Finite Group Databasebeta

← #36363953

Subgroups — #36363953

Order 22144.

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

normal subgroups only

This group has 424 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#26012039C211yesmaximal, characteristic
2C4⋊Q8×D173C211yesmaximal, characteristic, complemented
3#26011849C214yesmaximal, complemented
4#26011843C211yesmaximal, characteristic, complemented
5#26011842C211yesmaximal, characteristic, complemented
6#26011837C211yesmaximal, characteristic, complemented
7#26011489C214yesmaximal, complemented
8C4×#14583809C211yesmaximal, characteristic, complemented
9#486×C173C211yesmaximal, characteristic, complemented
10C2×#14583825C2212yes—
11C2×Q8×D173—21no—
12C2×#14583809C2212yes—
13#19149679C2211yescharacteristic
14#19149679C2211yescharacteristic
15#19149678—21no—
16#19149678C2212yes—
17#19149677—21no—
18#19149666C2212yes—
19#19149650—21no—
20#19149650—21no—
21D4×Dic173—21no—
22D4×Dic173—21no—
23C4⋊C4×D173—21no—
24C4⋊C4×D173C2212yes—
25C42×D173C2211yescharacteristic
26#19149604C2214yes—
27#19149602C2214yes—
28#19149600C2214yes—
29#19149589C2212yes—
30C4×#13063436C2211yescharacteristic
31C4⋊Q8×C173C2211yescharacteristic
32#127×C173C2211yescharacteristic
33#102×C173C2214yes—
34C4×C8×C173C2211yescharacteristic
35#14583825—28no—
36C2×Dic346—22no—
37C2×Dic346—21no—
38C2×Dic346C2312yes—
39Q8×D173—44no—
40C2×C22⋊D173—22no—
41C22×Dic173—22no—
42C2×C4×D173—22no—
43C2×C4×D173D411yescharacteristic
44C2×C4×D173D411yescharacteristic
45C2×C4×D173C2311yescharacteristic
46#14583809—24no—
47C2×#13063436C2312yes—
48C4⋊Dic173—21no—
49C4⋊Dic173—21no—
50C4⋊Dic173—21no—
51C4⋊Dic173C2312yes—
52#14583805—22no—
53#14583805—22no—
54C22⋊Dic173—22no—
55C22⋊Dic173—22no—
56C4×Dic173D411yescharacteristic
57C4×Dic173D411yescharacteristic
58C4×Dic173C2311yescharacteristic
59C2×Q8×C173—21no—
60C2×D4×C173—21no—
61C2×D4×C173—21no—
62C2×D4×C173C2312yes—
63C2×C8×C173C2312yes—
64C4⋊C4×C173—21no—
65C4⋊C4×C173C2312yes—
66C42×C173C2311yescharacteristic
67C22×D173C2×D411yescharacteristic
68Dic346—24no—
69Dic346—44no—
70Dic346—42no—
71C22⋊D173—44no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—44no—
75C2×Dic173C2×D411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C2×Dic173C2×D411yescharacteristic
78C4×D173—22no—
79C4×D173—22no—
80C4×D173—44no—
81C4×D173#4114yes—
82#13063436—22no—
83C23×C173—22no—
84Q8×C173—42no—
85D4×C173—24no—
86D4×C173—42no—
87D4×C173—42no—
88C2×C4×C173—22no—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2411yescharacteristic
92C8×C173—22no—
93C22×C173—42no—
94C22×C173—42no—
95C22×C173—42no—
96C22×C173C22×D411yescharacteristic
97C2×D173—21no—
98C2×D173—21no—
99C2×D173#12412yes—
100C4×C173—42no—
101C4×C173AGL(1,ℤ/8)12yes—
102C4×C173#13712yes—
103C4×C173C2×#4112yes—
104Dic173—21no—
105Dic173—21no—
106Dic173—42no—
107Dic173—42no—
108Dic173#12412yes—
109C2×C173—82no—
110C2×C173C2×#12411yescharacteristic
