Finite Group Databasebeta

← #31779706

Subgroups — #31779706

Order 17536.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#22577965C211yesmaximal, characteristic, complemented
2#22577537C212yesmaximal, complemented
3#22577136C212yesmaximal, complemented
4#22577127C211yesmaximal, characteristic, complemented
5#392×C137C211yesmaximal, characteristic, complemented
6C22×C22⋊D137—21no—
7C2×#14555495—22no—
8#17984072—22no—
9#17984062—21no—
10#17984062C2212yes—
11#17984058—22no—
12#17984009—22no—
13C22⋊#11764040C2212yescomplemented
14C22×D4×C137—21no—
15C22×D4×C137C2211yescharacteristic, complemented
16C2×C22⋊C4×C137—21no—
17C2×C22⋊C4×C137C2211yescharacteristic
18#95×C137C2211yescharacteristic, complemented
19C23×D137—21no—
20C2×D548—21no—
21C2×D548D412yescomplemented
22C2×C22⋊D137—22no—
23C2×C22⋊D137—22no—
24C2×C22⋊D137—42no—
25C2×C22⋊D137—42no—
26C22×Dic137—21no—
27#14555495—44no—
28C2×#11764040—22no—
29C4⋊Dic137—21no—
30C4⋊Dic137D412yescomplemented
31#14555487—22no—
32#14555487—22no—
33C24×C137—21no—
34C24×C137D411yescharacteristic
35C2×D4×C137—22no—
36C2×D4×C137—42no—
37C2×D4×C137—42no—
38C2×D4×C137—42no—
39C2×D4×C137—42no—
40C22×C4×C137—21no—
41C22×C4×C137D411yescharacteristic
42C22×C4×C137C2311yescharacteristic
43C22⋊C4×C137—22no—
44C22⋊C4×C137—42no—
45C22×D137—22no—
46C22×D137—42no—
47C22×D137—81no—
48C22×D137—81no—
49D548—42no—
50D548—81no—
51C22⋊D137—44no—
52C22⋊D137—42no—
53C22⋊D137—42no—
54C2×Dic137—22no—
55C2×Dic137—42no—
56#11764040—42no—
57C23×C137—21no—
58C23×C137—21no—
59C23×C137—22no—
60C23×C137—41no—
61C23×C137—41no—
62C23×C137—41no—
63C23×C137—41no—
64C23×C137—41no—
65C23×C137—41no—
66C23×C137—41no—
67C23×C137—41no—
68C23×C137D812yescomplemented
69C23×C137C2×D411yescharacteristic
70D4×C137—24no—
71D4×C137—42no—
72D4×C137—48no—
73C2×C4×C137—22no—
74C2×C4×C137—21no—
75C2×C4×C137—41no—
76C2×C4×C137—41no—
77C2×C4×C137—41no—
78C2×C4×C137—41no—
79C2×C4×C137C2×D412yes—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—22no—
83C22×C137—42no—
84C22×C137—42no—
85C22×C137—42no—
86C22×C137—41no—
87C22×C137—41no—
88C22×C137—42no—
89C22×C137—81no—
90C22×C137—81no—
91C22×C137—81no—
92C22×C137—81no—
93C22×C137—81no—
94C22×C137#12011yescharacteristic
95C22×C137C2×D811yescharacteristic
96C22×C137AGL(1,ℤ/8)11yescharacteristic
97C2×D137—42no—
98C2×D137—42no—
99C2×D137—42no—
100C2×D137—84no—
101C4×C137—22no—
102C4×C137—42no—
103C4×C137—41no—
104Dic137—42no—
105C2×C137—21no—
106C2×C137—21no—
107C2×C137—42no—
108C2×C137—82no—
109C2×C137#44711yescharacteristic
110C2×C137#45311yescharacteristic
111C2×C137B(4,𝔽2)11yescharacteristic
112D137—82no—
113C137#159511yescharacteristic, complemented, Sylow
114#1595—1371nomaximal, Sylow
115#534—1371no—
116#447—1372no—
117#392D13711yescharacteristic, complemented
118#327—1372no—
119#323—1371no—
120C22×D4—21no—
121C22×D4—2741no—
122C22×D4C2×D13711yescharacteristic, complemented
123C2×D8—2742no—
124#123—2742no—
125#121—2741no—
126#121—1372no—
127#120—2742no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C2×D13711yescharacteristic
130#102—2742no—
131C22⋊C8—1372no—
132#95C2×D13711yescharacteristic, complemented
133C24—21no—
134C24—2741no—
135C24C22⋊D13711yescharacteristic
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—2741no—
142C2×D4—2742no—
143C2×D4—2742no—
144C2×D4—5482no—
145C2×D4—5482no—
146C2×D4—1372no—
147C22×C4—21no—
148C22×C4—2741no—
149C22×C4C22⋊D13711yescharacteristic
150C22×C4C22×D13711yescharacteristic
151D8—5484no—
152C2×C8—2742no—
153C4⋊C4—2741no—
154C4⋊C4—1372no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C22⋊C4—2742no—
158C22⋊C4—2742no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—2742no—
171C23—5482no—
172C23—10961no—
173C23—10961no—
174C23#1455549512yescomplemented
175C23C2×C22⋊D13711yescharacteristic
176D4—24no—
177D4—42no—
178D4—48no—
179D4—5482no—
180D4—5484no—
181D4—5482no—
182D4—5482no—
183D4—10961no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—2742no—
191C2×C4—5482no—
192C2×C4D4×D13712yes—
193C8—5482no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—5482no—
209C22—5482no—
210C22—5482no—
211C22—10964no—
212C22#1798405611yescharacteristic
213C22C2×#1455549511yescharacteristic
214C22#1798411311yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—5482no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—10962no—
224C2#2257753611yescharacteristic
225C2#2257757011yescharacteristic
226C2#2257758911yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC22×C4×C137C23
FrattiniC22×C4C22×D137
Fitting#392×C137C2
socleC22×C137#120
hypercenter#392D137
hypocenterC137#1595
LCSG ▷ C22×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #22577965C2semidirect
2. #22577537C2semidirect
3. #22577136C2semidirect
4. #22577127C2semidirect
5. #392×C137C2semidirect
13. C22⋊#11764040C22semidirect
15. C22×D4×C137C22semidirect
18. #95×C137C22semidirect
21. C2×D548D4semidirect
30. C4⋊Dic137D4semidirect
68. C23×C137D8semidirect
113. C137#1595semidirect
117. #392D137semidirect
122. C22×D4C2×D137semidirect
132. #95C2×D137semidirect
174. C23#14555495semidirect

Maximal subgroups

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

subgroupindex
#225779652
#225775372
#225771362
#225771272
#392×C1372
#1595137