Finite Group Databasebeta

← #31779304

Subgroups — #31779304

Order 17536.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×#17984062C211yesmaximal, characteristic
2#22577367C211yesmaximal, characteristic, complemented
3#22577136C214yesmaximal, complemented
4C4×C22⋊C4×C137C211yesmaximal, characteristic, complemented
5C22×D548C411yescharacteristic, complemented
6C22×C22⋊D137—21no—
7C2×#14555494—21no—
8#17984062—21no—
9#17984062—21no—
10#17984062C412yescomplemented
11#17984062C2212yes—
12C2×C4⋊Dic137C411yescharacteristic, complemented
13C2×#14555487—21no—
14C22⋊#11764040—21no—
15C22⋊#11764040C2212yescomplemented
16C23×C4×C137C2211yescharacteristic
17C2×C22⋊C4×C137—21no—
18C2×C42×C137—21no—
19#95×C137C2212yescomplemented
20C23×D137—21no—
21C2×D548—21no—
22C2×D548—42no—
23C2×D548C2×C412yes—
24C2×C22⋊D137—22no—
25C2×C22⋊D137—22no—
26C2×C22⋊D137—44no—
27C22×Dic137—21no—
28#14555494—42no—
29C2×#11764040—22no—
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C412yes—
32#14555487—22no—
33#14555487—22no—
34C24×C137D411yescharacteristic
35C22×C4×C137—22no—
36C22×C4×C137—42no—
37C22×C4×C137C2×C411yescharacteristic
38C22×C4×C137C2×C411yescharacteristic
39C22×C4×C137D411yescharacteristic
40C22×C4×C137D411yescharacteristic
41C22×C4×C137D411yescharacteristic
42C22×C4×C137C2311yescharacteristic
43C22⋊C4×C137—24no—
44C42×C137—42no—
45C22×D137—22no—
46C22×D137—42no—
47C22×D137—42no—
48C22×D137—42no—
49D548—42no—
50D548—42no—
51C22⋊D137—44no—
52C22⋊D137—44no—
53C2×Dic137—22no—
54C2×Dic137—42no—
55#11764040—42no—
56C23×C137—22no—
57C23×C137—22no—
58C23×C137—41no—
59C23×C137C22⋊C411yescharacteristic
60C23×C137C22⋊C411yescharacteristic
61C23×C137C2×D411yescharacteristic
62C2×C4×C137—22no—
63C2×C4×C137—22no—
64C2×C4×C137—22no—
65C2×C4×C137—24no—
66C2×C4×C137—41no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137—42no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22×C411yescharacteristic
74C2×C4×C137C2×D411yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—21no—
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—44no—
83C22×C137#10412yes—
84C22×C137C2×C22⋊C411yescharacteristic
85C22×C137AGL(1,ℤ/8)12yes—
86C2×D137—42no—
87C2×D137—42no—
88C2×D137—42no—
89C2×D137—84no—
90C4×C137—21no—
91C4×C137—21no—
92C4×C137—21no—
93C4×C137—44no—
94C4×C137#9912yes—
95Dic137—42no—
96C2×C137—22no—
97C2×C137—22no—
98C2×C137—41no—
99C2×C137C2×#9911yescharacteristic
100C2×C137#41811yescharacteristic
101C2×C137C2×#10411yescharacteristic
102D137—82no—
103C137#150711yescharacteristic, complemented, Sylow
104#1507—1371nomaximal, Sylow
105C2×#121—1371no—
106#407—1371no—
107C4×C22⋊C4D13711yescharacteristic, complemented
108#327—1374no—
109C22×D4—2741no—
110C22×D4—1371no—
111C23×C4C2×D13711yescharacteristic
112C2×M16—2741no—
113#121—2741no—
114#121—2741no—
115#121—1372no—
116#121—1372no—
117C2×C4⋊C4—1371no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—2741no—
120C2×C42—21no—
121C22⋊C8—2741no—
122C22⋊C8—1372no—
123#95C2×D13712yescomplemented
124C24—2741no—
125C24C22⋊D13711yescharacteristic
126C2×D4—2742no—
127C2×D4—2742no—
128C2×D4—2741no—
129C2×D4—5482no—
130C2×D4—5484no—
131C2×D4—1372no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—2741no—
135C22×C4C4×D13711yescharacteristic
136C22×C4C4×D13711yescharacteristic
137C22×C4C22⋊D13711yescharacteristic
138C22×C4D54811yescharacteristic
139C22×C4D54811yescharacteristic
140C22×C4C22×D13711yescharacteristic
141M16—5482no—
142C2×C8—2742no—
143C4⋊C4—2741no—
144C4⋊C4—1372no—
145C22⋊C4—24no—
146C22⋊C4—2742no—
147C22⋊C4—2742no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—2742no—
153C23—5482no—
154C23—5482no—
155C23—5482no—
156C23#1455548711yescharacteristic
157C23#1455548711yescharacteristic
158C23C2×C22⋊D13711yescharacteristic
159D4—5482no—
160D4—5484no—
161D4—5484no—
162D4—5482no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—2742no—
173C2×C4—5482no—
174C2×C4#1455548711yescharacteristic
175C2×C4#1455548711yescharacteristic
176C2×C4C2×C4×D13711yescharacteristic
177C2×C4C2×D54811yescharacteristic
178C8—5482no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—5482no—
188C22—5482no—
189C22—5482no—
190C22—10964no—
191C22#1798401612yes—
192C22C2×#1455548711yescharacteristic
193C22#1798411312yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—5482no—
199C4#1798400212yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—10962no—
204C2C2×#1798400211yescharacteristic
205C2#2257740711yescharacteristic
206C2C2×#1798401611yescharacteristic
subgroupG/H
centerC22C2×#14555487
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
FittingC4×C22⋊C4×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984062C2
cosocleC22×C4×C137C23
hypercenterC4×C22⋊C4D137
hypocenterC137#1507
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22577367C2semidirect
3. #22577136C2semidirect
4. C4×C22⋊C4×C137C2semidirect
5. C22×D548C4semidirect
10. #17984062C4semidirect
12. C2×C4⋊Dic137C4semidirect
15. C22⋊#11764040C22semidirect
19. #95×C137C22semidirect
103. C137#1507semidirect
107. C4×C22⋊C4D137semidirect
123. #95C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840622
#225773672
#225771362
C4×C22⋊C4×C1372
#1507137