Finite Group Databasebeta

← #36356832

Subgroups — #36356832

Order 22144.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#19149644C211yesmaximal, characteristic, complemented
2C2×#19149635C211yesmaximal, characteristic
3#26011429C211yesmaximal, characteristic, complemented
4#26011422C212yesmaximal, complemented
5#26011414C212yesmaximal
6#26011412C211yesmaximal, characteristic, complemented
7#26011403C211yesmaximal, characteristic
8C4⋊C4×Dic173C212yesmaximal, complemented
9#26011342C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic173C212yesmaximal
11#399×C173C211yesmaximal, characteristic
12C22×Dic346C2211yescharacteristic
13C23×Dic173C2211yescharacteristic
14#19149644—24no—
15#19149635—24no—
16C2×C4⋊Dic173C2212yes—
17C2×#14583805C2211yescharacteristic
18C2×#14583805C2212yes—
19C2×#14583805C2212yes—
20C2×#14583805C2211yescharacteristic
21C2×C22⋊Dic173C2212yes—
22C2×C22⋊Dic173C2211yescharacteristic
23C2×C4×Dic173C2211yescharacteristic
24C2×C4×Dic173C2212yes—
25C2×C4×Dic173C2212yes—
26C2×C4×Dic173C2211yescharacteristic
27#19149586C2212yes—
28#19149586C2211yescharacteristic
29#19149586C2212yes—
30#19149586C2212yes—
31#19149586C2212yes—
32C2×C4⋊C4×C173C2212yes—
33C2×C4⋊C4×C173C2211yescharacteristic
34C2×C22⋊C4×C173C2211yescharacteristic
35C2×C22⋊C4×C173C2212yes—
36#95×C173C2211yescharacteristic
37C2×Dic346—24no—
38C2×Dic346—22no—
39C22×Dic173—22no—
40C22×Dic173—22no—
41C22×Dic173Q814yescomplemented
42C22×Dic173C2312yes—
43C22×Dic173C2311yescharacteristic
44C22×Dic173C2311yescharacteristic
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2312yes—
47C22×Dic173C2311yescharacteristic
48C4⋊Dic173—24no—
49#14583805—22no—
50#14583805—24no—
51#14583805—22no—
52#14583805—24no—
53C22⋊Dic173—22no—
54C22⋊Dic173—24no—
55C4×Dic173—22no—
56C4×Dic173—24no—
57C4×Dic173—24no—
58C4×Dic173D414yes—
59C24×C173C2311yescharacteristic
60C22×C4×C173C2311yescharacteristic
61C22×C4×C173C2312yes—
62C22×C4×C173C2311yescharacteristic
63C22×C4×C173C2312yes—
64C4⋊C4×C173—22no—
65C4⋊C4×C173—24no—
66C22⋊C4×C173—22no—
67C22⋊C4×C173—24no—
68Dic346—28no—
69C2×Dic173—21no—
70C2×Dic173—22no—
71C2×Dic173—21no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—28no—
75C2×Dic173—21no—
76C2×Dic173—22no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173C2×D412yes—
81C2×Dic173C2×D412yes—
82C2×Dic173C2×Q812yes—
83C2×Dic173C2×Q812yes—
84C2×Dic173#4114yes—
85C2×Dic173#4112yes—
86C23×C173—21no—
87C23×C173—21no—
88C23×C173—22no—
89C23×C173—22no—
90C23×C173C2×Q812yes—
91C23×C173C2411yescharacteristic
92C2×C4×C173—22no—
93C2×C4×C173—22no—
94C2×C4×C173—22no—
95C2×C4×C173—21no—
96C2×C4×C173—22no—
97C2×C4×C173—22no—
98C2×C4×C173—22no—
99C2×C4×C173—22no—
100C2×C4×C173C2×D412yes—
101C2×C4×C173#4114yes—
102C22×C173—22no—
103C22×C173—24no—
104C22×C173—41no—
105C22×C173—42no—
106C22×C173—41no—
107C22×C173C22×D411yescharacteristic
108C22×C173C22×Q811yescharacteristic
109C22×C173C2×#4111yescharacteristic
110C22×C173C2×#4112yes—
111C22×C173C2×#4112yes—
112C4×C173—22no—
113C4×C173—24no—
114C4×C173—41no—
115C4×C173—42no—
116Dic173—22no—
117Dic173—24no—
118Dic173—22no—
119Dic173—41no—
120Dic173—42no—
121Dic173C22⋊Q814yes—
