Finite Group Databasebeta

← #37507932

Subgroups — #37507932

Order 22912.

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

normal subgroups only

This group has 272 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#26039038C211yesmaximal, characteristic, complemented
2#26038589C211yesmaximal, characteristic, complemented
3#26038575C211yesmaximal, characteristic
4#26038357C211yesmaximal, characteristic, complemented
5#26038172C211yesmaximal, characteristic, complemented
6#26038164C211yesmaximal, characteristic, complemented
7#328×C179C211yesmaximal, characteristic, complemented
8C22×D716C2211yescharacteristic
9C2×#15668246—21no—
10C2×D1432—21no—
11#19153108—21no—
12#19153104—21no—
13#19153098—21no—
14#19153097—21no—
15#19153089—21no—
16#19153087C2211yescharacteristic
17(C22⋊D179)⋊C4—21no—
18C2×Dic179⋊C4C2211yescharacteristic
19C2×(C2×D179)⋊C4—21no—
20C2×(C2×D179)⋊C4—21no—
21#19153035—21no—
22#19153033—21no—
23C22⋊(C179⋊C8)C2211yescharacteristic
24C22⋊Q8×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27C23×D179—21no—
28C2×D716—21no—
29C2×D716—21no—
30C2×D716—42no—
31C2×D716D411yescharacteristic
32C2×C22⋊D179—21no—
33C22×Dic179D411yescharacteristic
34C22×Dic179D411yescharacteristic
35C2×C4×D179—21no—
36#15668246—42no—
37D1432—42no—
38C2×C179⋊C8—21no—
39C4⋊Dic179—21no—
40C4⋊Dic179—21no—
41C4⋊Dic179D411yescharacteristic
42Dic179⋊C4—21no—
43Dic179⋊C4—21no—
44Dic179⋊C4—21no—
45(C2×D179)⋊C4—21no—
46(C2×D179)⋊C4—21no—
47(C2×D179)⋊C4—21no—
48(C2×D179)⋊C4—21no—
49(C2×D179)⋊C4—42no—
50(C2×D179)⋊C4—42no—
51C4×Dic179—21no—
52C2×Q8×C179D411yescharacteristic
53C22×C4×C179—21no—
54C22×C4×C179C2311yescharacteristic
55C2×C8×C179—21no—
56C4⋊C4×C179—21no—
57C4⋊C4×C179D411yescharacteristic
58C22⋊C4×C179—21no—
59C22×D179—22no—
60C22×D179—21no—
61C22×D179—41no—
62C22×D179—41no—
63C22×D179—41no—
64C22×D179—41no—
65C22×D179—41no—
66C22×D179—41no—
67D716—22no—
68D716—22no—
69D716—41no—
70D716—41no—
71D716—41no—
72C22⋊D179—41no—
73C22⋊D179—41no—
74C2×Dic179—21no—
75C2×Dic179—21no—
76C2×Dic179—21no—
77C2×Dic179—21no—
78C2×Dic179—21no—
79C2×Dic179—41no—
80C4×D179—81no—
81C179⋊C8—41no—
82C23×C179C2×D411yescharacteristic
83Q8×C179—41no—
84C2×C4×C179—21no—
85C2×C4×C179—21no—
86C2×C4×C179—41no—
87C2×C4×C179—41no—
88C2×C4×C179—41no—
89C2×C4×C179—41no—
90C2×C4×C179C2×D411yescharacteristic
91C2×C4×C179C2×D411yescharacteristic
92C8×C179—41no—
93C22×C179—21no—
94C22×C179—21no—
95C22×C179#12011yescharacteristic
96C22×C179#13511yescharacteristic
97C22×C179AGL(1,ℤ/8)11yescharacteristic
98C2×D179—42no—
99C2×D179—42no—
100C2×D179—41no—
101C2×D179—41no—
102C2×D179—41no—
103C2×D179—42no—
104C2×D179—81no—
105C2×D179—81no—
106C2×D179—81no—
107C2×D179—81no—
108C4×C179—21no—
109C4×C179—21no—
110C4×C179—41no—
111C4×C179—41no—
112C4×C179—81no—
113Dic179—41no—
114Dic179—41no—
115Dic179—41no—
116C2×C179—21no—
117C2×C179—21no—
118C2×C179#44911yescharacteristic
