Finite Group Databasebeta

← #36369586

Subgroups — #36369586

Order 22144.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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×#19149758C211yesmaximal, characteristic, complemented
2C2×#19149737C212yesmaximal
3#26012434C211yesmaximal, characteristic, complemented
4#26012369C212yesmaximal, complemented
5C2×#19149661C211yesmaximal, characteristic
6#19149758—24no—
7C2×C4⋊#10621924—21no—
8C2×C22⋊#10621924C2212yes—
9C2×C22⋊#10621924C2212yes—
10C2×C4×#10621924—21no—
11#19149737—24no—
12(C2×#10621924)⋊C4—21no—
13(C2×#10621924)⋊C4—21no—
14C22×C22⋊D173C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16#19149661—21no—
17#19149661—21no—
18#19149661—21no—
19#19149661—21no—
20C2×C4⋊Dic173C411yescharacteristic, complemented
21C2×#14583805C411yescharacteristic, complemented
22C2×C22⋊Dic173C411yescharacteristic, complemented
23C2×#14583803C2211yescharacteristic
24C2×C22⋊C4×C173C411yescharacteristic, complemented
25C22×#10621924—22no—
26C22×#10621924—22no—
27C4⋊#10621924—24no—
28C22⋊#10621924—22no—
29C22⋊#10621924—22no—
30C22⋊#10621924—22no—
31C22⋊#10621924D414yes—
32C4×#10621924—24no—
33C23×D173C2311yescharacteristic
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—22no—
37C2×C22⋊D173—22no—
38C22×Dic173C2×C411yescharacteristic
39C22×Dic173C2×C411yescharacteristic
40C22×Dic173C2×C411yescharacteristic
41C2×C4×D173—22no—
42C2×C4×D173—22no—
43C2×C4×D173D411yescharacteristic
44C2×C4×D173D411yescharacteristic
45C2×C4×D173D411yescharacteristic
46C2×C4×D173D411yescharacteristic
47C4⋊Dic173—21no—
48C4⋊Dic173—21no—
49#14583805—21no—
50#14583805—21no—
51C22⋊Dic173—21no—
52C22⋊Dic173—21no—
53#14583803—21no—
54#14583803—21no—
55C24×C173C2×C411yescharacteristic
56C22×C4×C173C2×C411yescharacteristic
57C22×C4×C173C2×C411yescharacteristic
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C2×#10621924—24no—
61C2×#10621924—42no—
62C2×#10621924—42no—
63C2×#10621924—42no—
64C2×#10621924—42no—
65C2×#10621924—42no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173C2×D411yescharacteristic
69C22×D173C2×D411yescharacteristic
70C22×D173C2×D412yes—
71C22×D173#4112yes—
72C22⋊D173—41no—
73C22⋊D173—42no—
74C22⋊D173—41no—
75C2×Dic173—22no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—22no—
79C2×Dic173—41no—
80C2×Dic173C22⋊C411yescharacteristic
81C2×Dic173C22⋊C411yescharacteristic
82C4×D173—21no—
83C4×D173—21no—
84C4×D173—22no—
85C4×D173—22no—
86C4×D173—21no—
87C4×D173—21no—
88C23×C173—21no—
89C23×C173—22no—
90C23×C173—41no—
91C23×C173—41no—
92C23×C173C22×C411yescharacteristic
93C2×C4×C173—22no—
94C2×C4×C173—21no—
95C2×C4×C173—41no—
96C2×C4×C173C22⋊C411yescharacteristic
97C2×C4×C173C22⋊C411yescharacteristic
98C22×C173—21no—
99C22×C173—21no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173—82no—
104C22×C173C2×C22⋊C411yescharacteristic
105C22×C173C4×D412yes—
106C2×D173—21no—
107C2×D173—21no—
108C2×D173—21no—
109C2×D173—21no—
110C2×D173—42no—
111C2×D173—42no—
112C2×D173#12011yescharacteristic
113C2×D173#12112yes—
114C2×D173#12311yescharacteristic
115#10621924—44no—
116#10621924—82no—
117C4×C173—21no—
118C4×C173—21no—
119C4×C173—41no—
120Dic173—21no—
121Dic173—21no—
