Finite Group Databasebeta

← D173⋊#327 #36368929

Subgroups — D173⋊#327 #36368929

Order 22144.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic, complemented
3#26012343C212yesmaximal, complemented
4#26012342C212yesmaximal, complemented
5#121×D173C211yesmaximal, characteristic
6C23×#10621924—21no—
7C2×D173⋊C8—21no—
8#19149736C2212yescomplemented
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4—21no—
11(C2×#10621924)⋊C4C2212yescomplemented
12C2×D4×D173—21no—
13C2×D4×D173C411yescharacteristic, complemented
14C22×C4×D173C2211yescharacteristic
15#19149654C2211yescharacteristic
16#19149653C411yescharacteristic, complemented
17#19149652—21no—
18#19149651—21no—
19#19149650C2211yescharacteristic
20C4⋊C4×D173C411yescharacteristic, complemented
21C22⋊C4×D173—21no—
22#121×C173C411yescharacteristic, complemented
23C22×#10621924—24no—
24C22×#10621924—22no—
25C22×#10621924—42no—
26C22×#10621924—42no—
27D173⋊C8—42no—
28C2×#13063443—22no—
29C23×D173—21no—
30C23×D173D411yescharacteristic
31C2×D692—21no—
32C2×D692C2×C411yescharacteristic, complemented
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36D4×D173—42no—
37D4×D173—42no—
38D4×D173—42no—
39C22×Dic173D411yescharacteristic
40C2×C4×D173—21no—
41C2×C4×D173—42no—
42C2×C4×D173C2×C411yescharacteristic
43C2×C4×D173D411yescharacteristic
44C2×C4×D173D411yescharacteristic
45C2×C4×D173C2311yescharacteristic
46C4⋊Dic173C2×C411yescharacteristic, complemented
47#14583805—21no—
48C22⋊Dic173—21no—
49#14583803—21no—
50#14583803—21no—
51C2×D4×C173—21no—
52C2×D4×C173C2×C411yescharacteristic, complemented
53C22×C4×C173C2×C411yescharacteristic
54C4⋊C4×C173C2×C411yescharacteristic, complemented
55C22⋊C4×C173—21no—
56C2×#10621924—24no—
57C2×#10621924—42no—
58C2×#10621924—48no—
59C2×#10621924—42no—
60#13063443—42no—
61C22×D173—22no—
62C22×D173—22no—
63C22×D173—21no—
64C22×D173—42no—
65C22×D173—42no—
66C22×D173—42no—
67C22×D173C22⋊C411yescharacteristic
68C22×D173D812yes—
69C22×D173SD1612yes—
70C22×D173C2×D411yescharacteristic
71D692—41no—
72D692—41no—
73D692—41no—
74C22⋊D173—42no—
75C22⋊D173—42no—
76C22⋊D173—41no—
77C22⋊D173—41no—
78C2×Dic173—21no—
79C2×Dic173—41no—
80C2×Dic173C22⋊C411yescharacteristic
81C2×Dic173C2×D411yescharacteristic
82C4×D173—22no—
83C4×D173—22no—
84C4×D173—41no—
85C4×D173—41no—
86C4×D173—42no—
87C23×C173—21no—
88C23×C173C22⋊C411yescharacteristic
89D4×C173—41no—
90D4×C173—42no—
91C2×C4×C173—21no—
92C2×C4×C173—41no—
93C2×C4×C173C22⋊C411yescharacteristic
94C2×C4×C173C22×C411yescharacteristic
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173#10212yes—
101C22×C173C2×C22⋊C411yescharacteristic
102C2×D173—21no—
103C2×D173—21no—
104C2×D173—24no—
105C2×D173—24no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—41no—
109C2×D173—84no—
110C2×D173#9912yes—
111C2×D173#10212yes—
112C2×D173#10412yes—
113C2×D173C2×D811yescharacteristic
114C2×D173C2×SD1611yescharacteristic
115#10621924—44no—
116C4×C173—21no—
117C4×C173—21no—
118C4×C173—41no—
119Dic173—21no—
120Dic173—21no—
121Dic173—41no—
122C2×C173—22no—
123C2×C173—81no—
124C2×C173C2×#9911yescharacteristic
125C2×C173C2×#10211yescharacteristic
126C2×C173C2×#10411yescharacteristic
127D173—21no—
128D173—21no—
129D173—81no—
130D173#32714yescomplemented
131C173C2×#32711yescharacteristic, complemented, Sylow
