Finite Group Databasebeta

← #37507347

Subgroups — #37507347

Order 22912.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#19153098C211yesmaximal, characteristic
2#26038401C211yesmaximal, characteristic, complemented
3C22⋊C8×D179C211yesmaximal, characteristic, complemented
4#26038165C211yesmaximal, characteristic, complemented
5#26038162C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic, complemented
7#324×C179C211yesmaximal, characteristic, complemented
8C22×D716C411yescharacteristic
9C22×C4×D179C2211yescharacteristic
10C2×#15668242—21no—
11C2×C8×D179—21no—
12#19153098—21no—
13#19153098—21no—
14#19153098—21no—
15#19153098—21no—
16C2×Dic179⋊C4C411yescharacteristic
17C2×(C2×D179)⋊C4C411yescharacteristic
18C2×(C2×D179)⋊C4C411yescharacteristic
19C2×(C2×D179)⋊C4C2211yescharacteristic
20C22⋊(C179⋊C8)C2211yescharacteristic
21C22⋊(C179⋊C8)C2211yescharacteristic
22(C2×D179)⋊C8—21no—
23(C2×D179)⋊C8—21no—
24C2×C4⋊C4×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26C22⋊C8×C179C2211yescharacteristic
27C23×D179C2×C411yescharacteristic
28C23×D179C2×C411yescharacteristic
29C2×D716—21no—
30C2×D716—21no—
31C2×D716—42no—
32C22×Dic179C2×C411yescharacteristic
33C22×Dic179C2×C411yescharacteristic
34C2×C4×D179—42no—
35C2×C4×D179D411yescharacteristic
36C2×C4×D179D411yescharacteristic
37C2×C4×D179D411yescharacteristic
38C2×C4×D179D411yescharacteristic
39#15668242—42no—
40C2×C179⋊C8—21no—
41C2×C179⋊C8—21no—
42C8×D179—42no—
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—21no—
50(C2×D179)⋊C4—21no—
51C22×C4×C179C2×C411yescharacteristic
52C22×C4×C179C2×C411yescharacteristic
53C22×C4×C179C2311yescharacteristic
54C2×C8×C179—21no—
55C2×C8×C179—21no—
56C4⋊C4×C179—21no—
57C4⋊C4×C179—21no—
58C22×D179—21no—
59C22×D179—21no—
60C22×D179—22no—
61C22×D179—21no—
62C22×D179—21no—
63C22×D179—21no—
64C22×D179—41no—
65C22×D179—41no—
66C22×D179—41no—
67C22×D179C22⋊C411yescharacteristic
68C22×D179C22⋊C411yescharacteristic
69D716—42no—
70D716—42no—
71C2×Dic179—21no—
72C2×Dic179—41no—
73C2×Dic179—41no—
74C2×Dic179C22⋊C411yescharacteristic
75C2×Dic179C22⋊C411yescharacteristic
76C4×D179—22no—
77C4×D179—22no—
78C4×D179—41no—
79C4×D179—41no—
80C179⋊C8—41no—
81C179⋊C8—41no—
82C23×C179C22×C411yescharacteristic
83C2×C4×C179—21no—
84C2×C4×C179—21no—
85C2×C4×C179—21no—
86C2×C4×C179—21no—
87C2×C4×C179—41no—
88C2×C4×C179—41no—
89C2×C4×C179C2×D411yescharacteristic
90C2×C4×C179C2×D411yescharacteristic
91C8×C179—41no—
92C8×C179—41no—
93C22×C179—21no—
94C22×C179—21no—
95C22×C179C2×C22⋊C411yescharacteristic
96C22×C179#13111yescharacteristic
97C22×C179#13111yescharacteristic
98C2×D179—21no—
99C2×D179—21no—
100C2×D179—21no—
101C2×D179—21no—
102C2×D179—21no—
103C2×D179—42no—
104C2×D179—42no—
105C2×D179—41no—
106C2×D179—41no—
107C2×D179—41no—
108C2×D179—81no—
109C2×D179—81no—
110C2×D179#9912yes—
111C4×C179—21no—
112C4×C179—21no—
113C4×C179—41no—
114C4×C179—41no—
115Dic179—21no—
116Dic179—21no—
117Dic179—41no—
118C2×C179—21no—
119C2×C179—21no—
120C2×C179#40811yescharacteristic
121C2×C179C2×#9911yescharacteristic