122C2×C173—42no—
123C2×C173C2×C22⋊Q811yescharacteristic
124C2×C173#52511yescharacteristic
125C2×C173#52912yes—
126C2×C173#53211yescharacteristic
127C2×C173#53312yes—
128C173#239611yescharacteristic, complemented, Sylow
129#2396—1731nomaximal, Sylow
130C2×C4×Q8—1731no—
131C2×#117—1731no—
132#400—1731no—
133#399D17311yescharacteristic
134#398—1732no—
135#396—1732no—
136#395—1731no—
137#393—1731no—
138C4×C4⋊C4—1732no—
139C4×C4⋊C4—1731no—
140C4×C22⋊C4—1732no—
141C22×Q8—1731no—
142C23×C4—1731no—
143C4×Q8—3464no—
144#117—3464no—
145C2×C4⋊C4—1731no—
146C2×C4⋊C4—1731no—
147C2×C4⋊C4—1732no—
148C2×C4⋊C4—1732no—
149C2×C4⋊C4—1732no—
150C2×C4⋊C4C2×D17311yescharacteristic
151C2×C4⋊C4C2×D17312yes—
152C2×C22⋊C4—1731no—
153C2×C22⋊C4—1732no—
154C2×C22⋊C4C2×D17311yescharacteristic
155C2×C22⋊C4C2×D17312yes—
156C2×C42—1731no—
157C2×C42—1732no—
158C2×C42—1732no—
159C2×C42—1731no—
160#95—1732no—
161#95—1732no—
162#95—1732no—
163#95—1731no—
164#95—1732no—
165#95C2×D17311yescharacteristic
166C24C22×D17311yescharacteristic
167C2×Q8—3464no—
168C2×Q8—3462no—
169C22×C4—3462no—
170C22×C4—3462no—
171C22×C4—1732no—
172C22×C4—1731no—
173C22×C4—1734no—
174C22×C4—1731no—
175C22×C4—1732no—
176C22×C4—1731no—
177C22×C4—1731no—
178C22×C4C22×D17311yescharacteristic
179C22×C4C22×D17312yes—
180C22×C4C22×D17311yescharacteristic
181C22×C4C22×D17312yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—3462no—
185C4⋊C4—3464no—
186C4⋊C4—3464no—
187C4⋊C4—3462no—
188C4⋊C4—3464no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—3462no—
192C22⋊C4—3464no—
193C42—3462no—
194C42—3464no—
195C42—3464no—
196C42—1734no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D17312yes—
202C23C23×D17311yescharacteristic
203Q8—3468no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—3462no—
213C2×C4—3461no—
214C2×C4—3461no—
215C2×C4—3462no—
216C2×C4—3462no—
217C2×C4—3468no—
218C2×C4—3461no—
219C2×C4—3462no—
220C2×C4—3462no—
221C2×C4—3462no—
222C2×C4—3462no—
223C2×C4—1732no—
224C2×C4—1732no—
225C2×C4—1734no—
226C2×C4—1732no—
227C2×C4—1732no—
228C2×C4—1732no—
229C2×C4D4×D17312yes—
230C2×C4#1458382514yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D17311yescharacteristic
237C22C2×Q8×D17311yescharacteristic
238C22#41×D17311yescharacteristic
239C22#41×D17312yes—
240C22C2×#1458382512yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—3462no—
246C4—3464no—
247C4—3462no—
248C4—1734no—
249C4—6921no—
250C4—6922no—
251C2—42no—
252C2C22⋊Q8×D17311yescharacteristic
253C2#2601199911yescharacteristic
254C2#2601201812yes—
255C2#2601203911yescharacteristic
256C2#2601204312yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#399×C173C2
prime radicalC24×C173C23
hypercenter#399D173
hypocenterC173#2396
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#19149644C2semidirect
3. #26011429C2semidirect
4. #26011422C2semidirect
6. #26011412C2semidirect
8. C4⋊C4×Dic173C2semidirect
9. #26011342C2semidirect
41. C22×Dic173Q8semidirect
128. C173#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#191496442
C2×#191496352
#260114292
#260114222
#260114142
#260114122
#260114032
C4⋊C4×Dic1732
#260113422
C22⋊C4×Dic1732
#399×C1732
#2396173