132C2×#327—1731nomaximal, Sylow
133C2×#121—1731no—
134C2×C22⋊C8—1731no—
135C2×#95—1731no—
136#327—1732no—
137#327—1732no—
138C22×D4—3461no—
139C22×D4—1731no—
140C23×C4—3461no—
141C23×C4—1731no—
142C22×C8—3461no—
143#121—3461no—
144#121—3461no—
145#121—1731no—
146#121—1731no—
147#121—1731no—
148#121#1062192411yescharacteristic, complemented
149C2×C4⋊C4—1731no—
150C2×C22⋊C4—3461no—
151C22⋊C8—3461no—
152C22⋊C8—1732no—
153#95—3461no—
154#95—1732no—
155C24—3461no—
156C24—1731no—
157C2×D4—21no—
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—3461no—
161C2×D4—3461no—
162C2×D4—1731no—
163C2×D4—6922no—
164C2×D4—6922no—
165C2×D4—6922no—
166C2×D4C2×#1062192411yescharacteristic, complemented
167C22×C4—3461no—
168C22×C4—3464no—
169C22×C4—3462no—
170C22×C4—1731no—
171C22×C4—1731no—
172C22×C4—1731no—
173C22×C4—1731no—
174C22×C4—1731no—
175C22×C4—6922no—
176C22×C4—6922no—
177C22×C4—6922no—
178C22×C4C2×#1062192411yescharacteristic
179C2×C8—3462no—
180C2×C8—6922no—
181C4⋊C4—3461no—
182C4⋊C4—1731no—
183C4⋊C4C2×#1062192411yescharacteristic, complemented
184C22⋊C4—21no—
185C22⋊C4—3461no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C23—21no—
189C23—3462no—
190C23—3462no—
191C23—3461no—
192C23—1732no—
193C23—1732no—
194C23—1731no—
195C23—1731no—
196C23—6922no—
197C23—6922no—
198C23—6922no—
199C23C22⋊#1062192411yescharacteristic
200D4—41no—
201D4—42no—
202D4—6921no—
203D4—6921no—
204D4—6921no—
205D4—6922no—
206D4—6921no—
207D4—6921no—
208D4—6922no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—3464no—
214C2×C4—3461no—
215C2×C4—1731no—
216C2×C4—1731no—
217C2×C4—6921no—
218C2×C4—6921no—
219C2×C4—6922no—
220C2×C4—6921no—
221C2×C4—6928no—
222C2×C4—6922no—
223C2×C4—6922no—
224C2×C4C22⋊#1062192411yescharacteristic
225C2×C4C22×#1062192411yescharacteristic
226C8—6922no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—3464no—
233C22—3464no—
234C22—3461no—
235C22—3461no—
236C22—13844no—
237C22—1731no—
238C22—1731no—
239C22—1732no—
240C22—1732no—
241C22—1732no—
242C22—6921no—
243C22—6921no—
244C22—6921no—
245C22#1914974112yes—
246C22C2×C22⋊#1062192411yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—3461no—
251C4—3461no—
252C4—6924no—
253C4—6921no—
254C2—22no—
255C2—81no—
256C2—3461no—
257C2—3461no—
258C2—13841no—
259C2—1734no—
260C2D173⋊#9911yescharacteristic
261C2C2×#1914974111yescharacteristic
262C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#121×C173C4
socleC22×C173C2×C22⋊C4
prime radical#121×D173C2
cosocleC2×C4×D173C23
hypercenter#121#10621924
hypocenterC173C2×#327
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. D173⋊(C22⋊C8)C2semidirect
2. C2×(C2×#10621924)⋊C4C2semidirect
3. #26012343C2semidirect
4. #26012342C2semidirect
8. #19149736C22semidirect
11. (C2×#10621924)⋊C4C22semidirect
13. C2×D4×D173C4semidirect
16. #19149653C4semidirect
20. C4⋊C4×D173C4semidirect
22. #121×C173C4semidirect
32. C2×D692C2×C4semidirect
46. C4⋊Dic173C2×C4semidirect
52. C2×D4×C173C2×C4semidirect
54. C4⋊C4×C173C2×C4semidirect
130. D173#327semidirect
131. C173C2×#327semidirect
148. #121#10621924semidirect
166. C2×D4C2×#10621924semidirect
183. C4⋊C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
C2×(C2×#10621924)⋊C42
#260123432
#260123422
#121×D1732
C2×#327173