Finite Group Databasebeta

← #37508021

Subgroups — #37508021

Order 22912.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#26038997C211yesmaximal, characteristic
2#26038758C211yesmaximal, characteristic, complemented
3#26038755C211yesmaximal, characteristic, complemented
4#26038376C211yesmaximal, characteristic, complemented
5#26038172C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic, complemented
7#327×C179C211yesmaximal, characteristic, complemented
8C2×D4×D179—21no—
9#19153098—21no—
10#19153095—21no—
11#19153087C2211yescharacteristic
12#19153085—21no—
13#19153083—21no—
14#19153081—21no—
15C2×C4⋊Dic179—21no—
16C2×C4⋊Dic179C2211yescharacteristic, complemented
17C2×(C2×D179)⋊C4—21no—
18C2×(C2×D179)⋊C4C2211yescharacteristic
19#19153045—21no—
20#19153044—21no—
21#19153035—21no—
22#19153034—21no—
23C22⋊(C179⋊C8)C2211yescharacteristic, complemented
24#121×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic, complemented
26#95×C179C2211yescharacteristic, complemented
27C23×D179—21no—
28C23×D179D411yescharacteristic
29C2×D716—21no—
30C2×D716D411yescharacteristic
31C2×C22⋊D179—21no—
32C2×C22⋊D179—21no—
33C2×C22⋊D179—21no—
34D4×D179—42no—
35D4×D179—42no—
36C22×Dic179—21no—
37C22×Dic179D411yescharacteristic
38C2×C4×D179—21no—
39C2×C179⋊C8—21no—
40C4⋊Dic179—21no—
41C4⋊Dic179—21no—
42C4⋊Dic179—42no—
43C4⋊Dic179D411yescharacteristic
44Dic179⋊C4—21no—
45C22⋊Dic179—21no—
46(C2×D179)⋊C4—21no—
47(C2×D179)⋊C4—21no—
48(C2×D179)⋊C4—21no—
49(C2×D179)⋊C4—21no—
50(C2×D179)⋊C4—42no—
51C2×D4×C179—21no—
52C2×D4×C179D411yescharacteristic
53C22×C4×C179—21no—
54C22×C4×C179C2311yescharacteristic
55C2×C8×C179—21no—
56C4⋊C4×C179D411yescharacteristic
57C22⋊C4×C179—21no—
58C22×D179—21no—
59C22×D179—21no—
60C22×D179—21no—
61C22×D179—42no—
62C22×D179—41no—
63C22×D179—41no—
64C22×D179—81no—
65C22×D179—81no—
66C22×D179SD1612yes—
67D716—41no—
68D716—42no—
69C22⋊D179—42no—
70C22⋊D179—41no—
71C22⋊D179—41no—
72C22⋊D179—81no—
73C2×Dic179—21no—
74C2×Dic179—21no—
75C2×Dic179—41no—
76C2×Dic179—41no—
77C2×Dic179—41no—
78C2×Dic179—41no—
79C4×D179—42no—
80C179⋊C8—41no—
81C23×C179—21no—
82C23×C179C2×D411yescharacteristic
83D4×C179—41no—
84D4×C179—41no—
85D4×C179—41no—
86C2×C4×C179—22no—
87C2×C4×C179—21no—
88C2×C4×C179—41no—
89C2×C4×C179—41no—
90C2×C4×C179—41no—
91C2×C4×C179C2×D411yescharacteristic
92C2×C4×C179C2×D411yescharacteristic
93C8×C179—41no—
94C22×C179—21no—
95C22×C179—21no—
96C22×C179—41no—
97C22×C179—41no—
98C22×C179—41no—
99C22×C179#12011yescharacteristic
100C22×C179C2×SD1611yescharacteristic
101C22×C179AGL(1,ℤ/8)11yescharacteristic
102C2×D179—22no—
103C2×D179—41no—
104C2×D179—41no—
105C2×D179—42no—
106C2×D179—41no—
107C2×D179—41no—
108C2×D179—41no—
109C2×D179—81no—
110C2×D179—84no—
111C4×C179—21no—
112C4×C179—21no—
113C4×C179—42no—
114C4×C179—41no—
115Dic179—41no—
116Dic179—41no—
117Dic179—81no—
118C2×C179—21no—
119C2×C179—21no—
120C2×C179—81no—
121C2×C179#45011yescharacteristic
122C2×C179#45311yescharacteristic
123C2×C179#45811yescharacteristic
124D179—42no—
