Finite Group Databasebeta

← #36353945

Subgroups — #36353945

Order 22144.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#26012257C211yesmaximal, characteristic, complemented
2#26012041C211yesmaximal, characteristic
3#26011509C211yesmaximal, characteristic, complemented
4#26011507C211yesmaximal, characteristic, complemented
5#26011464C211yesmaximal, characteristic, complemented
6#26011251C211yesmaximal, characteristic, complemented
7#339×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C2×#14583826C2211yescharacteristic
10#19149687—22no—
11#19149685—22no—
12C2×#14583810C2211yescharacteristic
13M16×D173—22no—
14#19149679C411yescharacteristic, complemented
15#19149676C411yescharacteristic, complemented
16#19149667C411yescharacteristic, complemented
17#19149667C411yescharacteristic, complemented
18#19149653—22no—
19C4×D692—22no—
20#19149638C2211yescharacteristic
21#19149607—22no—
22#19149605—22no—
23#19149593—22no—
24C2×M16×C173C2211yescharacteristic
25#117×C173C2211yescharacteristic
26C2×C42×C173C2211yescharacteristic
27#14583826—21no—
28#14583826—21no—
29#14583826—21no—
30#14583826—21no—
31#14583826—21no—
32#14583826—21no—
33#14583826—21no—
34#14583826—21no—
35C2×Dic346C2×C411yescharacteristic
36C2×Dic346C2×C411yescharacteristic
37C2×D692—22no—
38C2×D692C2×C411yescharacteristic
39C2×D692C2×C411yescharacteristic
40C2×C22⋊D173—21no—
41C2×C22⋊D173D412yes—
42C2×C4×D173—21no—
43C2×C4×D173D412yes—
44#14583810—21no—
45#14583810D412yescomplemented
46#14583809—24no—
47C2×#13063436—21no—
48C8×D173—24no—
49C4⋊Dic173—22no—
50C4⋊Dic173—21no—
51C22⋊Dic173—21no—
52#14583803—22no—
53#14583803—22no—
54C4×Dic173—21no—
55C22×C4×C173—21no—
56C22×C4×C173C2311yescharacteristic
57M16×C173—21no—
58M16×C173D412yescomplemented
59C2×C8×C173—21no—
60C4⋊C4×C173—21no—
61C22⋊C4×C173—21no—
62C42×C173—21no—
63C42×C173—21no—
64C42×C173C2×C411yescharacteristic
65C42×C173C2×C411yescharacteristic
66C22×D173—21no—
67C22×D173C22⋊C412yes—
68Dic346—21no—
69Dic346—21no—
70Dic346—21no—
71Dic346—21no—
72D692—21no—
73D692—21no—
74D692—21no—
75D692—21no—
76D692—44no—
77C22⋊D173—22no—
78C22⋊D173—22no—
79C22⋊D173—41no—
80C22⋊D173—41no—
81C2×Dic173—21no—
82C2×Dic173—41no—
83C2×Dic173C22⋊C412yes—
84C4×D173—22no—
85C4×D173—22no—
86C4×D173—41no—
87C4×D173—41no—
88#13063436—22no—
89C23×C173C2×D411yescharacteristic
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—42no—
93C2×C4×C173—41no—
94C2×C4×C173C22×C411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173#4111yescharacteristic
99C2×C4×C173#4111yescharacteristic
100C8×C173—22no—
101C22×C173—41no—
102C22×C173C2×C22⋊C411yescharacteristic
103C22×C173#12111yescharacteristic
104C22×C173#12111yescharacteristic
105C2×D173—22no—
106C2×D173—42no—
107C2×D173—42no—
108C2×D173—41no—
109C4×C173—22no—
110C4×C173—41no—
111C4×C173—81no—
112C4×C173C4×D411yescharacteristic
113C4×C173C4×D411yescharacteristic
114C4×C173#12011yescharacteristic
115C4×C173#12311yescharacteristic
116Dic173—22no—
117Dic173—41no—
118Dic173—81no—
119C2×C173—21no—
120C2×C173—21no—
121C2×C173—21no—
122C2×C173#38611yescharacteristic
123D173—42no—
124D173—81no—
125C173#188411yescharacteristic, complemented, Sylow
