Finite Group Databasebeta

← #31796169

Subgroups — #31796169

Order 17536.

284 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578422C211yesmaximal, characteristic, complemented
2C4×C4⋊#10587739C211yesmaximal, characteristic, complemented
3#22578254C214yesmaximal, complemented
4#22577951C211yesmaximal, characteristic
5C2×C4⋊#10587739—21no—
6C2×C4×#10587739—21no—
7C4⋊#11764047—21no—
8#17984145C2212yes—
9C4×#11764047—21no—
10(C2×#10587739)⋊C4C2212yes—
11C2×#14555509—21no—
12#17984088C411yescharacteristic, complemented
13#17984075—21no—
14#17984059—21no—
15#17984059C2212yes—
16D4×Dic137—21no—
17D4×Dic137C412yescomplemented
18C42×D137C2211yescharacteristic
19#127×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4⋊#10587739—24no—
23C4×#10587739—42no—
24#14555509—42no—
25#14555509—42no—
26C2×Dic274—21no—
27C2×C22⋊D137—22no—
28C22×Dic137—22no—
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137C2311yescharacteristic
32C4⋊Dic137—21no—
33C4⋊Dic137C2×C412yes—
34C22⋊Dic137—22no—
35C22⋊Dic137—22no—
36C4×Dic137C2×C411yescharacteristic
37C4×Dic137D411yescharacteristic
38C4×Dic137D411yescharacteristic
39C2×D4×C137—21no—
40C2×D4×C137—21no—
41C2×D4×C137C2×C412yes—
42C42×C137C2×C411yescharacteristic
43C2×#10587739—24no—
44C2×#10587739—42no—
45C2×#10587739—42no—
46#11764047—42no—
47C22×D137C2×D411yescharacteristic
48Dic274—42no—
49C22⋊D137—42no—
50C22⋊D137—42no—
51C2×Dic137—22no—
52C2×Dic137—44no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C4×D137—22no—
57C4×D137—41no—
58C4×D137D812yes—
59C4×D137SD1612yes—
60C23×C137—22no—
61D4×C137—42no—
62D4×C137—42no—
63D4×C137—42no—
64C2×C4×C137C22⋊C411yescharacteristic
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22×C411yescharacteristic
67C22×C137—42no—
68C22×C137—42no—
69C22×C137—42no—
70C22×C137C2×C22⋊C411yescharacteristic
71C2×D137—21no—
72C2×D137—21no—
73C2×D137C2×D811yescharacteristic
74C2×D137C2×SD1611yescharacteristic
75#10587739—44no—
76C4×C137—21no—
77C4×C137#10212yes—
78C4×C137#10412yes—
79Dic137—21no—
80Dic137—21no—
81Dic137—42no—
82Dic137#10212yes—
83C2×C137—82no—
84C2×C137#41011yescharacteristic
85C2×C137C2×#10211yescharacteristic
86C2×C137C2×#10411yescharacteristic
87D137—21no—
88D137—21no—
89C137#150211yescharacteristic, complemented, Sylow
90#1502—1371nomaximal, Sylow
91#532—1371no—
92#431—1371no—
93C4×C4⋊C4—1371no—
94#327—1374no—
95C2×#41—2741no—
96C4⋊Q8—1371no—
97#127#1058773911yescharacteristic, complemented
98#124—2741no—
99#121—2741no—
100#121—1372no—
101C4×D4—2741no—
102C4×D4—1372no—
103C2×C4⋊C4—2741no—
104C2×C42—2741no—
105C2×C42—1371no—
106C4⋊C8—2741no—
107C22⋊C8—1372no—
108C4×C8—2741no—
109#95—1372no—
110#41—5482no—
111#41—5482no—
112C2×Q8—2741no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—2742no—
116C2×D4C2×#1058773912yes—
117C22×C4—2742no—
118C22×C4—2742no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4—1371no—
122C2×C8—2742no—
123C4⋊C4—2744no—
124C4⋊C4—2741no—
125C4⋊C4—1372no—
126C22⋊C4—2742no—
127C22⋊C4—2742no—
128C42—5482no—
129C42—1371no—
130C42—1371no—
131C42—1371no—
132C42C2×#1058773911yescharacteristic
133C23—22no—
134C23—1371no—
135Q8—5482no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—5482no—
140D4—5482no—
141C2×C4—2742no—
142C2×C4—2744no—
143C2×C4—2742no—
144C2×C4—5481no—
145C2×C4—5482no—
146C2×C4—5482no—
147C2×C4—5484no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4—1372no—
152C2×C4—1372no—
153C2×C4C22⋊#1058773911yescharacteristic
154C2×C4C22⋊#1058773911yescharacteristic
155C2×C4C22×#1058773911yescharacteristic
156C8—5482no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—2741no—
161C22—2741no—
162C22—1371no—
163C22—1371no—
164C22C2×C22⋊#1058773911yescharacteristic
165C4—21no—
166C4—2741no—
167C4—2741no—
168C4—5484no—
169C4—5482no—
170C4—1372no—
171C4#1798415012yes—
172C4#1798415212yes—
173C2—82no—
174C2—2741no—
175C2—2741no—
176C2#2257838311yescharacteristic
177C2C2×#1798415011yescharacteristic
178C2C2×#1798415211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1502
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. C4×C4⋊#10587739C2semidirect
3. #22578254C2semidirect
12. #17984088C4semidirect
17. D4×Dic137C4semidirect
19. #127×C137C4semidirect
89. C137#1502semidirect
97. #127#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784222
C4×C4⋊#105877392
#225782542
#225779512
#1502137