Finite Group Databasebeta

← #36369433

Subgroups — #36369433

Order 22144.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D173⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
2C2×#19149741C212yesmaximal
3#26012435C211yesmaximal, characteristic, complemented
4#26012364C212yesmaximal, complemented
5C4×D4×D173C211yesmaximal, characteristic
6C2×D173⋊C8C2212yes—
7C2×C4⋊#10621924C2212yes—
8#19149746—21no—
9C4⋊#13063443—21no—
10#19149741—24no—
11(C2×#10621924)⋊C4—21no—
12(C2×#10621924)⋊C4—21no—
13C2×D4×D173C2211yescharacteristic
14C22×C4×D173—21no—
15C4×D692C411yescharacteristic, complemented
16D4×Dic173C411yescharacteristic, complemented
17#19149641C411yescharacteristic, complemented
18C4×C22⋊D173—21no—
19#19149639—21no—
20C4⋊C4×D173C2211yescharacteristic
21C22⋊C4×D173—21no—
22C42×D173C2211yescharacteristic
23C4×D4×C173C411yescharacteristic, complemented
24C22×#10621924—22no—
25D173⋊C8—22no—
26D173⋊C8—22no—
27C2×#13063443—22no—
28C4⋊#10621924—22no—
29C4⋊#10621924—22no—
30C23×D173—21no—
31C2×D692C2×C411yescharacteristic
32C2×C22⋊D173—21no—
33D4×D173—21no—
34D4×D173—21no—
35D4×D173D412yes—
36D4×D173Q812yes—
37C22×Dic173—21no—
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40C2×C4×D173—21no—
41C2×C4×D173—44no—
42C2×C4×D173D411yescharacteristic
43C2×C4×D173D411yescharacteristic
44C2×C4×D173C2311yescharacteristic
45C4⋊Dic173C2×C411yescharacteristic
46#14583805—21no—
47C22⋊Dic173—21no—
48#14583803—21no—
49#14583803—21no—
50C4×Dic173C2×C411yescharacteristic
51C4×Dic173D411yescharacteristic
52C4×Dic173D411yescharacteristic
53C2×D4×C173C2×C411yescharacteristic
54C22×C4×C173—21no—
55C4⋊C4×C173C2×C411yescharacteristic
56C22⋊C4×C173—21no—
57C42×C173C2×C411yescharacteristic
58C2×#10621924—42no—
59C2×#10621924—42no—
60C2×#10621924—42no—
61#13063443—42no—
62C22×D173—21no—
63C22×D173—22no—
64C22×D173—22no—
65C22×D173—42no—
66C22×D173—42no—
67C22×D173C2×D411yescharacteristic
68D692—21no—
69D692C4⋊C412yes—
70C22⋊D173—22no—
71C22⋊D173—22no—
72C2×Dic173—21no—
73C2×Dic173—42no—
74C2×Dic173C22⋊C411yescharacteristic
75C2×Dic173C22⋊C411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C4×D173—22no—
78C4×D173—22no—
79C4×D173—42no—
80C4×D173—42no—
81C4×D173—42no—
82C4×D173C2×D411yescharacteristic
83C4×D173C2×Q811yescharacteristic
84C4×D173#4111yescharacteristic
85C4×D173#4111yescharacteristic
86C23×C173—21no—
87D4×C173—21no—
88D4×C173C4⋊C412yes—
89C2×C4×C173—21no—
90C2×C4×C173—42no—
91C2×C4×C173C22⋊C411yescharacteristic
92C2×C4×C173C22⋊C411yescharacteristic
93C2×C4×C173C22×C411yescharacteristic
94C22×C173—22no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173C2×C22⋊C411yescharacteristic
98C2×D173—22no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—44no—
102C2×D173—44no—
103C2×D173#12111yescharacteristic
104C2×D173C22⋊Q811yescharacteristic
105C2×D173#13512yes—
106C2×D173AGL(1,ℤ/8)12yes—
107#10621924—82no—
108C4×C173—21no—
109C4×C173—21no—
110C4×C173—41no—
111C4×C173C2×C4⋊C411yescharacteristic
112C4×C173#11711yescharacteristic
113Dic173—21no—
114Dic173—21no—
115Dic173—41no—
116Dic173#12111yescharacteristic
