Finite Group Databasebeta

← #36364507

Subgroups — #36364507

Order 22144.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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#26012046C211yesmaximal, characteristic
2C4⋊Q8×D173C211yesmaximal, characteristic
3#26011870C211yesmaximal, characteristic, complemented
4#26011842C211yesmaximal, characteristic, complemented
5#26011841C211yesmaximal, characteristic, complemented
6#26011699C214yesmaximal, complemented
7#26011645C214yesmaximal
8#26011542C211yesmaximal, characteristic
9#494×C173C211yesmaximal, characteristic, complemented
10C2×Q8×D173C2212yes—
11C2×#14583816C2214yes—
12#19149679C2211yescharacteristic
13#19149679C2211yescharacteristic
14#19149678—21no—
15#19149678—21no—
16#19149677C2212yes—
17#19149670—21no—
18#19149659—21no—
19#19149655C2212yes—
20Q8×Dic173C2212yes—
21#19149644—21no—
22#19149637—21no—
23C4⋊C4×D173—21no—
24C4⋊C4×D173—21no—
25C42×D173C2211yescharacteristic
26#19149608C2212yes—
27#19149604C2214yes—
28#19149603C2214yes—
29#19149591C2212yes—
30C4×#13063436C2211yescharacteristic
31C2×Q16×C173C2214yes—
32C4⋊Q8×C173C2211yescharacteristic
33C4⋊Q8×C173C2211yescharacteristic
34C4×C8×C173C2211yescharacteristic
35C2×Dic346—22no—
36C2×Dic346—21no—
37C2×Dic346C2312yes—
38Q8×D173—44no—
39C2×C4×D173—22no—
40C2×C4×D173D411yescharacteristic
41C2×C4×D173D411yescharacteristic
42C2×C4×D173C2311yescharacteristic
43#14583816—44no—
44C2×#13063436C2312yes—
45C4⋊Dic173—22no—
46C4⋊Dic173—21no—
47C4⋊Dic173—21no—
48C4⋊Dic173—21no—
49C4⋊Dic173C2312yes—
50#14583805—22no—
51#14583805—22no—
52#14583805—22no—
53#14583803—22no—
54C4×Dic173—22no—
55C4×Dic173D411yescharacteristic
56C4×Dic173D411yescharacteristic
57C4×Dic173C2311yescharacteristic
58C2×Q8×C173C2312yes—
59C2×Q8×C173C2312yes—
60Q16×C173—44no—
61C2×C8×C173C2312yes—
62C4⋊C4×C173—21no—
63C4⋊C4×C173—21no—
64C4⋊C4×C173—21no—
65C4⋊C4×C173—21no—
66C42×C173C2311yescharacteristic
67C22×D173C2×D411yescharacteristic
68Dic346—24no—
69Dic346—44no—
70Dic346—42no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173C2×D411yescharacteristic
74C2×Dic173C2×D411yescharacteristic
75C2×Dic173C2×D411yescharacteristic
76C4×D173—22no—
77C4×D173—22no—
78C4×D173—44no—
79C4×D173Q1614yescomplemented
80#13063436—22no—
81Q8×C173—24no—
82Q8×C173—24no—
83C2×C4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2×D411yescharacteristic
87C2×C4×C173C2411yescharacteristic
88C8×C173—22no—
89C22×C173C22×D411yescharacteristic
90C2×D173—21no—
91C2×D173—21no—
92C2×D173C2×Q1612yes—
93C4×C173—42no—
94C4×C173—42no—
95C4×C173C2×Q1612yes—
96C4×C173#13712yes—
97C4×C173H(5,𝔽2)12yes—
98Dic173—21no—
99Dic173—21no—
100Dic173—42no—
101Dic173—42no—
102Dic173C2×Q1612yes—
103C2×C173#53511yescharacteristic
104C2×C173C22×Q1611yescharacteristic
105C2×C173C2×#13711yescharacteristic
106D173—22no—
107C173#320911yescharacteristic, complemented, Sylow
108#3209—1731nomaximal, Sylow
109#533—1731no—
110C2×C4⋊Q8—1731no—
111#494—1731no—
112#494D17311yescharacteristic, complemented
113#487—1731no—
114#487—1731no—
115#462—1734no—
116#451—1734no—
117#431—1731no—
118C22×Q8—1732no—
119C2×Q16—1734no—
120C2×Q16C2×D17314yes—
121C4⋊Q8—3461no—
122C4⋊Q8—3461no—
123C4⋊Q8—1732no—
124C4⋊Q8—1731no—
125C4⋊Q8—1731no—
126C4⋊Q8C2×D17311yescharacteristic
127C4⋊Q8C2×D17311yescharacteristic
128#125—3461no—
129C22⋊Q8—3461no—
130C22⋊Q8—1732no—
131C4×Q8—3461no—
132C4×Q8—1732no—
133#117—3461no—
134C2×C4⋊C4—3461no—
135C2×C4⋊C4—3461no—
136C2×C42—1731no—
137C4⋊C8—1732no—
138#103—1734no—
139#103—1734no—
140C22⋊C8—1732no—
141C4×C8—1731no—
142C4×C8C2×D17311yescharacteristic
143C2×Q8—3462no—
144C2×Q8—3461no—
145C2×Q8—1732no—
146C2×Q8—6924no—
147C2×Q8C22×D17312yes—
148C2×Q8C22×D17312yes—
149C22×C4—3462no—
150C22×C4—1731no—
151C22×C4—1731no—
152C22×C4—1731no—
153Q16—44no—
154Q16—6924no—
155C2×C8—1732no—
156C2×C8C22×D17312yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3462no—
162C4⋊C4—3462no—
163C4⋊C4—3462no—
164C4⋊C4—3462no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—1732no—
169C22⋊C4—3462no—
170C42—3462no—
171C42—1731no—
172C42—1731no—
173C42—1731no—
174C42C22×D17311yescharacteristic
175C23—1731no—
176Q8—24no—
177Q8—24no—
178Q8—3464no—
179Q8—6924no—
180Q8—6922no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—3462no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—3462no—
187C2×C4—1731no—
188C2×C4—1731no—
189C2×C4—1731no—
190C2×C4—1734no—
191C2×C4—6924no—
192C2×C4D4×D17311yescharacteristic
193C2×C4D4×D17311yescharacteristic
194C2×C4C23×D17311yescharacteristic
195C8—22no—
196C8—3462no—
197C22—3461no—
198C22—3461no—
199C22—1732no—
200C22C2×D4×D17311yescharacteristic
201C4—42no—
202C4—42no—
203C4—3461no—
204C4—3461no—
205C4—1732no—
206C4—6922no—
207C4—6922no—
208C4Q16×D17312yes—
209C4#1914971112yes—
210C4#1914972412yes—
211C2—3462no—
212C2#2601205411yescharacteristic
213C2C2×Q16×D17311yescharacteristic
214C2C2×#1914971111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#494×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#494D173
hypocenterC173#3209
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. #26011870C2semidirect
4. #26011842C2semidirect
5. #26011841C2semidirect
6. #26011699C2semidirect
9. #494×C173C2semidirect
79. C4×D173Q16semidirect
107. C173#3209semidirect
112. #494D173semidirect

Maximal subgroups

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

subgroupindex
#260120462
C4⋊Q8×D1732
#260118702
#260118422
#260118412
#260116992
#260116452
#260115422
#494×C1732
#3209173