Finite Group Databasebeta

← #36369549

Subgroups — #36369549

Order 22144.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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
2C2×#19149742C212yesmaximal, complemented
3D173⋊(C22⋊C8)C211yesmaximal, characteristic
4#26012364C212yesmaximal, complemented
5C22⋊Q8×D173C211yesmaximal, characteristic
6C2×D173⋊C8C2212yes—
7#19149758—24no—
8C2×C4⋊#10621924C2212yes—
9C2×C22⋊#10621924—21no—
10C2×C4×#10621924—21no—
11#19149742—24no—
12#19149736—21no—
13C22⋊#13063443—21no—
14C2×Q8×D173C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16#19149659C411yescharacteristic, complemented
17#19149658C411yescharacteristic, complemented
18#19149657—21no—
19#19149656—21no—
20#19149655C411yescharacteristic, complemented
21C4⋊C4×D173—21no—
22C4⋊C4×D173C2211yescharacteristic
23C22⋊C4×D173—21no—
24C22⋊Q8×C173C411yescharacteristic, complemented
25C22×#10621924—22no—
26D173⋊C8—22no—
27D173⋊C8—22no—
28C2×#13063443—22no—
29C4⋊#10621924—22no—
30C4⋊#10621924D414yes—
31C22⋊#10621924—24no—
32C4×#10621924—24no—
33C23×D173D411yescharacteristic
34C2×Dic346—21no—
35C2×Dic346C2×C411yescharacteristic
36Q8×D173—22no—
37Q8×D173—22no—
38C22×Dic173D411yescharacteristic
39C2×C4×D173—22no—
40C2×C4×D173—22no—
41C2×C4×D173—21no—
42C2×C4×D173—21no—
43C2×C4×D173C2×C411yescharacteristic
44C2×C4×D173D411yescharacteristic
45C2×C4×D173D411yescharacteristic
46C2×C4×D173C2311yescharacteristic
47C4⋊Dic173—21no—
48C4⋊Dic173C2×C411yescharacteristic
49#14583805—21no—
50#14583805—21no—
51#14583805—21no—
52C22⋊Dic173—21no—
53#14583803—21no—
54#14583803—21no—
55C2×Q8×C173C2×C411yescharacteristic
56C22×C4×C173C2×C411yescharacteristic
57C4⋊C4×C173—21no—
58C4⋊C4×C173C2×C411yescharacteristic
59C22⋊C4×C173—21no—
60C2×#10621924—24no—
61C2×#10621924—42no—
62C2×#10621924—42no—
63#13063443—42no—
64C22×D173—21no—
65C22×D173—21no—
66C22×D173—22no—
67C22×D173—22no—
68C22×D173C22⋊C411yescharacteristic
69C22×D173C2×D411yescharacteristic
70Dic346—22no—
71Dic346—42no—
72C2×Dic173—22no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173C22⋊C411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C4×D173—21no—
78C4×D173—21no—
79C4×D173—22no—
80C4×D173—22no—
81C4×D173—41no—
82C4×D173—41no—
83C4×D173—42no—
84C4×D173C2×D412yes—
85C4×D173#4112yes—
86C23×C173C22⋊C411yescharacteristic
87Q8×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—21no—
90C2×C4×C173—21no—
91C2×C4×C173C22⋊C411yescharacteristic
92C2×C4×C173C22×C411yescharacteristic
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173C2×C22⋊C411yescharacteristic
97C2×D173—21no—
98C2×D173—21no—
99C2×D173—21no—
100C2×D173—42no—
101C2×D173—42no—
102C2×D173#12011yescharacteristic
103C2×D173#12311yescharacteristic
104C2×D173#13512yes—
105C2×D173#13712yes—
106#10621924—44no—
107C4×C173—21no—
108C4×C173—41no—
109C4×C173—41no—
110C4×C173C4×D412yes—
111Dic173—21no—
112Dic173—41no—
113Dic173—41no—
114Dic173#12112yes—
