Finite Group Databasebeta

← #31796161

Subgroups — #31796161

Order 17536.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578421C211yesmaximal, characteristic, complemented
2#22578339C211yesmaximal, characteristic, complemented
3#22578257C212yesmaximal, complemented
4#22578254C212yesmaximal
5#22577932C211yesmaximal, characteristic
6C2×C4×#10587739—21no—
7#17984155—21no—
8#17984145C2212yes—
9C4×#11764047—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—24no—
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—24no—
50C2×#10587739—42no—
51C2×#10587739—42no—
52#11764047—42no—
53C22×D137C2×D411yescharacteristic
54C22⋊D137—42no—
55C2×Dic137—21no—
56C2×Dic137—21no—
57C2×Dic137—42no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—22no—
62C4×D137—22no—
63C4×D137—41no—
64C23×C137—21no—
65Q8×C137—41no—
66D4×C137—41no—
67C2×C4×C137—21no—
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22×C411yescharacteristic
71C22×C137—41no—
72C22×C137—41no—
73C22×C137—41no—
74C22×C137C2×C22⋊C411yescharacteristic
75C2×D137—21no—
76C2×D137—21no—
77C2×D137#13512yes—
78#10587739—44no—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—41no—
83Dic137—21no—
84Dic137—21no—
85Dic137—41no—
86Dic137—41no—
87Dic137#10412yes—
88C2×C137—81no—
89C2×C137#41011yescharacteristic
90C2×C137#41611yescharacteristic
91C2×C137C2×#10411yescharacteristic
92D137—22no—
93C137#150011yescharacteristic, complemented, Sylow
94#1500—1371nomaximal, Sylow
95#529—1371no—
96#431—1371no—
97#376—1371no—
98#328—1372no—
99#327—1372no—
100#126—2741no—
101#125—1371no—
102#124#1058773911yescharacteristic, complemented
103#123—2741no—
104C22⋊Q8—1371no—
105#121—1371no—
106C4×Q8—1371no—
107C4×D4—2741no—
108C4×D4—1371no—
109#117—2741no—
110C2×C42—2741no—
111C2×C42—1371no—
112C4⋊C8—2741no—
113C22⋊C8—1372no—
114C4×C8—2741no—
115#95—2741no—
116#95—1372no—
117C2×Q8C2×#1058773911yescharacteristic
118C2×D4—2741no—
119C2×D4C2×#1058773911yescharacteristic
120C22×C4—2742no—
121C22×C4—2741no—
122C22×C4—1371no—
123C22×C4—1371no—
124C22×C4—1371no—
125C2×C8—2742no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—1371no—
131C4⋊C4—1371no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—2741no—
135C22⋊C4—2741no—
136C22⋊C4—2741no—
137C42—2741no—
138C42—2744no—
139C42—1371no—
140C42—1371no—
141C42—1371no—
142C42C2×#1058773911yescharacteristic
143C23—21no—
144C23—1371no—
145Q8—41no—
146D4—41no—
147D4—5482no—
148C2×C4—21no—
149C2×C4—2742no—
150C2×C4—2742no—
151C2×C4—2744no—
152C2×C4—2741no—
153C2×C4—2741no—
154C2×C4—5481no—
155C2×C4—5482no—
156C2×C4—5482no—
157C2×C4—5482no—
158C2×C4—1371no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4C22⋊#1058773911yescharacteristic
162C2×C4C22⋊#1058773911yescharacteristic
163C2×C4C22×#1058773911yescharacteristic
164C8—5482no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—2741no—
169C22—2741no—
170C22—1372no—
171C22C2×C22⋊#1058773911yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—2741no—
177C4—2741no—
178C4—5484no—
179C4—5481no—
180C4—5481no—
181C4—1372no—
182C2—81no—
183C2—2742no—
184C2#2257838311yescharacteristic
185C2#2257839311yescharacteristic
186C2D137⋊#10411yescharacteristic
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#1500
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. #22578421C2semidirect
2. #22578339C2semidirect
3. #22578257C2semidirect
13. #17984080C4semidirect
17. Q8×Dic137C4semidirect
18. D4×Dic137C4semidirect
22. #124×C137C4semidirect
93. C137#1500semidirect
102. #124#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225783392
#225782572
#225782542
#225779322
#1500137