119C2×C179#45411yescharacteristic
120C2×C179B(4,𝔽2)11yescharacteristic
121D179—81no—
122D179—82no—
123C179#159711yescharacteristic, complemented, Sylow
124#1597—1791nomaximal, Sylow
125#539—1791no—
126#450—1791no—
127#447—1791no—
128#394—1791no—
129#328D17911yescharacteristic, complemented
130#327—1791no—
131#324—1791no—
132C22×D4—1791no—
133C2×SD16—3581no—
134C2×D8—3581no—
135#126—3581no—
136#125—3581no—
137#123—3581no—
138#123—3581no—
139C22⋊Q8—3581no—
140C22⋊Q8C2×D17911yescharacteristic
141#121—1791no—
142C4×D4—3581no—
143C2×C4⋊C4—1791no—
144C2×C22⋊C4—3581no—
145C2×C22⋊C4—3581no—
146#102—3581no—
147#102—3581no—
148C22⋊C8—1791no—
149C22⋊C8C2×D17911yescharacteristic
150#95C2×D17911yescharacteristic
151C24—3581no—
152C2×Q8D71611yescharacteristic
153C2×D4—3581no—
154C2×D4—3581no—
155C2×D4—3581no—
156C2×D4—1791no—
157C2×D4—7162no—
158C22×C4—21no—
159C22×C4—3581no—
160C22×C4—1791no—
161C22×C4—1791no—
162C22×C4C22×D17911yescharacteristic
163SD16—7162no—
164D8—7162no—
165C2×C8—21no—
166C2×C8—3581no—
167C4⋊C4—21no—
168C4⋊C4—3581no—
169C4⋊C4—3581no—
170C4⋊C4—3581no—
171C4⋊C4—3581no—
172C4⋊C4—3581no—
173C4⋊C4—1791no—
174C4⋊C4D71611yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—3581no—
177C22⋊C4—3581no—
178C22⋊C4—3581no—
179C22⋊C4—3581no—
180C22⋊C4—7162no—
181C22⋊C4—7162no—
182C42—3581no—
183C23—3582no—
184C23—3581no—
185C23—7161no—
186C23—7161no—
187C23—7161no—
188C23—7161no—
189C23—7161no—
190C23—7161no—
191C23D4×D17911yescharacteristic
192Q8—41no—
193D4—3582no—
194D4—3582no—
195D4—7161no—
196D4—7161no—
197D4—7161no—
198D4—7161no—
199D4—7161no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—3581no—
207C2×C4—3581no—
208C2×C4—3581no—
209C2×C4—3581no—
210C2×C4—3581no—
211C2×C4—14321no—
212C2×C4—7161no—
213C2×C4D4×D17911yescharacteristic
214C2×C4C2×D71611yescharacteristic
215C8—41no—
216C8—7161no—
217C22—21no—
218C22—21no—
219C22—14321no—
220C22—14321no—
221C22—14321no—
222C22—14321no—
223C22—7162no—
224C22—7162no—
225C22—7161no—
226C22—7161no—
227C22—7161no—
228C22—7162no—
229C22#1915308311yescharacteristic
230C22#1915313611yescharacteristic
231C22#1915314011yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—7161no—
238C4—7161no—
239C4—7161no—
240C2—21no—
241C2—21no—
242C2—14322no—
243C2—14321no—
244C2#2603858711yescharacteristic
245C2#2603861311yescharacteristic
246C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#328×C179C2
socleC22×C179#120
prime radical#26038575C2
hypercenter#328D179
hypocenterC179#1597
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #26039038C2semidirect
2. #26038589C2semidirect
4. #26038357C2semidirect
5. #26038172C2semidirect
6. #26038164C2semidirect
7. #328×C179C2semidirect
123. C179#1597semidirect
129. #328D179semidirect

Maximal subgroups

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

subgroupindex
#260390382
#260385892
#260385752
#260383572
#260381722
#260381642
#328×C1792
#1597179