Finite Group Databasebeta

← #31796089

Subgroups — #31796089

Order 17536.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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
2#22578421C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5C4×D4×D137C211yesmaximal, characteristic
6#17984155—21no—
7C4⋊#11764047C2212yescomplemented
8#17984145—21no—
9#17984145—21no—
10C4×#11764047—21no—
11C4×#11764047C2212yescomplemented
12C2×D4×D137C411yescharacteristic
13C22×C4×D137—21no—
14C4×D548C411yescharacteristic
15D4×Dic137C2211yescharacteristic
16#17984050C2211yescharacteristic
17C4×C22⋊D137—21no—
18#17984048—21no—
19C4⋊C4×D137C411yescharacteristic
20C22⋊C4×D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C23×D137—21no—
26C2×D548C2×C411yescharacteristic
27C2×C22⋊D137—21no—
28D4×D137—22no—
29D4×D137C814yescomplemented
30C22×Dic137—21no—
31C2×C4×D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—44no—
35C2×C4×D137C2×C411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C4⋊Dic137C2×C411yescharacteristic
39#14555489—21no—
40C22⋊Dic137—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×D4×C137C2×C411yescharacteristic
47C22×C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50C42×C137C2×C411yescharacteristic
51#11764047—42no—
52#11764047—42no—
53C22×D137—21no—
54C22×D137—24no—
55C22×D137—42no—
56C22×D137—42no—
57C22×D137C22⋊C411yescharacteristic
58D548—21no—
59D548C2×C812yes—
60C22⋊D137—24no—
61C2×Dic137—21no—
62C2×Dic137—42no—
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C4×D137—22no—
67C4×D137—41no—
68C4×D137—42no—
69C4×D137—42no—
70C4×D137—42no—
71C4×D137C2×C812yes—
72C4×D137M1612yes—
73C23×C137—21no—
74D4×C137—21no—
75D4×C137C2×C812yes—
76C2×C4×C137—21no—
77C2×C4×C137—42no—
78C2×C4×C137C22⋊C411yescharacteristic
79C2×C4×C137C22⋊C411yescharacteristic
80C2×C4×C137C22×C411yescharacteristic
81C22×C137—22no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137C2×C22⋊C411yescharacteristic
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—22no—
88C2×D137—48no—
89C2×D137—41no—
90C2×D137—41no—
91C2×D137C22⋊C812yes—
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—41no—
95C4×C137C22×C811yescharacteristic
96C4×C137C2×M1611yescharacteristic
97Dic137—21no—
98Dic137—41no—
99Dic137C22⋊C812yes—
100Dic137AGL(1,ℤ/8)12yes—
101C2×C137—42no—
102C2×C137C2×C22⋊C811yescharacteristic
103C2×C137#41811yescharacteristic
104C2×C137#42111yescharacteristic
105D137—22no—
106D137—42no—
107C137#147811yescharacteristic, complemented, Sylow
108#1478—1371nomaximal, Sylow
109C2×C4×D4—1371no—
110#431—1371no—
111#431—1371no—
112D4⋊C8—1372no—
113D4⋊C8—1372no—
114C22×D4—1371no—
115C23×C4—2741no—
116C4×D4—2741no—
117C4×D4—2741no—
118C4×D4—1371no—
119C4×D4—1371no—
120C4×D4—1371no—
121C4×D4#1058773911yescharacteristic
122C2×C4⋊C4—1371no—
123C2×C22⋊C4—2741no—
124C2×C42—1371no—
125C4⋊C8—2741no—
126C4⋊C8—1372no—
127C22⋊C8—2741no—
128C22⋊C8—2741no—
129C4×C8—2741no—
130C4×C8—1372no—
131C24—2741no—
132C2×D4—2742no—
133C2×D4—2741no—
134C2×D4—1374no—
135C2×D4—1371no—
136C2×D4C2×#1058773911yescharacteristic
137C22×C4—21no—
138C22×C4—2741no—
139C22×C4—2741no—
140C22×C4—2741no—
141C22×C4—2741no—
142C22×C4—5484no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4—1371no—
146C2×C8—2742no—
147C2×C8—2742no—
148C4⋊C4—2741no—
149C4⋊C4—1371no—
150C4⋊C4C2×#1058773911yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—2741no—
153C22⋊C4—2741no—
154C22⋊C4—2741no—
155C42—1371no—
156C42—1371no—
157C42—1371no—
158C42C2×#1058773911yescharacteristic
159C23—21no—
160C23—2744no—
161C23—2741no—
162C23—5482no—
163C23—5482no—
164C23—1371no—
165D4—21no—
166D4—2744no—
167D4—2741no—
168D4—1372no—
169D4#1455551812yes—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—2742no—
173C2×C4—2741no—
174C2×C4—5481no—
175C2×C4—5482no—
176C2×C4—5482no—
177C2×C4—5482no—
178C2×C4—5482no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1372no—
183C2×C4—1372no—
184C2×C4C22⋊#1058773911yescharacteristic
185C2×C4C22⋊#1058773911yescharacteristic
186C2×C4C22×#1058773911yescharacteristic
187C8—5482no—
188C8—5482no—
189C22—22no—
190C22—41no—
191C22—41no—
192C22—2741no—
193C22—2741no—
194C22—2742no—
195C22—5488no—
196C22—5481no—
197C22—5481no—
198C22—1372no—
199C22C2×C22⋊#1058773911yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—2741no—
204C4—5481no—
205C4—1372no—
206C4—1372no—
207C4C2×#1455551811yescharacteristic
208C4D137⋊M1611yescharacteristic
209C2—42no—
210C2—2742no—
211C2—5482no—
212C2#2257837111yescharacteristic
213C2#2257839511yescharacteristic
214C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC4×C137C22×C8
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C4
cosocleC4×Dic137C23
hypercenterC4×D4#10587739
hypocenterC137#1478
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. #22578421C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
7. C4⋊#11764047C22semidirect
11. C4×#11764047C22semidirect
29. D4×D137C8semidirect
107. C137#1478semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225784212
#225782512
D4⋊#117640472
C4×D4×D1372
#1478137