Finite Group Databasebeta

← #36369598

Subgroups — #36369598

Order 22144.

310 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#26012559C211yesmaximal, characteristic, complemented
2#26012490C211yesmaximal, characteristic
3#26012484C211yesmaximal, characteristic, complemented
4#26012472C211yesmaximal, characteristic
5#26012373C211yesmaximal, characteristic, complemented
6#26012359C211yesmaximal, characteristic, complemented
7C2×#19149711C211yesmaximal, characteristic
8D173⋊M16C2211yescharacteristic
9Q8×#10621924—22no—
10Q8×#10621924—22no—
11#19149758—22no—
12#19149758C2211yescharacteristic
13#19149758C2211yescharacteristic
14#19149758C2211yescharacteristic
15#19149744—22no—
16#19149742—22no—
17#19149738—22no—
18C2×#14583827C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149711—22no—
21#19149711—22no—
22C2×#14583816C411yescharacteristic, complemented
23C2×#14583815C411yescharacteristic, complemented
24C2×#14583813C411yescharacteristic, complemented
25C2×#14583809C2211yescharacteristic
26C2×Q16×C173C411yescharacteristic, complemented
27#14583836—21no—
28#14583835—21no—
29D173⋊C8—21no—
30C4⋊#10621924—24no—
31C4⋊#10621924—21no—
32C4⋊#10621924—21no—
33C4⋊#10621924—21no—
34C22⋊#10621924—21no—
35C22⋊#10621924—21no—
36C22⋊#10621924D412yes—
37C4×#10621924—24no—
38C4×#10621924—21no—
39C4×#10621924—21no—
40C4×#10621924D412yes—
41#14583827—21no—
42#14583827—21no—
43#14583827—21no—
44#14583827—21no—
45C2×Dic346—21no—
46C2×Dic346C2×C411yescharacteristic
47Q8×D173—21no—
48Q8×D173—21no—
49Q8×D173D412yes—
50Q8×D173D412yes—
51C2×D692—21no—
52C2×D692C2×C411yescharacteristic
53C2×C4×D173—21no—
54C2×C4×D173—21no—
55C2×C4×D173C2311yescharacteristic
56#14583816—22no—
57#14583815—22no—
58#14583813—22no—
59#14583809—21no—
60#14583809—21no—
61C2×#13063436C2×C411yescharacteristic
62C2×Q8×C173C2×C411yescharacteristic
63C2×Q8×C173C2×C411yescharacteristic
64Q16×C173—22no—
65C2×C8×C173C2×C411yescharacteristic
66C2×#10621924—22no—
67C2×#10621924—22no—
68C2×#10621924—41no—
69#13063443—41no—
70C22×D173—21no—
71C22×D173C2×D411yescharacteristic
72Dic346—21no—
73Dic346—22no—
74Dic346—22no—
75Dic346C22⋊C412yes—
76D692—21no—
77D692—21no—
78D692—41no—
79D692—41no—
80C2×Dic173—21no—
81C2×Dic173#4111yescharacteristic
82C4×D173—22no—
83C4×D173—22no—
84C4×D173—41no—
85C4×D173—41no—
86C4×D173C2×D411yescharacteristic
87C4×D173C2×D411yescharacteristic
88C4×D173C2×D411yescharacteristic
89C4×D173#4111yescharacteristic
90#13063436—21no—
91Q8×C173—21no—
92Q8×C173—21no—
93Q8×C173—21no—
94Q8×C173C22⋊C412yes—
95C2×C4×C173—21no—
96C2×C4×C173—21no—
97C2×C4×C173C22×C411yescharacteristic
98C8×C173—21no—
99C22×C173C4×D411yescharacteristic
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—81no—
103C2×D173#12011yescharacteristic
104C2×D173#12111yescharacteristic
105#10621924—42no—
106#10621924—42no—
107#10621924—81no—
108C4×C173—22no—
109C4×C173—41no—
110C4×C173C2×C22⋊C411yescharacteristic
111C4×C173C4×D411yescharacteristic
112Dic173—22no—
