Finite Group Databasebeta

← #31779839

Subgroups — #31779839

Order 17536.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#22577991C211yesmaximal, characteristic
2#22577739C211yesmaximal, characteristic, complemented
3#22577722C211yesmaximal, characteristic, complemented
4#22577339C211yesmaximal, characteristic, complemented
5#22577141C211yesmaximal, characteristic, complemented
6#22577138C211yesmaximal, characteristic, complemented
7#323×C137C211yesmaximal, characteristic, complemented
8#17984081—21no—
9#17984074—21no—
10#17984070—21no—
11#17984067C2211yescharacteristic
12#17984062C2211yescharacteristic
13#17984061—21no—
14#17984057—21no—
15C4×C22⋊D137—21no—
16C2×C4⋊Dic137C2211yescharacteristic
17C2×#14555489—21no—
18C2×C22⋊Dic137—21no—
19#17984022—21no—
20#17984020—21no—
21#17984013—21no—
22#17984010—21no—
23#17983995C2211yescharacteristic
24C2×C22⋊C4×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26C22⋊C8×C137C2211yescharacteristic
27C2×Dic274D411yescharacteristic
28C2×D548D411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C22×Dic137—21no—
34C2×C4×D137—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137D411yescharacteristic
38C4⋊Dic137D411yescharacteristic
39#14555489—21no—
40#14555489—21no—
41#14555489—42no—
42C22⋊Dic137—21no—
43C22⋊Dic137—21no—
44C22⋊Dic137—21no—
45C22⋊Dic137—42no—
46#14555487—21no—
47#14555487—21no—
48C4×Dic137—21no—
49C24×C137D411yescharacteristic
50C22×C4×C137D411yescharacteristic
51C22×C4×C137C2311yescharacteristic
52C2×C8×C137—21no—
53C2×C8×C137—21no—
54C22⋊C4×C137—21no—
55C22⋊C4×C137—21no—
56C22×D137—21no—
57Dic274—41no—
58D548—41no—
59C22⋊D137—41no—
60C22⋊D137—41no—
61C22⋊D137—81no—
62C22⋊D137—81no—
63C2×Dic137—22no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—21no—
67C2×Dic137—41no—
68C2×Dic137—41no—
69C2×Dic137—41no—
70C2×Dic137—41no—
71C2×Dic137—41no—
72C4×D137—81no—
73C23×C137—21no—
74C23×C137—21no—
75C23×C137—21no—
76C23×C137—41no—
77C23×C137—41no—
78C23×C137C2×D411yescharacteristic
79C2×C4×C137—21no—
80C2×C4×C137—41no—
81C2×C4×C137—41no—
82C2×C4×C137C2×D411yescharacteristic
83C2×C4×C137C2×D411yescharacteristic
84C8×C137—41no—
85C8×C137—41no—
86C22×C137—21no—
87C22×C137—21no—
88C22×C137—42no—
89C22×C137—41no—
90C22×C137—41no—
91C22×C137—41no—
92C22×C137—81no—
93C22×C137#12011yescharacteristic
94C22×C137#13511yescharacteristic
95C22×C137#13711yescharacteristic
96C2×D137—41no—
97C2×D137—41no—
98C2×D137—41no—
99C4×C137—21no—
100C4×C137—21no—
101C4×C137—41no—
102Dic137—42no—
103Dic137—41no—
104Dic137—41no—
105Dic137—41no—
106Dic137—81no—
107C2×C137—21no—
108C2×C137—21no—
109C2×C137—81no—
110C2×C137#45211yescharacteristic
111C2×C137#45311yescharacteristic
112C2×C137#45811yescharacteristic
113D137—81no—
114C137#161811yescharacteristic, complemented, Sylow
115#1618—1371nomaximal, Sylow
116#538—1371no—
117#484—1371no—
118#481—1371no—
119#399—1371no—
120#328—1371no—
121#327—1371no—
122#323D13711yescharacteristic, complemented
123#126—2741no—
124#124—2741no—
125#123—2741no—
126C22⋊Q8—1371no—
127#121—2741no—
128#121—1371no—
129#120—2741no—
130C4×D4—2741no—
131C2×C4⋊C4—2741no—
132C2×C4⋊C4—1371no—
133C2×C22⋊C4—2741no—
134C2×C22⋊C4C2×D13711yescharacteristic
135#107—2741no—
136#106—2741no—
137#103—2741no—
138#102—2741no—
139C22⋊C8C2×D13711yescharacteristic
140C22⋊C8C2×D13711yescharacteristic
141#95—1371no—
142C24D54811yescharacteristic
143C2×Q8—1371no—
144C2×D4—2741no—
145C2×D4—2741no—
146C2×D4—2741no—
147C2×D4—1371no—
148C22×C4—2741no—
149C22×C4—2741no—
150C22×C4—2741no—
151C22×C4D54811yescharacteristic
152C22×C4C22×D13711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—2741no—
156C4⋊C4—2741no—
157C4⋊C4—2741no—
158C4⋊C4—2741no—
159C4⋊C4—5482no—
160C4⋊C4—1371no—
161C4⋊C4—1371no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—2741no—
165C22⋊C4—2741no—
166C22⋊C4—2741no—
167C22⋊C4—2741no—
168C22⋊C4—2741no—
169C22⋊C4—5482no—
170C42—2741no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—2741no—
177C23C2×D54811yescharacteristic
178Q8—5481no—
179D4—5481no—
180D4—5481no—
181D4—5481no—
182D4—10961no—
183D4—10961no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2742no—
188C2×C4—2741no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—5481no—
192C2×C4—5481no—
193C2×C4—5481no—
194C2×C4—5481no—
195C2×C4—5481no—
196C2×C4—10961no—
197C2×C4D4×D13711yescharacteristic
198C2×C4D4×D13711yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—5481no—
209C22—5481no—
210C22—5481no—
211C22#1798405811yescharacteristic
212C22#1798411011yescharacteristic
213C22#1798412111yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—5482no—
218C4—5481no—
219C4—5481no—
220C4—5481no—
221C4—10961no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—10961no—
226C2#2257756711yescharacteristic
227C2#2257757311yescharacteristic
228C2#2257759411yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC22×C4×C137C23
FrattiniC22×C4C22×D137
Fitting#323×C137C2
socleC22×C137#120
prime radical#22577991C2
hypercenter#323D137
hypocenterC137#1618
LCSG ▷ C22×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22577739C2semidirect
3. #22577722C2semidirect
4. #22577339C2semidirect
5. #22577141C2semidirect
6. #22577138C2semidirect
7. #323×C137C2semidirect
114. C137#1618semidirect
122. #323D137semidirect

Maximal subgroups

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

subgroupindex
#225779912
#225777392
#225777222
#225773392
#225771412
#225771382
#323×C1372
#1618137