122Dic173—41no—
123Dic173—41no—
124C2×C173—22no—
125C2×C173—81no—
126C2×C173#38611yescharacteristic
127C2×C173#41011yescharacteristic
128C2×C173#41011yescharacteristic
129D173—21no—
130D173—22no—
131D173—21no—
132C173#189911yescharacteristic, complemented, Sylow
133#1899—1731nomaximal, Sylow
134C2×#123—1731no—
135C2×#117—1731no—
136C2×#99—1732no—
137#381—1731no—
138#342—1732no—
139C22×D4—1731no—
140C23×C4—1731no—
141#123—3461no—
142#123—3461no—
143#123—3461no—
144#123—3461no—
145#117—3464no—
146C2×C4⋊C4—3461no—
147C2×C4⋊C4—1731no—
148C2×C4⋊C4—1731no—
149C2×C22⋊C4—1732no—
150C2×C22⋊C4—1731no—
151C2×C22⋊C4—1731no—
152C2×C22⋊C4—1732no—
153C2×C22⋊C4#1062192411yescharacteristic, complemented
154C2×C42—3461no—
155#99—3464no—
156#95—3461no—
157#95—3461no—
158C24—1731no—
159C24C2×#1062192411yescharacteristic
160C2×D4—3462no—
161C2×D4—3461no—
162C2×D4—3461no—
163C2×D4—3462no—
164C22×C4—3462no—
165C22×C4—3462no—
166C22×C4—3462no—
167C22×C4—3462no—
168C22×C4—1731no—
169C22×C4—1731no—
170C22×C4—1731no—
171C22×C4—1731no—
172C22×C4—1731no—
173C22×C4—1731no—
174C22×C4—1731no—
175C22×C4C2×#1062192411yescharacteristic
176C22×C4C2×#1062192411yescharacteristic
177C4⋊C4—3461no—
178C4⋊C4—3461no—
179C4⋊C4—3464no—
180C4⋊C4—3461no—
181C4⋊C4—3461no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—3462no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3462no—
190C22⋊C4—3462no—
191C22⋊C4—1734no—
192C42—3464no—
193C23—21no—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—1731no—
198C23—1731no—
199C23—1732no—
200C23—1732no—
201C23—6921no—
202C23—6921no—
203C23C22×#1062192411yescharacteristic
204D4—6921no—
205D4—6922no—
206D4—6921no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—3461no—
211C2×C4—3461no—
212C2×C4—3462no—
213C2×C4—3462no—
214C2×C4—3461no—
215C2×C4—3461no—
216C2×C4—3462no—
217C2×C4—3464no—
218C2×C4—3461no—
219C2×C4—3461no—
220C2×C4—3462no—
221C2×C4—1731no—
222C2×C4—1731no—
223C2×C4—6922no—
224C2×C4—6922no—
225C2×C4—6922no—
226C2×C4—6922no—
227C2×C4—6922no—
228C2×C4—6921no—
229C2×C4C22⋊#1062192411yescharacteristic
230C2×C4C22⋊#1062192411yescharacteristic
231C22—21no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—82no—
237C22—3461no—
238C22—3461no—
239C22—3461no—
240C22—3461no—
241C22—1731no—
242C22—1732no—
243C22—1731no—
244C22—6922no—
245C22—6922no—
246C22C2×C22⋊#1062192411yescharacteristic
247C22D4×#1062192412yes—
248C4—21no—
249C4—21no—
250C4—41no—
251C4—3461no—
252C4—3461no—
253C4—13842no—
254C4—6924no—
255C4—6921no—
256C4—6921no—
257C2—22no—
258C2—81no—
259C2—3462no—
260C2—3461no—
261C2—3461no—
262C2#2601243911yescharacteristic
263C2#2601247011yescharacteristic
264C2#2601247111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C22
cosocleC23×D173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1899
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19149758C2semidirect
3. #26012434C2semidirect
4. #26012369C2semidirect
20. C2×C4⋊Dic173C4semidirect
21. C2×#14583805C4semidirect
22. C2×C22⋊Dic173C4semidirect
24. C2×C22⋊C4×C173C4semidirect
132. C173#1899semidirect
153. C2×C22⋊C4#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497582
C2×#191497372
#260124342
#260123692
C2×#191496612
#1899173