113Dic173#12111yescharacteristic
114Dic173#12311yescharacteristic
115C2×C173—21no—
116C2×C173#38611yescharacteristic
117D173—21no—
118D173—21no—
119D173—81no—
120C173#190511yescharacteristic, complemented, Sylow
121#1905—1731nomaximal, Sylow
122C2×#137—1731no—
123#519—1731no—
124#421—1731no—
125#419—1731no—
126#410—1731no—
127#344—1731no—
128H(3,ℤ/4)—1731no—
129C2×#41—1731no—
130C22×Q8—1731no—
131#137—3462no—
132#137—3462no—
133C2×Q16—1731no—
134C2×Q16#1062192411yescharacteristic, complemented
135C2×SD16—1731no—
136C2×SD16—1731no—
137C2×M16—1731no—
138C2×M16—1731no—
139C4×Q8—3462no—
140C4×Q8—3462no—
141#117—3462no—
142#117—1731no—
143#117—1731no—
144#117—1731no—
145#104—3462no—
146#103—3462no—
147#99—3462no—
148#41—3461no—
149#41—3461no—
150#41—3461no—
151#41—3461no—
152C2×Q8—3461no—
153C2×Q8—3461no—
154C2×Q8—3461no—
155C2×Q8—1732no—
156C2×Q8—1732no—
157C2×Q8—1731no—
158C2×Q8C2×#1062192411yescharacteristic
159C2×Q8C2×#1062192411yescharacteristic
160C2×D4—3461no—
161C2×D4—1731no—
162C22×C4—3461no—
163C22×C4—3461no—
164C22×C4—1731no—
165Q16—22no—
166Q16—3462no—
167SD16—3462no—
168SD16—3462no—
169M16—3461no—
170M16—3461no—
171M16—3461no—
172M16—3461no—
173C2×C8—3461no—
174C2×C8—1731no—
175C2×C8C2×#1062192411yescharacteristic
176C4⋊C4—3464no—
177C4⋊C4—3461no—
178C4⋊C4—3461no—
179C4⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—1732no—
183C42—3464no—
184C42—3461no—
185C42—3461no—
186C42—1732no—
187C23—3461no—
188C23—1731no—
189Q8—21no—
190Q8—21no—
191Q8—21no—
192Q8—3462no—
193Q8—3462no—
194Q8—3461no—
195Q8—1732no—
196Q8C22⋊#1062192412yes—
197D4—3461no—
198D4—3461no—
199D4—6921no—
200D4—6921no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—3461no—
206C2×C4—3462no—
207C2×C4—3462no—
208C2×C4—1731no—
209C2×C4—1731no—
210C2×C4—1731no—
211C2×C4—1731no—
212C2×C4—1731no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4C22×#1062192411yescharacteristic
217C8—21no—
218C8—3461no—
219C8—6921no—
220C22—13841no—
221C22—1731no—
222C22—1731no—
223C22—6921no—
224C22—6921no—
225C22D4×#1062192411yescharacteristic
226C4—22no—
227C4—41no—
228C4—3462no—
229C4—13841no—
230C4—1731no—
231C4—1731no—
232C4—6922no—
233C4—6922no—
234C4C2×C22⋊#1062192411yescharacteristic
235C4D4×#1062192411yescharacteristic
236C2—21no—
237C2—3461no—
238C2—3461no—
239C2—13841no—
240C2#2601243911yescharacteristic
subgroupG/H
centerC2#26012439
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×Q16×C173C4
socleC2×C173#386
prime radicalC2×#14583827C22
cosocleC2×C4×D173C23
hypercenterC2×Q16#10621924
hypocenterC173#1905
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. #26012559C2semidirect
3. #26012484C2semidirect
5. #26012373C2semidirect
6. #26012359C2semidirect
22. C2×#14583816C4semidirect
23. C2×#14583815C4semidirect
24. C2×#14583813C4semidirect
26. C2×Q16×C173C4semidirect
120. C173#1905semidirect
134. C2×Q16#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125592
#260124902
#260124842
#260124722
#260123732
#260123592
C2×#191497112
#1905173