Finite Group Databasebeta

← #36354020

Subgroups — #36354020

Order 22144.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19149663C211yesmaximal, characteristic
2C2×#19149635C211yesmaximal, characteristic, complemented
3C2×#19149593C211yesmaximal, characteristic
4C2×#19149593C211yesmaximal, characteristic
5#26011352C211yesmaximal, characteristic, complemented
6#26011258C211yesmaximal, characteristic, complemented
7#342×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9C23×Dic173C2211yescharacteristic
10#19149663—22no—
11#19149663—21no—
12#19149663—21no—
13#19149635—22no—
14#19149635—22no—
15C2×C4⋊Dic173C412yescomplemented
16C2×#14583805—21no—
17C2×C22⋊Dic173C2211yescharacteristic
18C2×C22⋊Dic173C2211yescharacteristic
19C2×#14583803C412yescomplemented
20C2×C4×Dic173—21no—
21#19149593—22no—
22#19149593—22no—
23#19149586—22no—
24C2×C22⋊C4×C173C2211yescharacteristic
25C2×C22⋊C4×C173C2211yescharacteristic
26C2×C22⋊C4×C173C2211yescharacteristic
27C23×D173C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C22×Dic173—21no—
35C22×Dic173—21no—
36C22×Dic173—21no—
37C22×Dic173—21no—
38C22×Dic173—21no—
39C22×Dic173—21no—
40C22×Dic173C2×C411yescharacteristic
41C22×Dic173C2×C412yes—
42C22×Dic173D412yes—
43C22×Dic173D411yescharacteristic
44C22×Dic173D411yescharacteristic
45C4⋊Dic173—22no—
46C4⋊Dic173—22no—
47#14583805—24no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173D412yes—
52#14583803—22no—
53#14583803—22no—
54C4×Dic173—24no—
55C24×C173C2311yescharacteristic
56C22×C4×C173—21no—
57C22×C4×C173—21no—
58C22×C4×C173C2×C412yes—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173—21no—
62C22⋊C4×C173—21no—
63C22⋊C4×C173—21no—
64C22⋊C4×C173D412yes—
65C22×D173—21no—
66C22×D173—21no—
67C22×D173—21no—
68C22×D173—42no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C22⋊D173—41no—
72C22⋊D173—41no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—21no—
80C2×Dic173—21no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—21no—
84C2×Dic173—22no—
85C2×Dic173—42no—
86C2×Dic173—42no—
87C2×Dic173—41no—
88C2×Dic173C22⋊C412yes—
89C2×Dic173C22⋊C412yes—
90C23×C173—42no—
91C23×C173C22×C411yescharacteristic
92C23×C173C2×D411yescharacteristic
93C23×C173C2×D411yescharacteristic
94C23×C173C2×D411yescharacteristic
95C23×C173C2×D411yescharacteristic
96C23×C173#4111yescharacteristic
97C23×C173#4111yescharacteristic
98C2×C4×C173—22no—
99C2×C4×C173—22no—
100C2×C4×C173—42no—
101C2×C4×C173—42no—
102C2×C4×C173—42no—
103C2×C4×C173—41no—
104C22×C173—22no—
105C22×C173—22no—
106C22×C173—22no—
107C22×C173—41no—
108C22×C173—41no—
109C22×C173—41no—
110C22×C173—41no—
111C22×C173C2×C22⋊C411yescharacteristic
112C22×C173C4×D411yescharacteristic
113C22×C173C4×D411yescharacteristic
114C22×C173#12011yescharacteristic
115C22×C173#12111yescharacteristic
116C22×C173#12111yescharacteristic
117C22×C173#12311yescharacteristic
118C2×D173—41no—
119C2×D173—42no—
120C2×D173—81no—
121C2×D173—81no—
122C4×C173—42no—
123C4×C173—42no—
124C4×C173—81no—
125Dic173—22no—
126Dic173—22no—
127Dic173—42no—
128Dic173—41no—
129Dic173—81no—
130C2×C173—21no—
131C2×C173—21no—
132C2×C173—21no—
133C2×C173—21no—
134C2×C173—21no—
135C2×C173—21no—
