Finite Group Databasebeta

← #36369555

Subgroups — #36369555

Order 22144.

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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×#19149767C211yesmaximal, characteristic, complemented
2C2×#19149744C211yesmaximal, characteristic, complemented
3C2×#19149743C211yesmaximal, characteristic, complemented
4#26012465C211yesmaximal, characteristic, complemented
5#26012367C211yesmaximal, characteristic, complemented
6(C2×#10621924)⋊C8C211yesmaximal, characteristic, complemented
7C2×#19149700C211yesmaximal, characteristic
8#19149767—22no—
9#19149767—22no—
10C2×#14583836—21no—
11C2×#14583836C2211yescharacteristic
12C2×#14583836C2211yescharacteristic
13C2×D173⋊C8—21no—
14C2×D173⋊C8C2211yescharacteristic
15C2×C4×#10621924C2211yescharacteristic
16#19149744—22no—
17#19149743—22no—
18C22⋊#13063443—21no—
19C22⋊#13063443—21no—
20C2×#14583827C2211yescharacteristic
21C2×#14583825C2211yescharacteristic
22#19149700—22no—
23#19149700—22no—
24C2×#14583816C411yescharacteristic, complemented
25C2×#14583815C411yescharacteristic, complemented
26C2×#14583811C411yescharacteristic, complemented
27C2×C8×D173C2211yescharacteristic
28C2×SD16×C173C411yescharacteristic, complemented
29C22×#10621924—21no—
30#14583836—24no—
31#14583836—21no—
32#14583836—21no—
33#14583836—21no—
34#14583836D412yes—
35D173⋊C8—24no—
36D173⋊C8—21no—
37D173⋊C8—21no—
38C2×#13063443—21no—
39C2×#13063443—21no—
40C2×#13063443D411yescharacteristic
41C2×#13063443D411yescharacteristic
42C4×#10621924—21no—
43C4×#10621924—21no—
44#14583827—21no—
45#14583827—21no—
46#14583827D412yes—
47#14583827D412yes—
48#14583825—21no—
49#14583825—21no—
50#14583825—21no—
51#14583825—21no—
52C2×Dic346C2×C411yescharacteristic
53C2×D692—21no—
54C2×D692C2×C411yescharacteristic
55C2×C22⋊D173—21no—
56C22×Dic173—21no—
57C2×C4×D173—21no—
58C2×C4×D173C2311yescharacteristic
59#14583816—22no—
60#14583815—22no—
61#14583811—22no—
62C2×#13063436C2×C411yescharacteristic
63C8×D173—21no—
64C8×D173—21no—
65C2×Q8×C173C2×C411yescharacteristic
66C2×D4×C173C2×C411yescharacteristic
67SD16×C173—22no—
68C2×C8×C173C2×C411yescharacteristic
69C2×#10621924—22no—
70C2×#10621924—41no—
71C2×#10621924—41no—
72#13063443—22no—
73#13063443—21no—
74#13063443—21no—
75#13063443—41no—
76C22×D173—21no—
77C22×D173#4111yescharacteristic
78Dic346—21no—
79Dic346—21no—
80D692—22no—
81D692—22no—
82D692—21no—
83D692C22⋊C412yes—
84C22⋊D173—41no—
85C22⋊D173—41no—
86C2×Dic173—21no—
87C2×Dic173—41no—
88C2×Dic173—41no—
89C2×Dic173C2×D411yescharacteristic
90C4×D173—22no—
91C4×D173—22no—
92C4×D173C2×D411yescharacteristic
93C4×D173C2×D411yescharacteristic
94C4×D173C2×D411yescharacteristic
95C4×D173#4111yescharacteristic
96#13063436—21no—
97C23×C173—21no—
98Q8×C173—21no—
99Q8×C173C22⋊C412yes—
100D4×C173—21no—
101D4×C173—21no—
102C2×C4×C173—21no—
103C2×C4×C173C22×C411yescharacteristic
104C8×C173—21no—
105C22×C173—41no—
106C22×C173—41no—
107C22×C173—41no—
108C22×C173C4×D411yescharacteristic
109C2×D173—21no—
110C2×D173—21no—
111C2×D173—22no—
112C2×D173—41no—
113C2×D173—41no—
114C2×D173#12111yescharacteristic
115C2×D173#12311yescharacteristic
116#10621924—42no—
117C4×C173—22no—
118C4×C173C2×C22⋊C411yescharacteristic
119C4×C173C4×D411yescharacteristic
120Dic173—41no—
121Dic173#12011yescharacteristic
122Dic173#12111yescharacteristic
