Finite Group Databasebeta

← D173⋊#352 #36369812

Subgroups — D173⋊#352 #36369812

Order 22144.

278 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 54 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⋊#100C211yesmaximal, characteristic
2D173⋊#99C211yesmaximal, characteristic
3#26012394C212yesmaximal, complemented
4#26012393C212yesmaximal, complemented
5#123×D173C211yesmaximal, characteristic
6D173⋊M16—21no—
7C2×C22⋊#10621924—21no—
8#19149739—21no—
9#19149739C2212yes—
10#19149738—21no—
11#19149737C2212yes—
12C2×D4×D173C2211yescharacteristic
13C22×C4×D173C411yescharacteristic, complemented
14#19149664—21no—
15#19149663C2211yescharacteristic
16#19149662C2211yescharacteristic
17#19149661—21no—
18#19149660C411yescharacteristic, complemented
19C4⋊C4×D173—21no—
20C22⋊C4×D173—21no—
21C22⋊C4×D173C411yescharacteristic, complemented
22#123×C173C411yescharacteristic, complemented
23C22×#10621924—41no—
24#14583836—41no—
25#14583835—22no—
26D173⋊C8—41no—
27C22⋊#10621924—22no—
28C22⋊#10621924—41no—
29C23×D173C2×C411yescharacteristic
30C23×D173D411yescharacteristic
31C2×D692D411yescharacteristic
32C2×C22⋊D173D411yescharacteristic
33C2×C22⋊D173C2311yescharacteristic
34D4×D173—41no—
35D4×D173—41no—
36C22×Dic173C2×C411yescharacteristic, complemented
37C2×C4×D173—21no—
38C2×C4×D173—21no—
39C2×C4×D173—21no—
40C2×C4×D173—41no—
41C2×C4×D173—41no—
42C2×C4×D173D411yescharacteristic
43C4⋊Dic173—21no—
44#14583805—21no—
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173C2×C411yescharacteristic, complemented
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51C2×D4×C173C2×C411yescharacteristic
52C22×C4×C173C2×C411yescharacteristic, complemented
53C4⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173C2×C411yescharacteristic, complemented
56C2×#10621924—42no—
57C2×#10621924—82no—
58#13063443—42no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—42no—
64C22×D173—42no—
65C22×D173C22⋊C411yescharacteristic
66C22×D173C22⋊C411yescharacteristic
67C22×D173C2×D411yescharacteristic
68D692—41no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—41no—
74C2×Dic173C2×D411yescharacteristic
75C4×D173—21no—
76C4×D173—21no—
77C4×D173—42no—
78C4×D173—42no—
79C4×D173—41no—
80C4×D173—41no—
81C23×C173C22⋊C411yescharacteristic
82C23×C173C22×C411yescharacteristic
83D4×C173—41no—
84C2×C4×C173—21no—
85C2×C4×C173—21no—
86C2×C4×C173—41no—
87C2×C4×C173C22⋊C411yescharacteristic
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—41no—
91C22×C173—41no—
92C22×C173C2×C22⋊C411yescharacteristic
93C2×D173—22no—
94C2×D173—21no—
95C2×D173—21no—
96C2×D173—42no—
97C2×D173—42no—
98C2×D173—42no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—81no—
102C2×D173#9911yescharacteristic
103C2×D173#9911yescharacteristic
104#10621924—82no—
105C4×C173—21no—
106C4×C173—41no—
107C4×C173—41no—
108Dic173—21no—
109Dic173—41no—
110Dic173—41no—
111C2×C173—21no—
112C2×C173—41no—
113C2×C173—41no—
114C2×C173C2×#9911yescharacteristic
115D173—21no—
116D173—41no—
117D173—41no—
118D173#35212yescomplemented
119C173C2×#35211yescharacteristic, complemented, Sylow
120C2×#352—1731nomaximal, Sylow
121C2×#123—1731no—
122C2×#100—1731no—
123C2×#99—1731no—
124#352—1732no—
125#352—1732no—
126C22×D4—1731no—
127C23×C4—1731no—
128C2×M16—3461no—
129#123—3461no—
130#123—3461no—
131#123—1731no—
132#123—1731no—
133#123—1731no—
134#123#1062192411yescharacteristic, complemented
135C2×C4⋊C4—3461no—
136C2×C22⋊C4—3461no—
137C2×C22⋊C4—3461no—
138C2×C22⋊C4—1731no—
139#100—3461no—
140#100—1732no—
141#99—3461no—
142#99—1732no—
143C24—1731no—
144C24—1731no—
145C2×D4—1731no—
146C2×D4—1731no—
147C2×D4—1731no—
148C2×D4—6921no—
149C2×D4—6921no—
150C2×D4C2×#1062192411yescharacteristic
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—3461no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4—6921no—
157C22×C4—6921no—
158C22×C4—6921no—
159C22×C4C2×#1062192411yescharacteristic, complemented
160M16—3462no—
161M16—6921no—
162C2×C8—6921no—
163C4⋊C4—21no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C22⋊C4—21no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3462no—
173C22⋊C4—1731no—
174C22⋊C4—6921no—
175C22⋊C4C2×#1062192411yescharacteristic, complemented
176C23—3461no—
177C23—3461no—
178C23—3461no—
179C23—3461no—
180C23—1731no—
181C23—1731no—
182C23—1731no—
183C23—6922no—
184C23—6922no—
185C23C22⋊#1062192411yescharacteristic
186C23C22×#1062192411yescharacteristic
187D4—41no—
188D4—6921no—
189D4—6921no—
190D4—6921no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—13842no—
199C2×C4—1731no—
200C2×C4—6922no—
201C2×C4—6922no—
202C2×C4—6921no—
203C2×C4—6921no—
204C2×C4—6922no—
205C2×C4—6921no—
206C2×C4C22⋊#1062192411yescharacteristic
207C8—6922no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3462no—
213C22—3461no—
214C22—3461no—
215C22—13841no—
216C22—1731no—
217C22—1731no—
218C22—6921no—
219C22—6921no—
220C22—6922no—
221C22—6922no—
222C22—6922no—
223C22C2×C22⋊#1062192411yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—3461no—
228C4—13842no—
229C4—6921no—
230C4—6921no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—3461no—
235C2—1732no—
236C2—6921no—
237C2—6921no—
238C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173C2×#99
prime radicalC2×D4×D173C22
cosocleC2×C22⋊D173C23
hypercenter#123#10621924
hypocenterC173C2×#352
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #26012394C2semidirect
4. #26012393C2semidirect
13. C22×C4×D173C4semidirect
18. #19149660C4semidirect
21. C22⋊C4×D173C4semidirect
22. #123×C173C4semidirect
36. C22×Dic173C2×C4semidirect
47. C22⋊Dic173C2×C4semidirect
52. C22×C4×C173C2×C4semidirect
55. C22⋊C4×C173C2×C4semidirect
118. D173#352semidirect
119. C173C2×#352semidirect
134. #123#10621924semidirect
159. C22×C4C2×#10621924semidirect
175. C22⋊C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊#1002
D173⋊#992
#260123942
#260123932
#123×D1732
C2×#352173