115C2×C173—41no—
116C2×C173#38611yescharacteristic
117C2×C173#41611yescharacteristic
118C2×C173#41911yescharacteristic
119D173—41no—
120D173#45212yes—
121D173#48512yes—
122C173#187311yescharacteristic, complemented, Sylow
123#1873—1731nomaximal, Sylow
124C2×C22⋊Q8—1731no—
125C2×#117—1731no—
126C2×#103—1732no—
127C2×C22⋊C8—1731no—
128#340—1732no—
129C22×Q8—1731no—
130C23×C4—1731no—
131C22×C8—1732no—
132C22⋊Q8—3461no—
133C22⋊Q8—3461no—
134C22⋊Q8—1731no—
135C22⋊Q8—1731no—
136C22⋊Q8—1731no—
137C22⋊Q8#1062192411yescharacteristic, complemented
138#117—3464no—
139C2×C4⋊C4—3461no—
140C2×C4⋊C4—1732no—
141C2×C4⋊C4—1731no—
142C2×C22⋊C4—3461no—
143C2×C22⋊C4—3461no—
144C2×C42—3461no—
145#103—3464no—
146C22⋊C8—3461no—
147C22⋊C8—3461no—
148C24—1731no—
149C2×Q8—3461no—
150C2×Q8—3462no—
151C2×Q8—3462no—
152C2×Q8—1731no—
153C2×Q8C2×#1062192411yescharacteristic
154C22×C4—3462no—
155C22×C4—3462no—
156C22×C4—3461no—
157C22×C4—3461no—
158C22×C4—3462no—
159C22×C4—1731no—
160C22×C4—1731no—
161C22×C4—1731no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4C2×#1062192411yescharacteristic
165C2×C8—3462no—
166C2×C8—3462no—
167C2×C8—3462no—
168C4⋊C4—21no—
169C4⋊C4—3461no—
170C4⋊C4—3461no—
171C4⋊C4—3461no—
172C4⋊C4—3462no—
173C4⋊C4—3461no—
174C4⋊C4—1734no—
175C4⋊C4—1731no—
176C4⋊C4C2×#1062192411yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C22⋊C4—3464no—
181C22⋊C4—3461no—
182C42—3464no—
183C23—3461no—
184C23—3461no—
185C23—3462no—
186C23—3462no—
187C23—1731no—
188C23—1731no—
189C23C22⋊#1062192411yescharacteristic
190Q8—22no—
191Q8—3462no—
192Q8—6922no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—3462no—
199C2×C4—3462no—
200C2×C4—3462no—
201C2×C4—3464no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—1731no—
205C2×C4—1731no—
206C2×C4—1732no—
207C2×C4—1732no—
208C2×C4—6922no—
209C2×C4—6921no—
210C2×C4—6921no—
211C2×C4—6922no—
212C2×C4—6922no—
213C2×C4C22⋊#1062192411yescharacteristic
214C2×C4C22×#1062192411yescharacteristic
215C8—6922no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—3461no—
220C22—3461no—
221C22—3461no—
222C22—1731no—
223C22—1731no—
224C22—1732no—
225C22—1732no—
226C22—6922no—
227C22—6922no—
228C22C2×C22⋊#1062192411yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—3461no—
233C4—1732no—
234C4—6924no—
235C4—6921no—
236C4—6921no—
237C4D4×#1062192412yes—
238C2—41no—
239C2—1732no—
240C2—1732no—
241C2—6921no—
242C2#2601243911yescharacteristic
243C2#2601248111yescharacteristic
244C2#2601248411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC22⋊Q8#10621924
hypocenterC173#1873
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19149742C2semidirect
4. #26012364C2semidirect
16. #19149659C4semidirect
17. #19149658C4semidirect
20. #19149655C4semidirect
24. C22⋊Q8×C173C4semidirect
122. C173#1873semidirect
137. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497582
C2×#191497422
D173⋊(C22⋊C8)2
#260123642
C22⋊Q8×D1732
#1873173