Finite Group Databasebeta

← #31796201

Subgroups — #31796201

Order 17536.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#22578425C211yesmaximal, characteristic, complemented
2#22578347C211yesmaximal, characteristic, complemented
3#22578257C212yesmaximal, complemented
4#22578254C212yesmaximal
5#22577932C211yesmaximal, characteristic
6C2×C4⋊#10587739—21no—
7C4⋊#11764047—21no—
8#17984145C2212yes—
9#17984143—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12#17984081—21no—
13#17984080C411yescharacteristic, complemented
14#17984070—21no—
15#17984064C2211yescharacteristic
16#17984059C2211yescharacteristic
17Q8×Dic137C411yescharacteristic, complemented
18D4×Dic137C411yescharacteristic, complemented
19#17984048—21no—
20#17984044—21no—
21C42×D137C2211yescharacteristic
22#124×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C4⋊#10587739—42no—
26C2×C22⋊D137—21no—
27C22×Dic137—21no—
28C2×C4×D137D411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137C2311yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C411yescharacteristic
33C4⋊Dic137C2×C411yescharacteristic
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C2×Q8×C137C2×C411yescharacteristic
45C2×D4×C137C2×C411yescharacteristic
46C22⋊C4×C137—21no—
47C22⋊C4×C137—21no—
48C42×C137C2×C411yescharacteristic
49C2×#10587739—42no—
50C2×#10587739—42no—
51C2×#10587739—42no—
52#11764047—42no—
53C22×D137C2×D411yescharacteristic
54C22⋊D137—41no—
55C22⋊D137—41no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—42no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C4×D137—21no—
63C4×D137—21no—
64C4×D137—41no—
65C4×D137—41no—
66C23×C137—21no—
67Q8×C137—41no—
68D4×C137—41no—
69C2×C4×C137—21no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22×C411yescharacteristic
73C22×C137—41no—
74C22×C137—41no—
75C22×C137—41no—
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137AGL(1,ℤ/8)11yescharacteristic
80C2×D137#13711yescharacteristic
81#10587739—82no—
82C4×C137—21no—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137—41no—
86Dic137—21no—
87Dic137—21no—
88Dic137—21no—
89Dic137—41no—
90Dic137—41no—
91C2×C137—81no—
92C2×C137#41011yescharacteristic
93C2×C137#41711yescharacteristic
94C2×C137#42111yescharacteristic
95D137—21no—
96D137—21no—
97C137#151211yescharacteristic, complemented, Sylow
98#1512—1371nomaximal, Sylow
99#529—1371no—
100#432—1371no—
101#382—1371no—
102#328—1372no—
103#327—1372no—
104#126—2741no—
105#125—1371no—
106#124#1058773911yescharacteristic, complemented
107#123—2741no—
108C22⋊Q8—1371no—
109#121—1371no—
110C4×Q8—1371no—
111C4×D4—2741no—
112C4×D4—1371no—
113#117—2741no—
114C2×C4⋊C4—2741no—
115C2×C42—1371no—
116C4⋊C8—2741no—
117C22⋊C8—1372no—
118#97—2741no—
119#95—2741no—
120#95—1372no—
121C2×Q8C2×#1058773911yescharacteristic
122C2×D4—2741no—
123C2×D4C2×#1058773911yescharacteristic
124C22×C4—2741no—
125C22×C4—2742no—
126C22×C4—1371no—
127C22×C4—1371no—
128C22×C4—1371no—
129C2×C8—2742no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—5482no—
135C4⋊C4—1371no—
136C4⋊C4—1371no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—2741no—
140C22⋊C4—2741no—
141C22⋊C4—2741no—
142C42—2741no—
143C42—1371no—
144C42—1371no—
145C42—1371no—
146C42C2×#1058773911yescharacteristic
147C23—21no—
148C23—1371no—
149Q8—41no—
150D4—41no—
151D4—5481no—
152D4—5481no—
153C2×C4—21no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—5481no—
159C2×C4—5481no—
160C2×C4—5482no—
161C2×C4—5482no—
162C2×C4—5482no—
163C2×C4—5482no—
164C2×C4—1371no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4C22⋊#1058773911yescharacteristic
168C2×C4C22⋊#1058773911yescharacteristic
169C2×C4C22×#1058773911yescharacteristic
170C8—5482no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—2741no—
175C22—2741no—
176C22—1371no—
177C22—1371no—
178C22C2×C22⋊#1058773911yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—2741no—
184C4—2741no—
185C4—2741no—
186C4—5481no—
187C4—5481no—
188C4—10962no—
189C2—81no—
190C2—2741no—
191C2—2741no—
192C2#2257838311yescharacteristic
193C2#2257839411yescharacteristic
194C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984059C22
cosocleC2×C4×D137C23
hypercenter#124#10587739
hypocenterC137#1512
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #22578425C2semidirect
2. #22578347C2semidirect
3. #22578257C2semidirect
13. #17984080C4semidirect
17. Q8×Dic137C4semidirect
18. D4×Dic137C4semidirect
22. #124×C137C4semidirect
97. C137#1512semidirect
106. #124#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225783472
#225782572
#225782542
#225779322
#1512137