Finite Group Databasebeta

← #31779804

Subgroups — #31779804

Order 17536.

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

normal subgroups only

This group has 232 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#22577990C211yesmaximal, characteristic
2#22577734C211yesmaximal, characteristic, complemented
3#22577714C211yesmaximal, characteristic, complemented
4#22577330C211yesmaximal, characteristic, complemented
5#22577139C211yesmaximal, characteristic, complemented
6#22577126C211yesmaximal, characteristic, complemented
7#327×C137C211yesmaximal, characteristic, complemented
8#17984083—21no—
9#17984074—21no—
10#17984072—21no—
11#17984067C2211yescharacteristic
12#17984060—21no—
13#17984056—21no—
14D4×Dic137—21no—
15C2×C4⋊Dic137C2211yescharacteristic
16C2×#14555487—21no—
17C2×#14555487C2211yescharacteristic
18#17984021—21no—
19#17984020—21no—
20#17984013—21no—
21#17984009—21no—
22C22⋊#11764040C2211yescharacteristic
23#17983995—21no—
24#121×C137C2211yescharacteristic
25C22⋊C8×C137C2211yescharacteristic
26#95×C137C2211yescharacteristic
27C23×D137D411yescharacteristic
28C2×Dic274D411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C22×Dic137—21no—
32C22×Dic137—21no—
33C22×Dic137D411yescharacteristic
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137D411yescharacteristic
39#14555489—21no—
40C22⋊Dic137—21no—
41C22⋊Dic137—21no—
42#14555487—21no—
43#14555487—21no—
44#14555487—21no—
45#14555487—21no—
46#14555487—42no—
47C4×Dic137—21no—
48C2×D4×C137—21no—
49C2×D4×C137D411yescharacteristic
50C22×C4×C137—21no—
51C22×C4×C137C2311yescharacteristic
52C2×C8×C137—21no—
53C4⋊C4×C137D411yescharacteristic
54C22⋊C4×C137—21no—
55C22×D137—21no—
56C22×D137—21no—
57C22×D137—21no—
58C22×D137—41no—
59C22×D137—41no—
60Dic274—41no—
61C22⋊D137—81no—
62C22⋊D137—81no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—41no—
67C2×Dic137—41no—
68C2×Dic137—41no—
69C2×Dic137—41no—
70C2×Dic137—81no—
71#11764040—41no—
72C23×C137—21no—
73C23×C137C2×D411yescharacteristic
74D4×C137—41no—
75D4×C137—41no—
76D4×C137—41no—
77C2×C4×C137—21no—
78C2×C4×C137—41no—
79C2×C4×C137—41no—
80C2×C4×C137—41no—
81C2×C4×C137—41no—
82C2×C4×C137C2×D411yescharacteristic
83C2×C4×C137C2×D411yescharacteristic
84C8×C137—41no—
85C22×C137—21no—
86C22×C137—21no—
87C22×C137—41no—
88C22×C137—41no—
89C22×C137—41no—
90C22×C137#12011yescharacteristic
91C22×C137#13511yescharacteristic
92C22×C137AGL(1,ℤ/8)11yescharacteristic
93C2×D137—42no—
94C2×D137—41no—
95C2×D137—41no—
96C2×D137—41no—
97C2×D137—81no—
98C4×C137—21no—
99C4×C137—21no—
100C4×C137—41no—
101C4×C137—81no—
102Dic137—41no—
103Dic137—41no—
104Dic137—41no—
105Dic137—81no—
106C2×C137—21no—
107C2×C137—21no—
108C2×C137—81no—
109C2×C137#44911yescharacteristic
110C2×C137#45511yescharacteristic
111C2×C137#45811yescharacteristic
112D137—81no—
113C137#161211yescharacteristic, complemented, Sylow
114#1612—1371nomaximal, Sylow
115#538—1371no—
116#483—1371no—
117#480—1371no—
118#397—1371no—
119#328—1371no—
120#327D13711yescharacteristic, complemented
121#323—1371no—
122#126—2741no—
123#124—2741no—
124#123—2741no—
125C22⋊Q8—1371no—
126#121—2741no—
127#121C2×D13711yescharacteristic
128#120—2741no—
129C4×D4—2741no—
130C2×C4⋊C4—1371no—
131C2×C22⋊C4—2741no—
132C2×C22⋊C4—1371no—
133#107—2741no—
134#106—2741no—
135#103—2741no—
136#102—2741no—
137C22⋊C8—1371no—
138C22⋊C8C2×D13711yescharacteristic
139#95—2741no—
140#95C2×D13711yescharacteristic
141C24—1371no—
142C2×Q8—1371no—
143C2×D4—21no—
144C2×D4—2741no—
145C2×D4—2741no—
146C2×D4D54811yescharacteristic
147C22×C4—21no—
148C22×C4—2741no—
149C22×C4—2741no—
150C22×C4—1371no—
151C22×C4C22×D13711yescharacteristic
152C2×C8—21no—
153C2×C8—2741no—
154C4⋊C4—2741no—
155C4⋊C4—2741no—
156C4⋊C4—2741no—
157C4⋊C4—2741no—
158C4⋊C4—1371no—
159C4⋊C4D54811yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—2741no—
162C22⋊C4—2741no—
163C22⋊C4—2741no—
164C22⋊C4—2741no—
165C22⋊C4—2741no—
166C22⋊C4—2741no—
167C22⋊C4—5482no—
168C42—2741no—
169C23—21no—
170C23—2741no—
171C23—2741no—
172C23—2741no—
173C23—5481no—
174C23—5481no—
175C23D4×D13711yescharacteristic
176Q8—5481no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—10961no—
181D4—10961no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2741no—
188C2×C4—2741no—
189C2×C4—2741no—
190C2×C4—5481no—
191C2×C4—5481no—
192C2×C4—5481no—
193C2×C4—5481no—
194C2×C4—10961no—
195C2×C4D4×D13711yescharacteristic
196C2×C4C2×D54811yescharacteristic
197C8—41no—
198C8—5481no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—5482no—
205C22—5481no—
206C22—5481no—
207C22—5481no—
208C22—10961no—
209C22#1798405811yescharacteristic
210C22#1798410911yescharacteristic
211C22#1798411211yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—5481no—
217C4—5481no—
218C4—5481no—
219C4—10961no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—10961no—
224C2#2257754911yescharacteristic
225C2#2257758211yescharacteristic
226C2#2257759411yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC22×C4×C137C23
FrattiniC22×C4C22×D137
Fitting#327×C137C2
socleC22×C137#120
prime radical#22577990C2
hypercenter#327D137
hypocenterC137#1612
LCSG ▷ C22×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22577734C2semidirect
3. #22577714C2semidirect
4. #22577330C2semidirect
5. #22577139C2semidirect
6. #22577126C2semidirect
7. #327×C137C2semidirect
113. C137#1612semidirect
120. #327D137semidirect

Maximal subgroups

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

subgroupindex
#225779902
#225777342
#225777142
#225773302
#225771392
#225771262
#327×C1372
#1612137