Finite Group Databasebeta

← #36354741

Subgroups — #36354741

Order 22144.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012117C211yesmaximal, characteristic, complemented
2C2×#19149593C211yesmaximal, characteristic
3C2×#19149593C211yesmaximal, characteristic
4C2×#19149593C211yesmaximal, characteristic
5#26011258C211yesmaximal, characteristic, complemented
6#26011257C211yesmaximal, characteristic, complemented
7#342×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9#19149658—22no—
10#19149656—22no—
11#19149656—22no—
12C2×C22⋊Dic173C2211yescharacteristic
13C2×C22⋊Dic173C2211yescharacteristic
14C2×C22⋊Dic173C2211yescharacteristic
15#19149593—22no—
16#19149593—22no—
17#19149593—22no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C2×C22⋊C4×C173C2211yescharacteristic
20C2×C22⋊C4×C173C2211yescharacteristic
21C23×D173Q811yescharacteristic, complemented
22C2×Dic346—22no—
23C2×C22⋊D173—21no—
24C2×C22⋊D173—21no—
25C2×C22⋊D173—21no—
26C2×C22⋊D173—21no—
27C2×C22⋊D173—21no—
28C2×C22⋊D173—21no—
29C22×Dic173—21no—
30C22×Dic173—21no—
31C22×Dic173—21no—
32C22×Dic173Q811yescharacteristic, complemented
33C4⋊Dic173—22no—
34#14583805—24no—
35C22⋊Dic173—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C22⋊Dic173D412yes—
40C22⋊Dic173D412yes—
41C24×C173C2311yescharacteristic
42C22×C4×C173—21no—
43C22×C4×C173—21no—
44C22×C4×C173—21no—
45C22⋊C4×C173—21no—
46C22⋊C4×C173—21no—
47C22⋊C4×C173—21no—
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C22⋊C4×C173D412yes—
51C22×D173—21no—
52C22×D173—21no—
53C22×D173—21no—
54C22×D173—81no—
55Dic346—82no—
56C22⋊D173—41no—
57C22⋊D173—41no—
58C22⋊D173—41no—
59C22⋊D173—41no—
60C2×Dic173—22no—
61C2×Dic173—22no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173—81no—
67C2×Dic173—81no—
68C2×Dic173—81no—
69C23×C173—81no—
70C23×C173C2×D411yescharacteristic
71C23×C173C2×D411yescharacteristic
72C23×C173C2×D411yescharacteristic
73C23×C173C2×Q811yescharacteristic
74C23×C173#4111yescharacteristic
75C23×C173#4111yescharacteristic
76C23×C173#4111yescharacteristic
77C2×C4×C173—22no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—81no—
81C2×C4×C173—81no—
82C2×C4×C173—81no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—41no—
87C22×C173—41no—
88C22×C173—41no—
89C22×C173—41no—
90C22×C173#12011yescharacteristic
91C22×C173C22⋊Q811yescharacteristic
92C22×C173C22⋊Q811yescharacteristic
93C22×C173C22⋊Q811yescharacteristic
94C22×C173#12411yescharacteristic
95C22×C173#12411yescharacteristic
96C22×C173#12411yescharacteristic
97C2×D173—41no—
98C2×D173—81no—
99C2×D173—81no—
100C2×D173—81no—
101C4×C173—42no—
102C4×C173—81no—
103C4×C173—81no—
104Dic173—42no—
105Dic173—42no—
106Dic173—41no—
107Dic173—81no—
108C2×C173—21no—
109C2×C173—21no—
110C2×C173—21no—
111C2×C173—21no—
112C2×C173—21no—
113C2×C173—21no—
114C2×C173—21no—
115C2×C173#39311yescharacteristic
116D173—81no—
117C173#203211yescharacteristic, complemented, Sylow
118#2032—1731nomaximal, Sylow
119#543—1731no—
120C2×#99—1731no—
121C2×#99—1731no—
122C2×#99—1731no—
123#342—1731no—
124#342—1731no—
125#342D17311yescharacteristic, complemented
126C22×D4—1731no—
127C22⋊Q8—3462no—
128C22⋊Q8—3462no—
129C22⋊Q8—3462no—
130C2×C22⋊C4—1731no—
131C2×C22⋊C4—1731no—
132C2×C22⋊C4—1731no—
133C2×C22⋊C4C2×D17311yescharacteristic
134C2×C22⋊C4C2×D17311yescharacteristic
135C2×C22⋊C4C2×D17311yescharacteristic
136#99—3462no—
137#99—3462no—
138#99—3462no—
139C24—1731no—
140C24C22×D17311yescharacteristic
141C2×Q8—3462no—
142C2×D4—3461no—
143C2×D4—3461no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4—3461no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—3461no—
154C22×C4—1731no—
155C4⋊C4—3464no—
156C4⋊C4—3462no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—1732no—
167C22⋊C4—1732no—
168C22⋊C4D69212yes—
169C23—81no—
170C23—3461no—
171C23—3461no—
172C23—3461no—
173C23—13841no—
174C23D4×D17311yescharacteristic
175C23D4×D17311yescharacteristic
176C23C2×D69211yescharacteristic
177C23Q8×D17311yescharacteristic
178C23#1458382511yescharacteristic
179C23#1458382611yescharacteristic
180C23#1458382611yescharacteristic
181Q8—13842no—
182D4—6921no—
183D4—6921no—
184D4—6921no—
185D4—6921no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—3462no—
193C2×C4—3462no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—13841no—
198C2×C4—13841no—
199C2×C4—13841no—
200C2×C4—6921no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—13841no—
209C22—13841no—
210C22—13841no—
211C22—6921no—
212C22#1914964911yescharacteristic
213C22#1914965711yescharacteristic
214C22#1914965711yescharacteristic
215C22#1914965911yescharacteristic
216C22#1914966511yescharacteristic
217C22#1914966511yescharacteristic
218C22#1914966711yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—13841no—
223C4—6922no—
224C4—6922no—
225C4—6921no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—13841no—
234C2#2601140411yescharacteristic
subgroupG/H
centerC2#26011404
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
Fitting#342×C173C2
socleC2×C173#393
prime radicalC22×C22⋊D173C22
hypercenter#342D173
hypocenterC173#2032
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26012117C2semidirect
5. #26011258C2semidirect
6. #26011257C2semidirect
7. #342×C173C2semidirect
21. C23×D173Q8semidirect
32. C22×Dic173Q8semidirect
117. C173#2032semidirect
125. #342D173semidirect

Maximal subgroups

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

subgroupindex
#260121172
C2×#191495932
#260112582
#260112572
#342×C1732
#2032173