Finite Group Databasebeta

← #36354578

Subgroups — #36354578

Order 22144.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19149705C211yesmaximal, characteristic, complemented
2#26012058C211yesmaximal, characteristic, complemented
3C2×#19149607C211yesmaximal, characteristic, complemented
4#26011496C211yesmaximal, characteristic, complemented
5C2×#19149596C211yesmaximal, characteristic, complemented
6#26011251C211yesmaximal, characteristic, complemented
7C2×#127×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9#19149705—22no—
10#19149705—22no—
11C2×#14583814—21no—
12C2×#14583811—21no—
13C2×#14583810C2211yescharacteristic
14C2×#14583810C2211yescharacteristic
15#19149675—21no—
16#19149666—21no—
17#19149650—22no—
18#19149648—22no—
19#19149638C2211yescharacteristic
20#19149607—22no—
21#19149600—22no—
22#19149596—22no—
23C22×D4×C173—22no—
24C22×D4×C173C2211yescharacteristic
25C22×D4×C173C2211yescharacteristic
26#127×C173—22no—
27#127×C173—22no—
28C2×C42×C173C2211yescharacteristic
29#14583826—21no—
30#14583826—21no—
31#14583826—21no—
32#14583826—21no—
33C2×Dic346D411yescharacteristic, complemented
34C2×D692D411yescharacteristic, complemented
35C2×C22⋊D173—24no—
36C2×C22⋊D173—21no—
37C2×C4×D173—21no—
38#14583814—24no—
39#14583811—24no—
40#14583810—21no—
41#14583810—21no—
42#14583810—21no—
43#14583810D412yescomplemented
44C2×#13063436—21no—
45C2×#13063436—21no—
46C4⋊Dic173D412yescomplemented
47C22⋊Dic173—24no—
48C22⋊Dic173—21no—
49C4×Dic173—21no—
50C24×C173—21no—
51C24×C173D412yes—
52C2×D4×C173—28no—
53C2×D4×C173—21no—
54C2×D4×C173—21no—
55C2×D4×C173—21no—
56C2×D4×C173—21no—
57C2×D4×C173—21no—
58C2×D4×C173—21no—
59C2×D4×C173—21no—
60C2×D4×C173—21no—
61C2×D4×C173—21no—
62C2×D4×C173—21no—
63C2×D4×C173—21no—
64C2×D4×C173—48no—
65C2×D4×C173D412yescomplemented
66C22×C4×C173—21no—
67C22×C4×C173C2311yescharacteristic
68C42×C173—21no—
69C42×C173D412yescomplemented
70C22×D173—21no—
71Dic346—21no—
72Dic346—21no—
73D692—21no—
74D692—21no—
75C22⋊D173—41no—
76C22⋊D173—41no—
77C22⋊D173—84no—
78C2×Dic173—21no—
79C2×Dic173—22no—
80C4×D173—41no—
81C4×D173—41no—
82#13063436—22no—
83#13063436—41no—
84C23×C173—22no—
85C23×C173—22no—
86C23×C173—22no—
87C23×C173—22no—
88C23×C173—44no—
89C23×C173—44no—
90C23×C173—41no—
91C23×C173—41no—
92C23×C173C2×D411yescharacteristic
93D4×C173—22no—
94D4×C173—22no—
95D4×C173—416no—
96D4×C173—41no—
97D4×C173—41no—
98D4×C173—41no—
99D4×C173—41no—
100D4×C173—41no—
101D4×C173—41no—
102C2×C4×C173—24no—
103C2×C4×C173—22no—
104C2×C4×C173C2×D411yescharacteristic
105C2×C4×C173C2×D411yescharacteristic
106C2×C4×C173C2×D411yescharacteristic
107C2×C4×C173C2×D411yescharacteristic
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173#4111yescharacteristic
110C22×C173—22no—
111C22×C173—44no—
112C22×C173—44no—
113C22×C173—42no—
114C22×C173—42no—
115C22×C173—44no—
116C22×C173—82no—
117C22×C173—82no—
118C22×C173—82no—
119C22×C173#12011yescharacteristic
120C22×C173#12011yescharacteristic
121C22×C173#12111yescharacteristic
122C2×D173—41no—
123C2×D173—42no—
124C4×C173—24no—
125C4×C173#12011yescharacteristic
126C4×C173#12111yescharacteristic
127C4×C173#12111yescharacteristic
128C4×C173#12711yescharacteristic
