacteristic
161C2×C4C22×#1058773911yescharacteristic
162C8—5482no—
163C22—21no—
164C22—21no—
165C22—2741no—
166C22—2741no—
167C22—2741no—
168C22—2741no—
169C22—2741no—
170C22—5484no—
171C22—1372no—
172C22#1798414612yes—
173C22C2×C22⋊#1058773911yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—2741no—
178C4—2741no—
179C4—5482no—
180C4—10962no—
181C2—22no—
182C2—2742no—
183C2—5481no—
184C2C2×#1798414611yescharacteristic
185C2#2257839611yescharacteristic
186C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22×Q8#10587739
hypocenterC137#1511
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. #22578257C2semidirect
14. #17984064C4semidirect
16. C2×C4⋊Dic137C4semidirect
19. C22×Q8×C137C4semidirect
93. C137#1511semidirect
99. C22×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783752
#225783462
#225782572
C2×#179840642
#1511137