136C2×C173#38611yescharacteristic
137C2×C173#41012yes—
138D173—81no—
139C173#189911yescharacteristic, complemented, Sylow
140#1899—1731nomaximal, Sylow
141C2×#123—1731no—
142C2×#117—1731no—
143C2×#99—1731no—
144C2×#99—1731no—
145#381—1731no—
146#342—1731no—
147#342D17311yescharacteristic, complemented
148C22×D4—1731no—
149C23×C4—1731no—
150#123—3462no—
151#123—3461no—
152#123—3461no—
153#117—3462no—
154#117—3462no—
155C2×C4⋊C4—3461no—
156C2×C4⋊C4—1732no—
157C2×C22⋊C4—1731no—
158C2×C22⋊C4—1732no—
159C2×C22⋊C4—1731no—
160C2×C22⋊C4C2×D17311yescharacteristic
161C2×C22⋊C4C2×D17311yescharacteristic
162C2×C22⋊C4C2×D17311yescharacteristic
163C2×C42—3461no—
164#99—3462no—
165#99—3462no—
166#95—3462no—
167C24—1731no—
168C24C22×D17311yescharacteristic
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—3461no—
173C2×D4—3461no—
174C2×D4—3461no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—3461no—
178C22×C4—3461no—
179C22×C4—3461no—
180C22×C4—3461no—
181C22×C4—3461no—
182C22×C4—3461no—
183C22×C4—1731no—
184C22×C4—1732no—
185C22×C4—1732no—
186C22×C4—1731no—
187C22×C4—1731no—
188C22×C4C4×D17312yes—
189C4⋊C4—3464no—
190C4⋊C4—3462no—
191C4⋊C4—3462no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—3462no—
198C22⋊C4—3462no—
199C22⋊C4—3461no—
200C22⋊C4—3461no—
201C22⋊C4—3461no—
202C22⋊C4—1732no—
203C22⋊C4D69212yes—
204C42—3464no—
205C23—42no—
206C23—3461no—
207C23—3461no—
208C23—3461no—
209C23—6922no—
210C23C2×C4×D17311yescharacteristic
211C23D4×D17311yescharacteristic
212C23D4×D17311yescharacteristic
213C23D4×D17311yescharacteristic
214C23C2×D69211yescharacteristic
215C23#1458382611yescharacteristic
216C23#1458382711yescharacteristic
217D4—6921no—
218D4—6921no—
219D4—6921no—
220D4—6921no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—3462no—
228C2×C4—3462no—
229C2×C4—3461no—
230C2×C4—3461no—
231C2×C4—3462no—
232C2×C4—3462no—
233C2×C4—3461no—
234C2×C4—3461no—
235C2×C4—3461no—
236C2×C4—3461no—
237C2×C4—3461no—
238C2×C4—3462no—
239C2×C4—1732no—
240C2×C4—1732no—
241C2×C4—6922no—
242C2×C4—6921no—
243C2×C4—6922no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—13841no—
252C22—13841no—
253C22—6922no—
254C22—6921no—
255C22C22⋊C4×D17311yescharacteristic
256C22#1914964111yescharacteristic
257C22C4×D69211yescharacteristic
258C22#1914964911yescharacteristic
259C22#1914965211yescharacteristic
260C22#1914965411yescharacteristic
261C22#1914966411yescharacteristic
262C4—42no—
263C4—42no—
264C4—81no—
265C4—3462no—
266C4—3462no—
267C4—13841no—
268C4—6922no—
269C4—6921no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—13841no—
277C2#2601136911yescharacteristic
278C2#2601146312yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC23×C173C22×C4
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C22
cosocleC24×C173C23
hypercenter#342D173
hypocenterC173#1899
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#19149635C2semidirect
5. #26011352C2semidirect
6. #26011258C2semidirect
7. #342×C173C2semidirect
15. C2×C4⋊Dic173C4semidirect
19. C2×#14583803C4semidirect
139. C173#1899semidirect
147. #342D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496632
C2×#191496352
C2×#191495932
#260113522
#260112582
#342×C1732
#1899173