Finite Group Databasebeta

← #36369578

Subgroups — #36369578

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×D173⋊M16C211yesmaximal, characteristic
2C2×#19149742C212yesmaximal, complemented
3#26012433C211yesmaximal, characteristic
4#26012364C212yesmaximal, complemented
5C22⋊Q8×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7D173⋊M16—24no—
8C2×#14583835—21no—
9C2×D173⋊C8C2212yes—
10C2×C4⋊#10621924C2212yes—
11C2×C22⋊#10621924—21no—
12#19149742—24no—
13(C2×#10621924)⋊C4—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—
26#14583836—24no—
27#14583835—24no—
28D173⋊C8—22no—
29D173⋊C8D414yes—
30C2×#13063443—22no—
31C4⋊#10621924—22no—
32C4⋊#10621924—22no—
33C22⋊#10621924—42no—
34C23×D173D411yescharacteristic
35C2×Dic346—21no—
36C2×Dic346C2×C411yescharacteristic
37Q8×D173—22no—
38Q8×D173—22no—
39C22×Dic173D411yescharacteristic
40C2×C4×D173—22no—
41C2×C4×D173—22no—
42C2×C4×D173—21no—
43C2×C4×D173—21no—
44C2×C4×D173C2×C411yescharacteristic
45C2×C4×D173D411yescharacteristic
46C2×C4×D173D411yescharacteristic
47C2×C4×D173C2311yescharacteristic
48C4⋊Dic173—21no—
49C4⋊Dic173C2×C411yescharacteristic
50#14583805—21no—
51#14583805—21no—
52#14583805—21no—
53C22⋊Dic173—21no—
54#14583803—21no—
55#14583803—21no—
56C2×Q8×C173C2×C411yescharacteristic
57C22×C4×C173C2×C411yescharacteristic
58C4⋊C4×C173—21no—
59C4⋊C4×C173C2×C411yescharacteristic
60C22⋊C4×C173—21no—
61C2×#10621924—42no—
62C2×#10621924—42no—
63C2×#10621924—42no—
64#13063443—24no—
65C22×D173—22no—
66C22×D173—21no—
67C22×D173—21no—
68C22×D173—22no—
69C22×D173C22⋊C411yescharacteristic
70C22×D173C2×D411yescharacteristic
71Dic346—22no—
72Dic346—41no—
73Dic346—41no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173C22⋊C411yescharacteristic
79C2×Dic173C2×D411yescharacteristic
80C4×D173—22no—
81C4×D173—22no—
82C4×D173—21no—
83C4×D173—21no—
84C4×D173—42no—
85C4×D173—41no—
86C4×D173—41no—
87C4×D173C2×D412yes—
88C4×D173#4112yes—
89C23×C173C22⋊C411yescharacteristic
90Q8×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173—21no—
93C2×C4×C173—21no—
94C2×C4×C173C22⋊C411yescharacteristic
95C2×C4×C173C22×C411yescharacteristic
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173C2×C22⋊C411yescharacteristic
100C2×D173—21no—
101C2×D173—21no—
102C2×D173—21no—
103C2×D173—42no—
104C2×D173—42no—
105C2×D173#12112yes—
106C2×D173AGL(1,ℤ/8)11yescharacteristic
107C2×D173#13712yes—
108C2×D173#13711yescharacteristic
109#10621924—82no—
110C4×C173—21no—
111C4×C173—41no—
112C4×C173—41no—
113C4×C173C4×D412yes—
114Dic173—21no—
115Dic173—41no—
116Dic173—41no—
117Dic173#12011yescharacteristic
118Dic173#12311yescharacteristic
119C2×C173—41no—
120C2×C173#38611yescharacteristic
121C2×C173#41711yescharacteristic
122C2×C173#41911yescharacteristic
123D173—41no—
124D173#46812yes—
125D173#47012yes—
126C173#189411yescharacteristic, complemented, Sylow
127#1894—1731nomaximal, Sylow
128C22×M16—1731no—
129C2×C22⋊Q8—1731no—
130C2×#103—1732no—
131#380—1731no—
132#340—1732no—
133C22×Q8—1731no—
134C23×C4—1731no—
135C2×M16—3461no—
136C2×M16—3461no—
137C2×M16—3464no—
138C22×C8—1732no—
139C22⋊Q8—3461no—
140C22⋊Q8—3461no—
141C22⋊Q8—1731no—
142C22⋊Q8—1731no—
143C22⋊Q8—1731no—
144C22⋊Q8#1062192411yescharacteristic, complemented
145C2×C4⋊C4—3461no—
146C2×C4⋊C4—1732no—
147C2×C4⋊C4—1731no—
148C2×C22⋊C4—3461no—
149C2×C22⋊C4—3461no—
150#103—3464no—
151#95—3461no—
152C24—1731no—
153C2×Q8—3461no—
154C2×Q8—3462no—
155C2×Q8—3462no—
156C2×Q8—1731no—
157C2×Q8C2×#1062192411yescharacteristic
158C22×C4—3462no—
159C22×C4—3462no—
160C22×C4—3461no—
161C22×C4—3461no—
162C22×C4—3462no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4—1731no—
166C22×C4—1731no—
167C22×C4—1731no—
168C22×C4C2×#1062192411yescharacteristic
169M16—3464no—
170M16—3464no—
171C2×C8—3462no—
172C2×C8—3462no—
173C2×C8—1734no—
174C4⋊C4—21no—
175C4⋊C4—3461no—
176C4⋊C4—3461no—
177C4⋊C4—3461no—
178C4⋊C4—3461no—
179C4⋊C4—3462no—
180C4⋊C4—3462no—
181C4⋊C4—1731no—
182C4⋊C4C2×#1062192411yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—3461no—
187C22⋊C4—6922no—
188C23—3461no—
189C23—3462no—
190C23—3461no—
191C23—3462no—
192C23—1731no—
193C23—1731no—
194C23C22⋊#1062192411yescharacteristic
195Q8—22no—
196Q8—3462no—
197Q8—6921no—
198Q8—6921no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—3462no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3462no—
208C2×C4—3461no—
209C2×C4—3461no—
210C2×C4—1731no—
211C2×C4—1731no—
212C2×C4—1732no—
213C2×C4—1732no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6922no—
217C2×C4—6922no—
218C2×C4—6922no—
219C2×C4—6922no—
220C2×C4C22⋊#1062192411yescharacteristic
221C2×C4C22×#1062192411yescharacteristic
222C8—3464no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—3461no—
227C22—3461no—
228C22—3461no—
229C22—1732no—
230C22—1731no—
231C22—1731no—
232C22—1732no—
233C22—6922no—
234C22—6922no—
235C22C2×C22⋊#1062192411yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—3461no—
240C4—13842no—
241C4—1731no—
242C4—1731no—
243C4—6921no—
244C4—6921no—
245C4D4×#1062192412yes—
246C2—41no—
247C2—1732no—
248C2—1732no—
249C2—6921no—
250C2#2601243911yescharacteristic
251C2#2601248211yescharacteristic
252C2#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#1894
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
126. C173#1894semidirect
144. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×D173⋊M162
C2×#191497422
#260124332
#260123642
C22⋊Q8×D1732
#1894173