123C2×C173—81no—
124C2×C173#38611yescharacteristic
125C2×C173#44311yescharacteristic
126C2×C173#44411yescharacteristic
127D173—21no—
128D173—21no—
129D173—42no—
130C173#187611yescharacteristic, complemented, Sylow
131#1876—1731nomaximal, Sylow
132C2×#135—1731no—
133C2×#131—1731no—
134C2×#104—1731no—
135C2×#104—1731no—
136#408—1731no—
137#341—1731no—
138#336—1731no—
139C2×#41—1731no—
140C2×#41—1731no—
141#135—3462no—
142#135—3462no—
143C2×Q16—1731no—
144C2×SD16—1731no—
145C2×SD16#1062192411yescharacteristic, complemented
146C2×D8—1731no—
147#131—3462no—
148#131—3462no—
149C2×M16—3461no—
150C2×M16—1731no—
151C2×M16—1731no—
152C22×C8—3461no—
153C22×C8—1731no—
154C22×C8—1731no—
155C2×C42—1731no—
156#104—3462no—
157#104—3462no—
158C22⋊C8—3461no—
159C22⋊C8—3461no—
160#41—3461no—
161#41—3461no—
162#41—3461no—
163#41—3461no—
164#41—3461no—
165#41—3461no—
166#41—1732no—
167#41—1732no—
168C2×Q8—1731no—
169C2×Q8C2×#1062192411yescharacteristic
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—1731no—
173C2×D4C2×#1062192411yescharacteristic
174C22×C4—3461no—
175C22×C4—3461no—
176C22×C4—3461no—
177C22×C4—1731no—
178Q16—3462no—
179SD16—22no—
180SD16—3462no—
181D8—3462no—
182M16—3464no—
183M16—3461no—
184M16—3461no—
185M16—3461no—
186M16—1732no—
187C2×C8—3461no—
188C2×C8—3461no—
189C2×C8—3464no—
190C2×C8—3461no—
191C2×C8—3461no—
192C2×C8—3461no—
193C2×C8—3461no—
194C2×C8—1731no—
195C2×C8—1731no—
196C2×C8—1731no—
197C2×C8C2×#1062192411yescharacteristic
198C42—3461no—
199C42—3461no—
200C23—21no—
201C23—3461no—
202C23—1731no—
203Q8—21no—
204Q8—3461no—
205Q8—3461no—
206Q8C22⋊#1062192412yes—
207D4—21no—
208D4—21no—
209D4—3462no—
210D4—3462no—
211D4—3461no—
212D4—1732no—
213D4—6921no—
214D4—6921no—
215C2×C4—21no—
216C2×C4—3462no—
217C2×C4—3462no—
218C2×C4—3462no—
219C2×C4—3461no—
220C2×C4—1731no—
221C2×C4—1731no—
222C2×C4—1731no—
223C2×C4—1731no—
224C2×C4—1731no—
225C2×C4—6921no—
226C2×C4—6921no—
227C2×C4—6921no—
228C2×C4—6921no—
229C2×C4C22×#1062192411yescharacteristic
230C8—21no—
231C8—3461no—
232C8—3462no—
233C8—3461no—
234C8—3461no—
235C8—6921no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—3461no—
240C22—3461no—
241C22—3462no—
242C22—1731no—
243C22—1731no—
244C22—6921no—
245C22—6921no—
246C22D4×#1062192411yescharacteristic
247C4—22no—
248C4—1731no—
249C4—1731no—
250C4—6922no—
251C4—6921no—
252C4C2×C22⋊#1062192411yescharacteristic
253C4D4×#1062192411yescharacteristic
254C2—81no—
255C2—3461no—
256C2—3461no—
257C2—6922no—
258C2#2601243911yescharacteristic
259C2#2601253411yescharacteristic
260C2#2601253611yescharacteristic
subgroupG/H
centerC22D4×#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×SD16×C173C4
socleC22×C173C4×D4
prime radicalC2×#19149700C2
cosocleC2×C4×D173C23
hypercenterC2×SD16#10621924
hypocenterC173#1876
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19149767C2semidirect
2. C2×#19149744C2semidirect
3. C2×#19149743C2semidirect
4. #26012465C2semidirect
5. #26012367C2semidirect
6. (C2×#10621924)⋊C8C2semidirect
24. C2×#14583816C4semidirect
25. C2×#14583815C4semidirect
26. C2×#14583811C4semidirect
28. C2×SD16×C173C4semidirect
130. C173#1876semidirect
145. C2×SD16#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497672
C2×#191497442
C2×#191497432
#260124652
#260123672
(C2×#10621924)⋊C82
C2×#191497002
#1876173