Finite Group Databasebeta

← #36353759

Subgroups — #36353759

Order 22144.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#19149637C211yesmaximal, characteristic
2C2×#19149610C211yesmaximal, characteristic, complemented
3C2×#19149591C211yesmaximal, characteristic
4#26011348C211yesmaximal, characteristic
5#26011254C211yesmaximal, characteristic, complemented
6#26011253C211yesmaximal, characteristic, complemented
7C2×#107×C173C211yesmaximal, characteristic, complemented
8C22×C4×D173C2211yescharacteristic
9C22×#13063436C2211yescharacteristic
10#19149637—22no—
11#19149637—22no—
12C2×C4⋊Dic173C2211yescharacteristic
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583803—21no—
15C2×#14583803—21no—
16C2×C4×Dic173—21no—
17#19149610—22no—
18#19149591C414yescomplemented
19#19149586—21no—
20C22×C8×C173C2211yescharacteristic
21C2×C4⋊C4×C173C2211yescharacteristic
22C2×C4⋊C4×C173C2211yescharacteristic
23#107×C173—22no—
24C23×D173D411yescharacteristic
25C22×Dic173—21no—
26C22×Dic173—21no—
27C22×Dic173D411yescharacteristic
28C2×C4×D173—21no—
29C2×C4×D173—21no—
30C2×C4×D173—21no—
31C2×C4×D173—21no—
32C2×C4×D173C2×C412yes—
33C2×C4×D173Q812yes—
34C2×#13063436—21no—
35C2×#13063436—21no—
36C2×#13063436C2×C412yes—
37C4⋊Dic173—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173D412yes—
41#14583803—24no—
42#14583803—42no—
43C4×Dic173—24no—
44C22×C4×C173—21no—
45C22×C4×C173—21no—
46C22×C4×C173C2311yescharacteristic
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C2×C8×C173C2×C412yes—
50C4⋊C4×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D412yes—
54C22×D173—21no—
55C22×D173—21no—
56C22×D173—21no—
57C22×D173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173C4⋊C412yes—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—22no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173C4⋊C412yes—
70C4×D173—22no—
71C4×D173—22no—
72C4×D173—22no—
73C4×D173—22no—
74#13063436—22no—
75C23×C173C2×D411yescharacteristic
76C2×C4×C173—22no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173C22×C411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×Q811yescharacteristic
86C2×C4×C173#4111yescharacteristic
87C2×C4×C173#4111yescharacteristic
88C8×C173—22no—
89C22×C173C2×C4⋊C411yescharacteristic
90C22×C173#12011yescharacteristic
91C22×C173#12311yescharacteristic
92C22×C173#13511yescharacteristic
93C22×C173#13511yescharacteristic
94C22×C173AGL(1,ℤ/8)11yescharacteristic
95C22×C173#13711yescharacteristic
96C2×D173—22no—
97C2×D173—22no—
98C2×D173—22no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C4×C173—42no—
104C4×C173—81no—
105C4×C173C4×D411yescharacteristic
106C4×C173C4×D411yescharacteristic
107C4×C173C22⋊Q811yescharacteristic
108C4×C173C22⋊Q811yescharacteristic
109Dic173—22no—
110Dic173—42no—
111Dic173—81no—
112C2×C173#38511yescharacteristic
113C2×C173#42711yescharacteristic
114C2×C173#42811yescharacteristic
115C2×C173#44911yescharacteristic
116C2×C173#45211yescharacteristic
117C2×C173#48211yescharacteristic
118C2×C173#48511yescharacteristic
119D173—42no—
120C173#185411yescharacteristic, complemented, Sylow
121#1854—1731nomaximal, Sylow
122C2×#117—1731no—
123C2×#107D17311yescharacteristic, complemented
124C2×#106—1731no—
125C2×C22⋊C8—1731no—
126#380—1731no—
127#340—1731no—
128#340—1731no—
129C23×C4—1731no—
130C22×C8—1731no—
131C22×C8C2×D17311yescharacteristic
132#117—3462no—
133#117—3462no—
134C2×C4⋊C4—1731no—
135C2×C4⋊C4—1731no—
136C2×C4⋊C4C2×D17311yescharacteristic
137C2×C4⋊C4C2×D17311yescharacteristic
138C2×C22⋊C4—3461no—
139C2×C22⋊C4—3461no—
140C2×C42—3461no—
141#107—22no—
142#106—3462no—
143C22⋊C8—1734no—
144#95—3461no—
145C24—1731no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3461no—
149C22×C4—3461no—
150C22×C4—3461no—
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—3461no—
154C22×C4—1731no—
155C22×C4—1732no—
156C22×C4—1732no—
157C22×C4C22×D17311yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—3461no—
161C2×C8—3461no—
162C2×C8—1732no—
163C2×C8C4×D17312yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—3461no—
170C4⋊C4—1732no—
171C4⋊C4C22⋊D17312yes—
172C22⋊C4—3464no—
173C22⋊C4—6922no—
174C42—3464no—
175C23—3461no—
176C23—3461no—
177C23—3461no—
178C23—3461no—
179C23—3461no—
180C23—3461no—
181C23—1732no—
182C23D4×D17311yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3461no—
190C2×C4—3461no—
191C2×C4—3462no—
192C2×C4—3462no—
193C2×C4—3462no—
194C2×C4—3462no—
195C2×C4—3462no—
196C2×C4—1732no—
197C2×C4—6921no—
198C2×C4—6921no—
199C2×C4—6921no—
200C2×C4—6921no—
201C2×C4—6921no—
202C2×C4C2×C4×D17311yescharacteristic
203C2×C4D4×D17311yescharacteristic
204C2×C4C2×C22⋊D17311yescharacteristic
205C2×C4Q8×D17311yescharacteristic
206C2×C4#1458382611yescharacteristic
207C2×C4#1458382711yescharacteristic
208C8—22no—
209C8—3462no—
210C22—3462no—
211C22—3462no—
212C22—3462no—
213C22—6921no—
214C22—6921no—
215C22—6921no—
216C22—6921no—
217C22C4⋊C4×D17311yescharacteristic
218C22#1914964711yescharacteristic
219C22#1914966411yescharacteristic
220C22#1914969811yescharacteristic
221C22#1914970211yescharacteristic
222C22#1914970311yescharacteristic
223C22#1914971111yescharacteristic
224C4—42no—
225C4—81no—
226C4—3462no—
227C4—13841no—
228C4—6922no—
229C4C4×C22⋊D17311yescharacteristic
230C4#1914964111yescharacteristic
231C4#1914965511yescharacteristic
232C4#1914965711yescharacteristic
233C2—6922no—
234C2#2601136211yescharacteristic
235C2#2601152811yescharacteristic
236C2#2601153211yescharacteristic
237C2#2601163311yescharacteristic
238C2#2601165211yescharacteristic
239C2#2601181811yescharacteristic
240C2#2601183411yescharacteristic
subgroupG/H
centerC23D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC2×#107×C173C2
socleC23×C173C2×D4
prime radicalC23×D173D4
cosocleC22×C4×C173C23
hypercenterC2×#107D173
hypocenterC173#1854
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#19149610C2semidirect
5. #26011254C2semidirect
6. #26011253C2semidirect
7. C2×#107×C173C2semidirect
18. #19149591C4semidirect
120. C173#1854semidirect
123. C2×#107D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496372
C2×#191496102
C2×#191495912
#260113482
#260112542
#260112532
C2×#107×C1732
#1854173