126#1884—1731nomaximal, Sylow
127#568—1731no—
128#532—1731no—
129#421—1731no—
130#421—1731no—
131#410—1731no—
132#339—1731no—
133#339D17311yescharacteristic, complemented
134C2×#41—1731no—
135C2×#41—1731no—
136#131—3462no—
137#131—3462no—
138C2×M16—3462no—
139C2×M16—1731no—
140C2×M16C2×D17311yescharacteristic
141C4⋊Q8—1731no—
142#127—1731no—
143#124—1731no—
144#124—1731no—
145#121—3462no—
146C4×D4—3462no—
147#117—1731no—
148#117C2×D17311yescharacteristic
149C2×C42C2×D17311yescharacteristic
150#104—3462no—
151#104—3462no—
152#99—3462no—
153#41—3461no—
154#41—3461no—
155#41—3461no—
156#41—3461no—
157#41—3461no—
158#41—3461no—
159#41—3461no—
160#41—3461no—
161C2×Q8—1731no—
162C2×Q8—1731no—
163C2×D4—3462no—
164C2×D4—3461no—
165C2×D4—1732no—
166C2×D4—1731no—
167C2×D4—1731no—
168C22×C4—21no—
169C22×C4—3461no—
170C22×C4—1732no—
171C22×C4C22×D17311yescharacteristic
172M16—21no—
173M16—3464no—
174M16—3461no—
175M16—1732no—
176M16D69212yescomplemented
177C2×C8—21no—
178C2×C8—3464no—
179C2×C8—3461no—
180C4⋊C4—21no—
181C4⋊C4—3462no—
182C4⋊C4—3461no—
183C22⋊C4—21no—
184C22⋊C4—3462no—
185C22⋊C4—3462no—
186C22⋊C4—3461no—
187C42—21no—
188C42—21no—
189C42—3461no—
190C42C4×D17311yescharacteristic
191C42C4×D17311yescharacteristic
192C23—3461no—
193C23—1732no—
194C23D4×D17311yescharacteristic
195Q8—3461no—
196Q8—3461no—
197Q8—3461no—
198Q8—3461no—
199D4—3462no—
200D4—3462no—
201D4—3461no—
202D4—3461no—
203D4—3461no—
204D4—3461no—
205D4—6924no—
206D4—6921no—
207D4—6921no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—3462no—
213C2×C4—3462no—
214C2×C4—3461no—
215C2×C4—1732no—
216C2×C4—6921no—
217C2×C4—6921no—
218C2×C4—6921no—
219C2×C4C2×C4×D17311yescharacteristic
220C2×C4D4×D17311yescharacteristic
221C2×C4D4×D17311yescharacteristic
222C2×C4C2×D69211yescharacteristic
223C2×C4#1458382611yescharacteristic
224C2×C4#1458382711yescharacteristic
225C8—22no—
226C8—3462no—
227C22—41no—
228C22—3462no—
229C22—6922no—
230C22—6922no—
231C22—6921no—
232C22C22⋊C4×D17311yescharacteristic
233C22#1914965211yescharacteristic
234C22#1914965411yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—3462no—
239C4—13841no—
240C4—6921no—
241C4#1914964111yescharacteristic
242C4C4×D69211yescharacteristic
243C4#1914964911yescharacteristic
244C4#1914966411yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—13841no—
249C2—6922no—
250C2#2601136911yescharacteristic
subgroupG/H
centerC4#19149649
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#339×C173C2
socleC2×C173#386
prime radical#26012041C2
cosocleC22×C4×C173C23
hypercenter#339D173
hypocenterC173#1884
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #26012257C2semidirect
3. #26011509C2semidirect
4. #26011507C2semidirect
5. #26011464C2semidirect
6. #26011251C2semidirect
7. #339×C173C2semidirect
14. #19149679C4semidirect
15. #19149676C4semidirect
16. #19149667C4semidirect
17. #19149667C4semidirect
45. #14583810D4semidirect
58. M16×C173D4semidirect
125. C173#1884semidirect
133. #339D173semidirect
176. M16D692semidirect

Maximal subgroups

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

subgroupindex
#260122572
#260120412
#260115092
#260115072
#260114642
#260112512
#339×C1732
#1884173