125D179—81no—
126C179#161311yescharacteristic, complemented, Sylow
127#1613—1791nomaximal, Sylow
128#534—1791no—
129#481—1791no—
130#481—1791no—
131#399—1791no—
132#327—1791no—
133#327D17911yescharacteristic, complemented
134#323—1791no—
135C22×D4—3581no—
136#123—3581no—
137#123—3581no—
138#121—3581no—
139#121—1791no—
140#121C2×D17911yescharacteristic
141#120—3581no—
142#120—3581no—
143C2×C4⋊C4—3581no—
144C2×C4⋊C4—1791no—
145C2×C22⋊C4—3581no—
146C2×C22⋊C4—1791no—
147#106—3581no—
148#106—3581no—
149#102—3581no—
150#102—3581no—
151C22⋊C8—1791no—
152C22⋊C8C2×D17911yescharacteristic, complemented
153#95C2×D17911yescharacteristic, complemented
154C24—3581no—
155C24—1791no—
156C2×D4—21no—
157C2×D4—3581no—
158C2×D4—3581no—
159C2×D4—3581no—
160C2×D4—3581no—
161C2×D4—1791no—
162C2×D4—7162no—
163C2×D4—7162no—
164C2×D4D71611yescharacteristic
165C22×C4—21no—
166C22×C4—3581no—
167C22×C4—3581no—
168C22×C4—1791no—
169C22×C4C22×D17911yescharacteristic
170C2×C8—21no—
171C2×C8—3581no—
172C4⋊C4—3581no—
173C4⋊C4—3581no—
174C4⋊C4—3581no—
175C4⋊C4—1791no—
176C4⋊C4—7162no—
177C4⋊C4D71611yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3581no—
180C22⋊C4—3581no—
181C22⋊C4—3581no—
182C22⋊C4—3581no—
183C22⋊C4—3581no—
184C22⋊C4—7162no—
185C23—21no—
186C23—3581no—
187C23—3581no—
188C23—3581no—
189C23—14321no—
190C23—14321no—
191C23—1792no—
192C23—7161no—
193C23—7161no—
194C23—7162no—
195C23D4×D17911yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—14321no—
200D4—7161no—
201D4—7162no—
202D4—7162no—
203D4—7161no—
204D4—7161no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3581no—
211C2×C4—3581no—
212C2×C4—7162no—
213C2×C4—7161no—
214C2×C4—7161no—
215C2×C4—7161no—
216C2×C4—7161no—
217C2×C4D4×D17911yescharacteristic
218C2×C4C2×D71611yescharacteristic
219C8—41no—
220C8—7161no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—3582no—
227C22—14321no—
228C22—14324no—
229C22—7162no—
230C22—7161no—
231C22—7161no—
232C22—7161no—
233C22—7161no—
234C22—7161no—
235C22#1915308311yescharacteristic
236C22SD16×D17911yescharacteristic
237C22#1915313711yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—14321no—
243C4—7161no—
244C4—7161no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—14321no—
249C2—7162no—
250C2#2603859111yescharacteristic
251C2#2603860811yescharacteristic
252C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#327×C179C2
socleC22×C179#120
prime radical#26038997C2
hypercenter#327D179
hypocenterC179#1613
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26038758C2semidirect
3. #26038755C2semidirect
4. #26038376C2semidirect
5. #26038172C2semidirect
6. #26038162C2semidirect
7. #327×C179C2semidirect
16. C2×C4⋊Dic179C22semidirect
23. C22⋊(C179⋊C8)C22semidirect
25. C22⋊C8×C179C22semidirect
26. #95×C179C22semidirect
126. C179#1613semidirect
133. #327D179semidirect
152. C22⋊C8C2×D179semidirect
153. #95C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260389972
#260387582
#260387552
#260383762
#260381722
#260381622
#327×C1792
#1613179