Finite Group Databasebeta

← #36369575

Subgroups — #36369575

Order 22144.

342 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012569C211yesmaximal, characteristic, complemented
2#26012478C211yesmaximal, characteristic, complemented
3#26012463C212yesmaximal, complemented
4#26012372C212yesmaximal, complemented
5#26011999C211yesmaximal, characteristic
6#19149766—24no—
7D173⋊M16—21no—
8C2×#14583835—21no—
9C2×#14583835C2212yes—
10C2×#14583835C2212yes—
11#19149740—21no—
12#19149739—21no—
13#19149736—24no—
14C2×#14583825C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16#19149662C411yescharacteristic
17#19149657—21no—
18#19149652—21no—
19#19149649C411yescharacteristic
20#19149647C411yescharacteristic
21#19149641—21no—
22#19149639—21no—
23#19149635C2211yescharacteristic
24#123×C173C411yescharacteristic
25#14583836—22no—
26#14583835—22no—
27#14583835—22no—
28#14583835—22no—
29#14583835—22no—
30#14583835—22no—
31D173⋊C8—22no—
32C2×#13063443—22no—
33C2×#13063443—22no—
34C2×#13063443D414yescomplemented
35C23×D173C2×C411yescharacteristic
36#14583825—22no—
37#14583825—22no—
38C2×Dic346D411yescharacteristic
39C2×D692—21no—
40C2×C22⋊D173—21no—
41C2×C22⋊D173C2×C411yescharacteristic
42C2×C22⋊D173D411yescharacteristic
43C22×Dic173D411yescharacteristic
44C22×Dic173C2311yescharacteristic
45C2×C4×D173—21no—
46C2×C4×D173—21no—
47C2×C4×D173—22no—
48C2×C4×D173—22no—
49C2×C4×D173D411yescharacteristic
50#14583805—21no—
51#14583805—21no—
52C22⋊Dic173C2×C411yescharacteristic
53#14583803—21no—
54#14583803—21no—
55#14583803—21no—
56C4×Dic173—21no—
57C2×D4×C173C2×C411yescharacteristic
58C22×C4×C173C2×C411yescharacteristic
59C4⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173C2×C411yescharacteristic
62#13063443—24no—
63#13063443—42no—
64C22×D173—21no—
65C22×D173—22no—
66C22×D173—42no—
67C22×D173C22⋊C411yescharacteristic
68Dic346—22no—
69D692—42no—
70C22⋊D173—22no—
71C22⋊D173—22no—
72C22⋊D173—42no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173C22⋊C411yescharacteristic
77C2×Dic173C2×D412yes—
78C2×Dic173C2×D411yescharacteristic
79C2×Dic173C2×D411yescharacteristic
80C2×Dic173#4112yes—
81C4×D173—21no—
82C4×D173—21no—
83C4×D173—41no—
84C4×D173—42no—
85C4×D173—41no—
86C23×C173C22⋊C411yescharacteristic
87C23×C173C22×C411yescharacteristic
88D4×C173—22no—
89C2×C4×C173—21no—
90C2×C4×C173—21no—
91C2×C4×C173—22no—
92C2×C4×C173C22⋊C411yescharacteristic
93C22×C173—21no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173C2×C22⋊C411yescharacteristic
97C22×C173C4×D412yes—
98C2×D173—22no—
99C2×D173—24no—
100C2×D173—21no—
101C2×D173—41no—
102C2×D173—42no—
103C2×D173—41no—
104C2×D173—41no—
105C4×C173—21no—
106C4×C173—41no—
107C4×C173—41no—
108Dic173—21no—
109Dic173—41no—
110Dic173#12011yescharacteristic
111Dic173#12112yes—
112Dic173#12311yescharacteristic
113C2×C173—21no—
114C2×C173—22no—
115C2×C173—41no—
116C2×C173#38611yescharacteristic
117D173—42no—
118D173—41no—
119C173#189111yescharacteristic, complemented, Sylow
120#1891—1731nomaximal, Sylow
121#568—1731no—
122#525—1731no—
123#413—1731no—
124#407—1732no—
125#344—1732no—
126C2×#41—1731no—
127C23×C4—1731no—
128#131—3464no—
129C2×M16—3461no—
130C2×M16—3461no—
131C2×M16—1732no—
132C2×M16—1732no—
133#123—1731no—
134#123#1062192411yescharacteristic
135C22⋊Q8—3461no—
136#121—3461no—
137#120—1731no—
138#120—1731no—
139C4×D4—3461no—
140C4×D4—3461no—
141#117—1731no—
142#101—3461no—
143#100—3461no—
144C22⋊C8—3464no—
145C24—1731no—
146#41—3462no—
147#41—3462no—
148C2×Q8—1731no—
149C2×D4—3461no—
150C2×D4—3461no—
151C2×D4—1731no—
152C2×D4—1731no—
153C2×D4C2×#1062192411yescharacteristic
154C22×C4—3462no—
155C22×C4—3461no—
156C22×C4—3461no—
157C22×C4—3462no—
158C22×C4—1731no—
159C22×C4—1731no—
160C22×C4—1731no—
161C22×C4C2×#1062192411yescharacteristic
162M16—3462no—
163M16—3462no—
164M16—3462no—
165M16—3462no—
166M16—3462no—
167M16—3462no—
168C2×C8—3462no—
169C2×C8—3462no—
170C2×C8—3462no—
171C2×C8—1734no—
172C4⋊C4—21no—
173C4⋊C4—3461no—
174C4⋊C4—3461no—
175C22⋊C4—21no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—1731no—
180C22⋊C4C2×#1062192411yescharacteristic
181C42—3461no—
182C23—3461no—
183C23—3462no—
184C23—1731no—
185C23—6922no—
186C23C22⋊#1062192411yescharacteristic
187C23C22×#1062192411yescharacteristic
188Q8—3462no—
189D4—22no—
190D4—3462no—
191D4—3462no—
192D4—6922no—
193D4—6922no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—3461no—
198C2×C4—3461no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—1731no—
203C2×C4—1731no—
204C2×C4—1731no—
205C2×C4—1732no—
206C2×C4—1732no—
207C2×C4—6921no—
208C2×C4—6922no—
209C2×C4—6921no—
210C2×C4C22⋊#1062192411yescharacteristic
211C8—3464no—
212C8—6922no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—3462no—
217C22—3464no—
218C22—3461no—
219C22—6921no—
220C22—6921no—
221C22—6922no—
222C22—6921no—
223C22C2×C22⋊#1062192411yescharacteristic
224C22D4×#1062192412yes—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—3461no—
229C4—1731no—
230C4—1732no—
231C4—1731no—
232C4—6921no—
233C2—21no—
234C2—22no—
235C2—41no—
236C2—6922no—
237C2—6921no—
238C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173#386
prime radical#26011999C2
cosocleC22×Dic173C23
hypercenter#123#10621924
hypocenterC173#1891
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26012569C2semidirect
2. #26012478C2semidirect
3. #26012463C2semidirect
4. #26012372C2semidirect
34. C2×#13063443D4semidirect
119. C173#1891semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260124782
#260124632
#260123722
#260119992
#1891173