Finite Group Databasebeta

← #31796301

Subgroups — #31796301

Order 17536.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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×#17984155C211yesmaximal, characteristic, complemented
3#22578262C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8#17984155—21no—
9#17984155C2212yes—
10#17984155C2212yes—
11C4⋊#11764047—21no—
12C2×#14555509—21no—
13#17984087C2211yescharacteristic
14#17984084C2211yescharacteristic
15#17984074—21no—
16#17984062C411yescharacteristic
17#17984060—21no—
18D4×Dic137C411yescharacteristic
19#17984050C411yescharacteristic
20#17984048—21no—
21C2×C4×Dic137C2211yescharacteristic
22#121×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
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—42no—
53#11764047—42no—
54C22×D137—21no—
55Dic274—41no—
56Dic274—41no—
57D548—41no—
58C22⋊D137—42no—
59C22⋊D137—41no—
60C22⋊D137—41no—
61C22⋊D137—41no—
62C22⋊D137—41no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—42no—
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C22⋊C411yescharacteristic
69C2×Dic137D812yes—
70C2×Dic137SD1612yes—
71C2×Dic137C2×D411yescharacteristic
72C2×Dic137C2×D411yescharacteristic
73C4×D137—42no—
74C23×C137—21no—
75C23×C137C22⋊C411yescharacteristic
76D4×C137—41no—
77D4×C137—42no—
78C2×C4×C137—21no—
79C2×C4×C137—41no—
80C2×C4×C137C22⋊C411yescharacteristic
81C2×C4×C137C22×C411yescharacteristic
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—41no—
85C22×C137—41no—
86C22×C137—41no—
87C22×C137#10212yes—
88C22×C137C2×C22⋊C411yescharacteristic
89C2×D137—41no—
90C2×D137—41no—
91C2×D137—41no—
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—41no—
95Dic137—21no—
96Dic137—41no—
97Dic137#10212yes—
98Dic137C2×D811yescharacteristic
99Dic137C2×SD1611yescharacteristic
100Dic137AGL(1,ℤ/8)11yescharacteristic
101Dic137#13711yescharacteristic
102C2×C137—22no—
103C2×C137—81no—
104C2×C137#41311yescharacteristic
105C2×C137C2×#10211yescharacteristic
106C2×C137#41711yescharacteristic
107D137—81no—
108C137#154711yescharacteristic, complemented, Sylow
109#1547—1371nomaximal, Sylow
110#532—1371no—
111#423—1371no—
112C2×C4⋊C8—1371no—
113#332—1372no—
114#331—1372no—
115C2×#41—2741no—
116C2×M16—2741no—
117C22×C8—2741no—
118C4⋊Q8—1371no—
119#127—1371no—
120#124—2741no—
121#121—2741no—
122#121—1371no—
123#121#1058773911yescharacteristic
124C4×D4—2741no—
125C4×D4—1371no—
126C4×D4—1371no—
127C2×C42—1371no—
128C4⋊C8—2741no—
129C4⋊C8—2741no—
130C4⋊C8—1372no—
131C4⋊C8—1372no—
132#41—5482no—
133#41—5482no—
134C2×Q8—2741no—
135C2×D4—21no—
136C2×D4—2741no—
137C2×D4—2741no—
138C2×D4—2741no—
139C2×D4—1371no—
140C2×D4C2×#1058773911yescharacteristic
141C22×C4—2741no—
142C22×C4—2741no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4C2×#1058773911yescharacteristic
146M16—5482no—
147C2×C8—2742no—
148C2×C8—2742no—
149C2×C8—5482no—
150C4⋊C4—2741no—
151C4⋊C4—1371no—
152C4⋊C4C2×#1058773911yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C22⋊C4—2741no—
157C42—1371no—
158C42—1371no—
159C42—1371no—
160C42—1371no—
161C23—21no—
162C23—2741no—
163C23C22⋊#1058773911yescharacteristic
164Q8—5481no—
165Q8—5481no—
166D4—41no—
167D4—42no—
168D4—5481no—
169D4—5482no—
170D4—5481no—
171D4—5481no—
172D4—5481no—
173D4—5481no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—2741no—
177C2×C4—2741no—
178C2×C4—2741no—
179C2×C4—5482no—
180C2×C4—5482no—
181C2×C4—1371no—
182C2×C4—1372no—
183C2×C4—1372no—
184C2×C4—1371no—
185C2×C4—1371no—
186C2×C4—1371no—
187C2×C4C22⋊#1058773911yescharacteristic
188C2×C4C22×#1058773911yescharacteristic
189C8—5482no—
190C8—5482no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—5481no—
197C22—5481no—
198C22—5481no—
199C22#1798415012yes—
200C22C2×C22⋊#1058773911yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—2741no—
205C4—5481no—
206C4—1371no—
207C4—1372no—
208C4—1371no—
209C4—1371no—
210C4—1371no—
211C2—22no—
212C2—81no—
213C2—10961no—
214C2#2257839011yescharacteristic
215C2C2×#1798415011yescharacteristic
216C2#2257839411yescharacteristic
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#1547
LCSG ▷ C2×C4×C137 ▷ C22×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×#17984155C2semidirect
3. #22578262C2semidirect
4. #22578261C2semidirect
108. C137#1547semidirect

Maximal subgroups

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

subgroupindex
#225784062
C2×#179841552
#225782622
#225782612
#225779482
#1547137