122C2×C179#41311yescharacteristic
123D179—22no—
124D179—41no—
125D179—81no—
126C179#146811yescharacteristic, complemented, Sylow
127#1468—1791nomaximal, Sylow
128C2×#123—1791no—
129#407—1791no—
130C2×C22⋊C8—1791no—
131#324—1791no—
132#324D17911yescharacteristic, complemented
133#323—1791no—
134#323—1791no—
135C22×D4—1791no—
136C23×C4—1791no—
137C2×M16—3581no—
138C22×C8—3581no—
139#123—3581no—
140#123—3581no—
141#123—3581no—
142#123—3581no—
143C2×C4⋊C4—1791no—
144C2×C4⋊C4C2×D17911yescharacteristic
145C2×C22⋊C4—1791no—
146C2×C22⋊C4—1791no—
147C2×C22⋊C4—1791no—
148C22⋊C8—3581no—
149C22⋊C8—3581no—
150C22⋊C8—1791no—
151C22⋊C8—1791no—
152C22⋊C8C2×D17911yescharacteristic
153C22⋊C8C2×D17911yescharacteristic
154C24—1791no—
155C24—1791no—
156C2×D4—3581no—
157C2×D4—3581no—
158C2×D4—7162no—
159C22×C4—1791no—
160C22×C4—1791no—
161C22×C4—1791no—
162C22×C4—1791no—
163C22×C4—1791no—
164C22×C4—1791no—
165C22×C4—7162no—
166C22×C4C4×D17911yescharacteristic
167C22×C4C4×D17911yescharacteristic
168C22×C4C22×D17911yescharacteristic
169M16—7162no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—3581no—
173C2×C8—3581no—
174C2×C8—7162no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3581no—
178C4⋊C4—3581no—
179C22⋊C4—3581no—
180C22⋊C4—3581no—
181C22⋊C4—3581no—
182C22⋊C4—3581no—
183C22⋊C4—3581no—
184C22⋊C4—3581no—
185C23—3582no—
186C23—3581no—
187C23—3581no—
188C23—3581no—
189C23—3581no—
190C23—3581no—
191C23—1791no—
192C23—1791no—
193C23—7161no—
194C23—7161no—
195C23—7161no—
196C23C2×C4×D17911yescharacteristic
197D4—7162no—
198D4—7162no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—3582no—
206C2×C4—3582no—
207C2×C4—3581no—
208C2×C4—1791no—
209C2×C4—1791no—
210C2×C4—7161no—
211C2×C4—7161no—
212C2×C4—7161no—
213C2×C4—7161no—
214C2×C4D4×D17911yescharacteristic
215C2×C4D4×D17911yescharacteristic
216C8—41no—
217C8—41no—
218C8—7161no—
219C8—7161no—
220C22—21no—
221C22—21no—
222C22—3581no—
223C22—3581no—
224C22—3581no—
225C22—3581no—
226C22—3581no—
227C22—14321no—
228C22—14321no—
229C22—1792no—
230C22—7162no—
231C22—7162no—
232C22—7161no—
233C22—7161no—
234C22—7161no—
235C22C22⋊C4×D17911yescharacteristic
236C22#1915311911yescharacteristic
237C22#1915312111yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—3581no—
243C4—3581no—
244C4—7161no—
245C2—21no—
246C2—21no—
247C2—3582no—
248C2—14321no—
249C2—7161no—
250C2#2603840411yescharacteristic
251C2#99×D17911yescharacteristic
252C2#2603842211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D179
commutatorC23×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179C2×C22⋊C4
prime radicalC22×D716C4
cosocleC22×C4×C179C23
hypercenter#324D179
hypocenterC179#1468
LCSG ▷ C23×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26038401C2semidirect
3. C22⋊C8×D179C2semidirect
4. #26038165C2semidirect
5. #26038162C2semidirect
6. #26038162C2semidirect
7. #324×C179C2semidirect
126. C179#1468semidirect
132. #324D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191530982
#260384012
C22⋊C8×D1792
#260381652
#260381622
#324×C1792
#1468179