117Dic173C22⋊Q811yescharacteristic
118C2×C173—42no—
119C2×C173#38011yescharacteristic
120C2×C173#41611yescharacteristic
121C2×C173#41811yescharacteristic
122D173—42no—
123D173#46312yes—
124D173#47812yes—
125C173#180411yescharacteristic, complemented, Sylow
126#1804—1731nomaximal, Sylow
127C2×C4×D4—1731no—
128C2×C4⋊C8—1731no—
129C2×#102—1732no—
130#382—1731no—
131#340—1732no—
132C22×D4—1731no—
133C23×C4—3461no—
134C22×C8—1732no—
135C4×D4—3461no—
136C4×D4—3461no—
137C4×D4—1731no—
138C4×D4—1731no—
139C4×D4—1731no—
140C4×D4#1062192411yescharacteristic, complemented
141C2×C4⋊C4—1732no—
142C2×C4⋊C4—1731no—
143C2×C22⋊C4—3461no—
144C2×C42—1731no—
145C4⋊C8—3461no—
146C4⋊C8—3461no—
147#102—3464no—
148#95—3461no—
149#95—3461no—
150C24—3461no—
151C2×D4—3461no—
152C2×D4—3461no—
153C2×D4—3461no—
154C2×D4—1732no—
155C2×D4—1732no—
156C2×D4—1731no—
157C2×D4C2×#1062192411yescharacteristic
158C22×C4—21no—
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4—3461no—
162C22×C4—3461no—
163C22×C4—3462no—
164C22×C4—1731no—
165C22×C4—1731no—
166C22×C4—1731no—
167C22×C4—6924no—
168C2×C8—3462no—
169C2×C8—3462no—
170C2×C8—3462no—
171C4⋊C4—3461no—
172C4⋊C4—3462no—
173C4⋊C4—3462no—
174C4⋊C4—1731no—
175C4⋊C4C2×#1062192411yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C42—1731no—
181C42—1731no—
182C42—1731no—
183C42C2×#1062192411yescharacteristic
184C23—21no—
185C23—3462no—
186C23—3462no—
187C23—3461no—
188C23—1731no—
189C23—6922no—
190C23—6922no—
191D4—21no—
192D4—3462no—
193D4—3462no—
194D4—3461no—
195D4—1732no—
196D4C4⋊#1062192412yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—3462no—
200C2×C4—3462no—
201C2×C4—3461no—
202C2×C4—1731no—
203C2×C4—1731no—
204C2×C4—1731no—
205C2×C4—1731no—
206C2×C4—1731no—
207C2×C4—1731no—
208C2×C4—1731no—
209C2×C4—6922no—
210C2×C4—6922no—
211C2×C4—6922no—
212C2×C4—6922no—
213C2×C4—6922no—
214C2×C4—6922no—
215C2×C4—6922no—
216C2×C4C22⋊#1062192411yescharacteristic
217C2×C4C22⋊#1062192411yescharacteristic
218C2×C4C22×#1062192411yescharacteristic
219C8—6922no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—3462no—
224C22—1731no—
225C22—1731no—
226C22—1732no—
227C22—1732no—
228C22—6924no—
229C22—6924no—
230C22—6921no—
231C22—6921no—
232C22C2×C22⋊#1062192411yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—3461no—
237C4—3461no—
238C4—13842no—
239C4—1731no—
240C4—1731no—
241C4—6921no—
242C4C2×C4⋊#1062192411yescharacteristic
243C4#1914975811yescharacteristic
244C2—42no—
245C2—1732no—
246C2—1732no—
247C2—6922no—
248C2#2601243311yescharacteristic
249C2#2601248111yescharacteristic
250C2#2601248311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C22
cosocleC2×C4×D173C23
hypercenterC4×D4#10621924
hypocenterC173#1804
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. D173⋊(C4⋊C8)C2semidirect
3. #26012435C2semidirect
4. #26012364C2semidirect
15. C4×D692C4semidirect
16. D4×Dic173C4semidirect
17. #19149641C4semidirect
23. C4×D4×C173C4semidirect
125. C173#1804semidirect
140. C4×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C4⋊C8)2
C2×#191497412
#260124352
#260123642
C4×D4×D1732
#1804173