Finite Group Databasebeta

← #37509542

Subgroups — #37509542

Order 22912.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#26038998C211yesmaximal, characteristic
2C2×#19153028C211yesmaximal, characteristic, complemented
3C2×#19153027C212yesmaximal, complemented
4#26038205C212yesmaximal, complemented
5#518×C179C211yesmaximal, characteristic, complemented
6C22×C22⋊D179C2211yescharacteristic
7#19153096—21no—
8#19153094C412yescomplemented
9#19153085—22no—
10#19153082—21no—
11#19153081C412yescomplemented
12C2×C22⋊Dic179C2212yes—
13C2×C22⋊Dic179C2211yescharacteristic
14#19153028—22no—
15#19153027—24no—
16C22×D4×C179C2211yescharacteristic
17C4×D4×C179—24no—
18#117×C179—21no—
19C2×C22⋊C4×C179—21no—
20C2×C22⋊C4×C179C2212yes—
21C23×D179C2×C411yescharacteristic, complemented
22C2×C22⋊D179—22no—
23C2×C22⋊D179—22no—
24C2×C22⋊D179—21no—
25C2×C22⋊D179—21no—
26C2×C22⋊D179—22no—
27C22×Dic179—21no—
28C22×Dic179—22no—
29C22×Dic179C2×C411yescharacteristic, complemented
30Dic179⋊C4—22no—
31C22⋊Dic179—22no—
32C22⋊Dic179—21no—
33C22⋊Dic179—22no—
34C22⋊Dic179—22no—
35C22⋊Dic179C2×C412yescomplemented
36(C2×D179)⋊C4—22no—
37C24×C179D411yescharacteristic
38C24×C179C2311yescharacteristic
39C2×D4×C179—22no—
40C2×D4×C179—22no—
41C2×D4×C179C2×C412yes—
42C2×D4×C179D412yes—
43C22×C4×C179—22no—
44C22×C4×C179—22no—
45C22×C4×C179D411yescharacteristic
46C4⋊C4×C179—22no—
47C22⋊C4×C179—22no—
48C22⋊C4×C179—22no—
49C22⋊C4×C179D414yescomplemented
50C42×C179—22no—
51C22×D179—21no—
52C22×D179—22no—
53C22×D179—81no—
54C22⋊D179—41no—
55C22⋊D179—42no—
56C22⋊D179—41no—
57C22⋊D179—82no—
58C2×Dic179—22no—
59C2×Dic179—21no—
60C2×Dic179—22no—
61C2×Dic179—42no—
62C2×Dic179—81no—
63C2×Dic179—82no—
64C23×C179—21no—
65C23×C179—21no—
66C23×C179—81no—
67C23×C179—81no—
68C23×C179C22⋊C412yes—
69C23×C179C22×C411yescharacteristic
70C23×C179C2×D411yescharacteristic
71C23×C179C2×D412yes—
72C23×C179C2×D411yescharacteristic
73C23×C179#4112yes—
74D4×C179—24no—
75D4×C179—24no—
76C2×C4×C179—21no—
77C2×C4×C179—21no—
78C2×C4×C179—24no—
79C2×C4×C179—82no—
80C2×C4×C179—82no—
81C2×C4×C179C22⋊C412yes—
82C22×C179—22no—
83C22×C179—42no—
84C22×C179—41no—
85C22×C179—41no—
86C22×C179—41no—
87C22×C179—42no—
88C22×C179—42no—
89C22×C179—42no—
90C22×C179—81no—
91C22×C179C2×C22⋊C411yescharacteristic
92C22×C179C4×D412yes—
93C22×C179#12011yescharacteristic
94C22×C179#12112yes—
95C22×C179#12311yescharacteristic
96C2×D179—42no—
97C2×D179—41no—
98C2×D179—82no—
99C4×C179—22no—
100C4×C179—44no—
101Dic179—42no—
102Dic179—41no—
103Dic179—82no—
104C2×C179—21no—
105C2×C179—22no—
106C2×C179—21no—
107C2×C179—22no—
108C2×C179—21no—
109C2×C179—42no—
110C2×C179#38611yescharacteristic
111D179—81no—
112C179#191311yescharacteristic, complemented, Sylow
113#1913—1791nomaximal, Sylow
114#534—1791no—
115#518D17911yescharacteristic, complemented
116C2×#99—1791no—
117C2×#99—1792no—
118#342—1792no—
119C22×D4—1791no—
120C22×D4C2×D17911yescharacteristic
121#123—3581no—
122#123—1792no—
123#121—3582no—
124#120—3581no—
125#120—1792no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—1792no—
130C2×C22⋊C4—1791no—
131C2×C22⋊C4C2×D17912yes—
132#99—3582no—
133#99—3584no—
134C24—1791no—
135C24C22⋊D17911yescharacteristic
136C24C22×D17911yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—3582no—
140C2×D4—3582no—
141C2×D4—3581no—
142C2×D4—3581no—
143C2×D4—3582no—
144C2×D4C4×D17912yes—
145C2×D4D71612yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—3581no—
149C22×C4—3582no—
150C22×C4—1791no—
151C22×C4C22⋊D17911yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—3582no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—3582no—
157C22⋊C4—3581no—
158C22⋊C4—3582no—
159C22⋊C4—3582no—
160C22⋊C4—3582no—
161C22⋊C4—1792no—
162C22⋊C4C22⋊D17914yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—3582no—
169C23—3581no—
170C23—14321no—
171C23(C2×D179)⋊C412yes—
172C23C2×C4×D17911yescharacteristic
173C23C2×C22⋊D17911yescharacteristic
174C23C2×C22⋊D17912yes—
175C23C2×D71611yescharacteristic
176C23#1566825912yes—
177D4—24no—
178D4—24no—
179D4—14322no—
180D4—7161no—
181D4—7162no—
182D4—7161no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—3582no—
189C2×C4—3581no—
190C2×C4—3582no—
191C2×C4—14322no—
192C2×C4—14321no—
193C2×C4—7162no—
194C2×C4(C2×D179)⋊C412yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—14322no—
205C22—7162no—
206C22—7161no—
207C22C2×(C2×D179)⋊C411yescharacteristic
208C22C4×C22⋊D17912yes—
209C22#1915308211yescharacteristic
210C22#1915308712yes—
211C22#1915309611yescharacteristic
212C4—22no—
213C4—44no—
214C4—14322no—
215C4—7162no—
216C4—7161no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—14321no—
224C2#2603831811yescharacteristic
subgroupG/H
centerC2#26038318
commutatorC23×C179C22×C4
FrattiniC24C22×D179
Fitting#518×C179C2
socleC2×C179#386
prime radical#26038998C2
cosocleC24×C179C23
hypercenter#518D179
hypocenterC179#1913
LCSG ▷ C23×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#19153028C2semidirect
3. C2×#19153027C2semidirect
4. #26038205C2semidirect
5. #518×C179C2semidirect
8. #19153094C4semidirect
11. #19153081C4semidirect
21. C23×D179C2×C4semidirect
29. C22×Dic179C2×C4semidirect
35. C22⋊Dic179C2×C4semidirect
49. C22⋊C4×C179D4semidirect
112. C179#1913semidirect
115. #518D179semidirect
162. C22⋊C4C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260389982
C2×#191530282
C2×#191530272
#260382052
#518×C1792
#1913179