Finite Group Databasebeta

← #31796134

Subgroups — #31796134

Order 17536.

276 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 72 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#22578406C211yesmaximal, characteristic, complemented
2C2×C4×#11764047C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8#17984155—21no—
9C4⋊#11764047C2212yes—
10C4×#11764047—21no—
11C4×#11764047C2212yes—
12C2×#14555509—21no—
13#17984087C411yescharacteristic
14#17984084C411yescharacteristic
15#17984074—21no—
16#17984062C411yescharacteristic
17#17984060—21no—
18D4×Dic137C2211yescharacteristic
19#17984050C2211yescharacteristic
20#17984048—21no—
21C2×C4×Dic137C2211yescharacteristic
22#121×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—24no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27#14555509—42no—
28#14555509—42no—
29C2×Dic274—21no—
30C2×D548C2×C411yescharacteristic
31C2×C22⋊D137—21no—
32C2×C22⋊D137—21no—
33C2×C22⋊D137—21no—
34C22×Dic137—21no—
35C22×Dic137D411yescharacteristic
36C22×Dic137D411yescharacteristic
37C2×C4×D137—21no—
38C4⋊Dic137C2×C411yescharacteristic
39#14555489—21no—
40C22⋊Dic137—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137C2×C411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137D411yescharacteristic
46C4×Dic137C2311yescharacteristic
47C2×D4×C137—21no—
48C2×D4×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C4⋊C4×C137C2×C411yescharacteristic
51C22⋊C4×C137—21no—
52#11764047—24no—
53#11764047—42no—
54C22×D137—21no—
55Dic274—42no—
56D548—22no—
57C22⋊D137—42no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C2×Dic137—22no—
61C2×Dic137—21no—
62C2×Dic137—41no—
63C2×Dic137—41no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137D812yes—
67C2×Dic137SD1612yes—
68C2×Dic137C2×D411yescharacteristic
69C2×Dic137C2×D411yescharacteristic
70C4×D137—41no—
71C4×D137—41no—
72C23×C137—21no—
73C23×C137C22⋊C411yescharacteristic
74D4×C137—22no—
75D4×C137—42no—
76C2×C4×C137—22no—
77C2×C4×C137—21no—
78C2×C4×C137C22⋊C411yescharacteristic
79C2×C4×C137C22×C411yescharacteristic
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137—41no—
85C22×C137#10212yes—
86C22×C137C2×C22⋊C411yescharacteristic
87C2×D137—41no—
88C2×D137—41no—
89C2×D137—41no—
90C4×C137—21no—
91C4×C137—41no—
92C4×C137#13112yes—
93Dic137—21no—
94Dic137—41no—
95Dic137#10212yes—
96Dic137#10412yes—
97Dic137C2×D811yescharacteristic
98Dic137C2×SD1611yescharacteristic
99C2×C137—22no—
100C2×C137—81no—
101C2×C137#40811yescharacteristic
102C2×C137C2×#10211yescharacteristic
103C2×C137C2×#10411yescharacteristic
104D137—81no—
105C137#149211yescharacteristic, complemented, Sylow
106#1492—1371nomaximal, Sylow
107#532—1371no—
108#423—1371no—
109C2×C4×C8—1371no—
110D4⋊C8—1372no—
111D4⋊C8—1372no—
112C2×#41—2741no—
113C2×M16—2741no—
114C22×C8—2741no—
115C4⋊Q8—1371no—
116#127—1371no—
117#124—2741no—
118#121—2741no—
119#121—1371no—
120#121#1058773911yescharacteristic
121C4×D4—2741no—
122C4×D4—1371no—
123C4×D4—1371no—
124C2×C42—1371no—
125C4⋊C8—2741no—
126C4⋊C8—1372no—
127C4×C8—2741no—
128C4×C8—1372no—
129#41—5482no—
130#41—5482no—
131C2×Q8—2741no—
132C2×D4—21no—
133C2×D4—2741no—
134C2×D4—2741no—
135C2×D4—2741no—
136C2×D4—1371no—
137C2×D4C2×#1058773911yescharacteristic
138C22×C4—2741no—
139C22×C4—2741no—
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4C2×#1058773911yescharacteristic
143M16—5482no—
144C2×C8—2744no—
145C2×C8—2742no—
146C2×C8—2742no—
147C4⋊C4—2741no—
148C4⋊C4—1371no—
149C4⋊C4C2×#1058773911yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—2741no—
152C22⋊C4—2741no—
153C22⋊C4—2741no—
154C42—1371no—
155C42—1371no—
156C42—1371no—
157C42—1371no—
158C23—21no—
159C23—2741no—
160C23C22⋊#1058773911yescharacteristic
161Q8—5482no—
162D4—22no—
163D4—42no—
164D4—2742no—
165D4—5482no—
166D4—5482no—
167D4—5482no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—2742no—
171C2×C4—2741no—
172C2×C4—5481no—
173C2×C4—5481no—
174C2×C4—5481no—
175C2×C4—5481no—
176C2×C4—1371no—
177C2×C4—1372no—
178C2×C4—1372no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4C22⋊#1058773911yescharacteristic
183C2×C4C22×#1058773911yescharacteristic
184C8—2744no—
185C8—5482no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—5481no—
192C22—5481no—
193C22—5481no—
194C22#1798415012yes—
195C22C2×C22⋊#1058773911yescharacteristic
196C4—21no—
197C4—41no—
198C4—2741no—
199C4—5481no—
200C4—1371no—
201C4—1371no—
202C4—1372no—
203C4—1372no—
204C4#1798417512yes—
205C2—22no—
206C2—81no—
207C2—10961no—
208C2#2257837611yescharacteristic
209C2C2×#1798415011yescharacteristic
210C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577948C2
cosocleC4×Dic137C23
hypercenter#121#10587739
hypocenterC137#1492
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578406C2semidirect
2. C2×C4×#11764047C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
105. C137#1492semidirect

Maximal subgroups

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

subgroupindex
#225784062
C2×C4×#117640472
#225782512
D4⋊#117640472
#225779482
#1492137