129Dic173—41no—
130Dic173—42no—
131C2×C173—21no—
132C2×C173—21no—
133C2×C173—82no—
134C2×C173—82no—
135C2×C173#39211yescharacteristic
136C2×C173#45312yes—
137D173—81no—
138C173#200211yescharacteristic, complemented, Sylow
139#2002—1731nomaximal, Sylow
140C2×AGL(1,ℤ/8)—1731no—
141#535—1731no—
142C2×#127D17311yescharacteristic, complemented
143C2×#104—1731no—
144#418—1731no—
145C2×#100—1731no—
146#339—1731no—
147C2×#41—1731no—
148C22×D4—22no—
149C22×D4C2×D17311yescharacteristic
150C22×D4C2×D17311yescharacteristic
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—3462no—
153C2×SD16—3461no—
154C2×D8—3461no—
155C2×M16—1731no—
156C2×M16—1731no—
157#127—22no—
158#127—22no—
159#127—3461no—
160#124—3461no—
161#121—3462no—
162#120—3462no—
163#117—1731no—
164C2×C42C2×D17311yescharacteristic
165#104—3462no—
166#102—3462no—
167#100—3462no—
168C24—21no—
169C24C22⋊D17312yes—
170#41—3461no—
171#41—3461no—
172#41—3461no—
173#41—3461no—
174C2×Q8—1731no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—3464no—
189C2×D4—3461no—
190C2×D4—1731no—
191C2×D4C22⋊D17312yescomplemented
192C22×C4—21no—
193C22×C4—3461no—
194C22×C4C22×D17311yescharacteristic
195SD16—3464no—
196D8—3464no—
197M16—3461no—
198M16—3461no—
199M16—3461no—
200M16—1732no—
201C2×C8—3461no—
202C2×C8—3461no—
203C4⋊C4—1732no—
204C22⋊C4—3464no—
205C22⋊C4—3461no—
206C42—21no—
207C42—3461no—
208C42C22⋊D17312yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—3461no—
218C23C2×C22⋊D17311yescharacteristic
219Q8—3461no—
220Q8—3461no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—3461no—
231D4—3461no—
232D4—13844no—
233D4—6921no—
234D4—6921no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—3462no—
238C2×C4—3461no—
239C2×C4—6921no—
240C2×C4—6921no—
241C2×C4D4×D17311yescharacteristic
242C2×C4D4×D17311yescharacteristic
243C2×C4D4×D17311yescharacteristic
244C2×C4C2×C22⋊D17311yescharacteristic
245C2×C4C2×C22⋊D17311yescharacteristic
246C2×C4#1458382511yescharacteristic
247C8—3462no—
248C8—6921no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—6921no—
259C22—6922no—
260C22#1914964711yescharacteristic
261C22#1914964811yescharacteristic
262C22#1914965111yescharacteristic
263C4—24no—
264C4—6921no—
265C4—6922no—
266C4#1914964711yescharacteristic
267C4#1914965011yescharacteristic
268C4#1914965111yescharacteristic
269C4#1914967511yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—13841no—
275C2#2601139911yescharacteristic
276C2#2601165812yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#127×C173C2
socleC22×C173#120
hypercenterC2×#127D173
hypocenterC173#2002
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. #26012058C2semidirect
3. C2×#19149607C2semidirect
4. #26011496C2semidirect
5. C2×#19149596C2semidirect
6. #26011251C2semidirect
7. C2×#127×C173C2semidirect
33. C2×Dic346D4semidirect
34. C2×D692D4semidirect
43. #14583810D4semidirect
46. C4⋊Dic173D4semidirect
65. C2×D4×C173D4semidirect
69. C42×C173D4semidirect
138. C173#2002semidirect
142. C2×#127D173semidirect
191. C2×D4C22⋊D173semidirect
208. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260120582
C2×#191496072
#260114962
C2×#191495962
#260112512
C2×#127×C1732
#2002173