Finite Group Databasebeta

← #37507982

Subgroups — #37507982

Order 22912.

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

normal subgroups only

This group has 260 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#26039027C211yesmaximal, characteristic
2#26038597C211yesmaximal, characteristic, complemented
3#26038580C211yesmaximal, characteristic, complemented
4#26038351C211yesmaximal, characteristic, complemented
5#26038177C211yesmaximal, characteristic, complemented
6#26038174C211yesmaximal, characteristic, complemented
7#323×C179C211yesmaximal, characteristic, complemented
8C22×Dic358C2211yescharacteristic
9C2×Dic716—21no—
10C2×#15668248—21no—
11#19153106—21no—
12#19153099—21no—
13#19153095—21no—
14#19153092C2211yescharacteristic
15#19153087C2211yescharacteristic
16#19153086—21no—
17#19153082—21no—
18C4×C22⋊D179—21no—
19C2×C22⋊Dic179—21no—
20#19153038—21no—
21#19153038—21no—
22C179⋊#95—21no—
23C179⋊#95C2211yescharacteristic
24C2×C22⋊C4×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26C22⋊C8×C179C2211yescharacteristic
27C2×Dic358—21no—
28C2×Dic358—21no—
29C2×Dic358—42no—
30C2×Dic358D411yescharacteristic
31C2×D716D411yescharacteristic
32C2×C22⋊D179—21no—
33C2×C22⋊D179—21no—
34C2×C22⋊D179—21no—
35C22×Dic179—21no—
36C22×Dic179—21no—
37C2×C4×D179—21no—
38Dic716—42no—
39#15668248—42no—
40C4⋊Dic179D411yescharacteristic
41C4⋊Dic179D411yescharacteristic
42Dic179⋊C4—21no—
43Dic179⋊C4—21no—
44C22⋊Dic179—21no—
45C22⋊Dic179—21no—
46C22⋊Dic179—21no—
47C22⋊Dic179—42no—
48(C2×D179)⋊C4—21no—
49(C2×D179)⋊C4—21no—
50C4×Dic179—21no—
51C24×C179D411yescharacteristic
52C22×C4×C179D411yescharacteristic
53C22×C4×C179C2311yescharacteristic
54C2×C8×C179—21no—
55C2×C8×C179—21no—
56C22⋊C4×C179—21no—
57C22⋊C4×C179—21no—
58C22×D179—21no—
59Dic358—22no—
60Dic358—22no—
61Dic358—41no—
62Dic358—41no—
63Dic358—41no—
64D716—41no—
65C22⋊D179—41no—
66C22⋊D179—41no—
67C22⋊D179—81no—
68C22⋊D179—81no—
69C2×Dic179—22no—
70C2×Dic179—21no—
71C2×Dic179—21no—
72C2×Dic179—21no—
73C2×Dic179—41no—
74C2×Dic179—41no—
75C2×Dic179—41no—
76C2×Dic179—41no—
77C2×Dic179—41no—
78C4×D179—81no—
79C23×C179—21no—
80C23×C179—21no—
81C23×C179—21no—
82C23×C179—41no—
83C23×C179—41no—
84C23×C179C2×D411yescharacteristic
85C2×C4×C179—21no—
86C2×C4×C179—41no—
87C2×C4×C179—41no—
88C2×C4×C179C2×D411yescharacteristic
89C2×C4×C179C2×D411yescharacteristic
90C8×C179—41no—
91C8×C179—41no—
92C22×C179—21no—
93C22×C179—21no—
94C22×C179—42no—
95C22×C179—41no—
96C22×C179—41no—
97C22×C179—41no—
98C22×C179—81no—
99C22×C179#12011yescharacteristic
100C22×C179#13511yescharacteristic
101C22×C179#13711yescharacteristic
102C2×D179—41no—
103C2×D179—41no—
104C2×D179—41no—
105C4×C179—21no—
106C4×C179—21no—
107C4×C179—41no—
108Dic179—41no—
109Dic179—41no—
110Dic179—42no—
111Dic179—41no—
112Dic179—81no—
113C2×C179—21no—
114C2×C179—21no—
115C2×C179—81no—
116C2×C179#45211yescharacteristic
117C2×C179#45511yescharacteristic
118C2×C179B(4,𝔽2)11yescharacteristic
119D179—81no—
120C179#160611yescharacteristic, complemented, Sylow
121#1606—1791nomaximal, Sylow
122#538—1791no—
123#451—1791no—
124#448—1791no—
125#393—1791no—
126#328—1791no—
127#327—1791no—
128#323D17911yescharacteristic, complemented
129C22×Q8—1791no—
130C2×Q16—3581no—
131C2×SD16—3581no—
132#126—3581no—
133#124—3581no—
134#123—3581no—
135C22⋊Q8—1791no—
136#121—3581no—
137#121—1791no—
138#120—3581no—
139C4×D4—3581no—
140C2×C22⋊C4—3581no—
141C2×C22⋊C4C2×D17911yescharacteristic
142#103—3581no—
143#103—3581no—
144C22⋊C8C2×D17911yescharacteristic
145C22⋊C8C2×D17911yescharacteristic
146#95—3581no—
147#95—1791no—
148C24D71611yescharacteristic
149C2×Q8—3581no—
150C2×Q8—3581no—
151C2×Q8—1791no—
152C2×Q8—7162no—
153C2×D4—3581no—
154C2×D4—3581no—
155C2×D4—3581no—
156C2×D4—1791no—
157C22×C4—3581no—
158C22×C4—3581no—
159C22×C4—3581no—
160C22×C4D71611yescharacteristic
161C22×C4C22×D17911yescharacteristic
162Q16—7162no—
163SD16—7162no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—3581no—
167C4⋊C4—3581no—
168C4⋊C4—1791no—
169C4⋊C4—1791no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3581no—
173C22⋊C4—3581no—
174C22⋊C4—3581no—
175C22⋊C4—3581no—
176C22⋊C4—3581no—
177C22⋊C4—7162no—
178C42—3581no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3581no—
185C23C2×D71611yescharacteristic
186Q8—3582no—
187Q8—3582no—
188Q8—7161no—
189Q8—7161no—
190Q8—7161no—
191D4—14321no—
192D4—14321no—
193D4—7161no—
194D4—7161no—
195D4—7161no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3582no—
200C2×C4—3581no—
201C2×C4—3581no—
202C2×C4—3581no—
203C2×C4—14321no—
204C2×C4—7161no—
205C2×C4—7161no—
206C2×C4—7161no—
207C2×C4—7161no—
208C2×C4—7161no—
209C2×C4D4×D17911yescharacteristic
210C2×C4D4×D17911yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—7161no—
221C22—7161no—
222C22—7161no—
223C22#1915308311yescharacteristic
224C22#1915313511yescharacteristic
225C22#1915314611yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—14321no—
230C4—7161no—
231C4—7161no—
232C4—7162no—
233C4—7161no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—14321no—
238C2#2603860311yescharacteristic
239C2#2603861911yescharacteristic
240C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#323×C179C2
socleC22×C179#120
prime radical#26039027C2
hypercenter#323D179
hypocenterC179#1606
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26038597C2semidirect
3. #26038580C2semidirect
4. #26038351C2semidirect
5. #26038177C2semidirect
6. #26038174C2semidirect
7. #323×C179C2semidirect
120. C179#1606semidirect
128. #323D179semidirect

Maximal subgroups

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

subgroupindex
#260390272
#260385972
#260385802
#260383512
#260381772
#260381742
#323×C1792
#1606179