Finite Group Databasebeta

← D137⋊#354 #31797058

Subgroups — D137⋊#354 #31797058

Order 17536.

284 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D137⋊#99C212yesmaximal
2#22578310C212yesmaximal, complemented
3#22578309C212yesmaximal, complemented
4#124×D137C211yesmaximal, characteristic
5C2×C22⋊#10587739—22no—
6#17984147C2214yes—
7#17984146—22no—
8C2×Q8×D137C411yescharacteristic, complemented
9C2×D4×D137C2211yescharacteristic
10#17984077C2211yescharacteristic
11#17984076C2211yescharacteristic
12#17984075C411yescharacteristic, complemented
13#17984074—22no—
14C22⋊C4×D137—22no—
15C42×D137C411yescharacteristic, complemented
16#124×C137C411yescharacteristic, complemented
17C22×#10587739—42no—
18C22⋊#10587739—24no—
19C22⋊#10587739—42no—
20C23×D137D412yes—
21C2×Dic274—21no—
22C2×Dic274C2×C411yescharacteristic, complemented
23Q8×D137—41no—
24Q8×D137—41no—
25C2×D548C2311yescharacteristic
26C2×C22⋊D137D412yes—
27D4×D137—41no—
28D4×D137—41no—
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137C2×C411yescharacteristic
32C22⋊Dic137—22no—
33#14555487—22no—
34#14555487—22no—
35C4×Dic137—21no—
36C4×Dic137C2×C411yescharacteristic, complemented
37C2×Q8×C137C2×C411yescharacteristic, complemented
38C2×D4×C137C2×C411yescharacteristic
39C22⋊C4×C137—22no—
40C42×C137C2×C411yescharacteristic, complemented
41C2×#10587739—44no—
42C2×#10587739—84no—
43C22×D137—22no—
44C22×D137—22no—
45C22×D137—44no—
46C22×D137C22⋊C411yescharacteristic
47C22×D137C2×D412yes—
48Dic274—42no—
49Dic274—41no—
50D548—41no—
51C22⋊D137—42no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137C22⋊C411yescharacteristic
55C4×D137—21no—
56C4×D137—21no—
57C4×D137—42no—
58C4×D137—41no—
59C4×D137—41no—
60C23×C137C22⋊C412yes—
61Q8×C137—41no—
62D4×C137—41no—
63C2×C4×C137—21no—
64C2×C4×C137—21no—
65C2×C4×C137C22×C411yescharacteristic
66C22×C137—22no—
67C22×C137—42no—
68C22×C137C2×C22⋊C411yescharacteristic
69C2×D137—22no—
70C2×D137—22no—
71C2×D137—44no—
72C2×D137—42no—
73C2×D137—82no—
74C2×D137#9911yescharacteristic
75C2×D137#9911yescharacteristic
76#10587739—84no—
77C4×C137—21no—
78C4×C137—41no—
79C4×C137—41no—
80Dic137—21no—
81Dic137—41no—
82Dic137—41no—
83C2×C137—21no—
84C2×C137—42no—
85C2×C137C2×#9911yescharacteristic
86D137—21no—
87D137—42no—
88D137#35412yescomplemented
89C137C2×#35411yescharacteristic, complemented, Sylow
90C2×#354—1371nomaximal, Sylow
91C2×#124—1371no—
92C2×#99—1372no—
93#354—1372no—
94#354—1372no—
95C22×Q8—1371no—
96C22×D4—1371no—
97#124—2742no—
98#124—1371no—
99#124—1371no—
100#124—1371no—
101#124#1058773911yescharacteristic, complemented
102C2×C22⋊C4—2742no—
103C2×C22⋊C4—2742no—
104C2×C42—1371no—
105#99—2742no—
106#99—1374no—
107C24—1372no—
108C2×Q8—2741no—
109C2×Q8—5481no—
110C2×Q8—5481no—
111C2×Q8—1371no—
112C2×Q8C2×#1058773911yescharacteristic, complemented
113C2×D4—5481no—
114C2×D4—5481no—
115C2×D4—1372no—
116C2×D4—1371no—
117C2×D4C2×#1058773911yescharacteristic
118C22×C4—2741no—
119C22×C4—2741no—
120C22×C4—5482no—
121C22×C4—1371no—
122C22⋊C4—22no—
123C22⋊C4—2742no—
124C22⋊C4—2742no—
125C22⋊C4—2742no—
126C22⋊C4—2744no—
127C22⋊C4—5482no—
128C42—2741no—
129C42—1371no—
130C42C2×#1058773911yescharacteristic, complemented
131C23—2742no—
132C23—2742no—
133C23—5484no—
134C23—1371no—
135C23—1372no—
136C23C22⋊#1058773912yes—
137Q8—41no—
138Q8—5481no—
139Q8—5482no—
140D4—41no—
141D4—5481no—
142D4—5482no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—2741no—
146C2×C4—2741no—
147C2×C4—2741no—
148C2×C4—2741no—
149C2×C4—5482no—
150C2×C4—5481no—
151C2×C4—5481no—
152C2×C4—5484no—
153C2×C4—10964no—
154C2×C4—1371no—
155C2×C4C22×#1058773911yescharacteristic
156C22—22no—
157C22—42no—
158C22—2742no—
159C22—2742no—
160C22—5482no—
161C22—5484no—
162C22—10962no—
163C22—1371no—
164C22—1371no—
165C22C2×C22⋊#1058773911yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—2741no—
170C4—5481no—
171C4—5481no—
172C4—10964no—
173C2—21no—
174C2—42no—
175C2—2741no—
176C2—5482no—
177C2—1372no—
178C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137C2×#99
prime radicalC2×D4×D137C22
cosocleC2×D548C23
hypercenter#124#10587739
hypocenterC137C2×#354
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #22578310C2semidirect
3. #22578309C2semidirect
8. C2×Q8×D137C4semidirect
12. #17984075C4semidirect
15. C42×D137C4semidirect
16. #124×C137C4semidirect
22. C2×Dic274C2×C4semidirect
36. C4×Dic137C2×C4semidirect
37. C2×Q8×C137C2×C4semidirect
40. C42×C137C2×C4semidirect
88. D137#354semidirect
89. C137C2×#354semidirect
101. #124#10587739semidirect
112. C2×Q8C2×#10587739semidirect
130. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#992
#225783102
#225783092
#124×D1372
C2×#354137