111C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C173C2×#13711yescharacteristic
113D173—22no—
114C173#314111yescharacteristic, complemented, Sylow
115#3141—1731nomaximal, Sylow
116#532—1731no—
117C2×C4⋊Q8—1731no—
118#489—1734no—
119#487—1731no—
120#487—1731no—
121#486—1731no—
122#486D17311yescharacteristic, complemented
123#417—1734no—
124C4×M16—1731no—
125C2×#41—1732no—
126C22×Q8—3461no—
127C2×M16—1732no—
128C4⋊Q8—3461no—
129C4⋊Q8—3461no—
130C4⋊Q8—1732no—
131C4⋊Q8—1731no—
132C4⋊Q8—1731no—
133C4⋊Q8C2×D17311yescharacteristic
134#127C2×D17311yescharacteristic
135#124—1732no—
136#121—3461no—
137#121—3461no—
138C4×D4—3461no—
139C4×D4—3461no—
140C2×C4⋊C4—3461no—
141C2×C4⋊C4—1732no—
142C2×C42—1731no—
143#103—1734no—
144#103—1734no—
145#102—1734no—
146#102C2×D17314yes—
147#97—1732no—
148C4×C8—1731no—
149C4×C8C2×D17311yescharacteristic
150#41—3468no—
151C2×Q8—21no—
152C2×Q8—3462no—
153C2×Q8—3461no—
154C2×Q8—1732no—
155C2×Q8—6924no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—3462no—
159C2×D4C22×D17312yes—
160C22×C4—3462no—
161C22×C4—3462no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4—1731no—
165M16—3464no—
166C2×C8—1732no—
167C2×C8C22×D17312yes—
168C4⋊C4—21no—
169C4⋊C4—3462no—
170C4⋊C4—3462no—
171C4⋊C4—3461no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C4⋊C4—1732no—
175C4⋊C4C22×D17312yes—
176C22⋊C4—3462no—
177C22⋊C4—3462no—
178C42—1731no—
179C42—1731no—
180C42—1731no—
181C42C22×D17311yescharacteristic
182C23—22no—
183C23—1731no—
184Q8—42no—
185Q8—3464no—
186Q8—6924no—
187Q8—6922no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—6924no—
192C2×C4—22no—
193C2×C4—3462no—
194C2×C4—3462no—
195C2×C4—3462no—
196C2×C4—3462no—
197C2×C4—1731no—
198C2×C4—1731no—
199C2×C4—1731no—
200C2×C4—1734no—
201C2×C4—6924no—
202C2×C4—6924no—
203C2×C4D4×D17311yescharacteristic
204C2×C4D4×D17311yescharacteristic
205C2×C4C23×D17311yescharacteristic
206C8—22no—
207C8—3462no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—3461no—
212C22—3461no—
213C22—1732no—
214C22C2×D4×D17311yescharacteristic
215C4—42no—
216C4—3461no—
217C4—3461no—
218C4—1732no—
219C4—6922no—
220C4—6922no—
221C4#1914970312yes—
222C4#1914970812yes—
223C4#41×D17312yes—
224C2—82no—
225C2—3462no—
226C2#124×D17311yescharacteristic
227C2C2×#1914970311yescharacteristic
228C2C2×#1914970811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#486×C173C2
socleC22×C173C22×D4
prime radical#26012039C2
hypercenter#486D173
hypocenterC173#3141
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D173C2semidirect
3. #26011849C2semidirect
4. #26011843C2semidirect
5. #26011842C2semidirect
6. #26011837C2semidirect
7. #26011489C2semidirect
8. C4×#14583809C2semidirect
9. #486×C173C2semidirect
114. C173#3141semidirect
122. #486D173semidirect

Maximal subgroups

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

subgroupindex
#260120392
C4⋊Q8×D1732
#260118492
#260118432
#260118422
#260118372
#260114892
C4×#145838092
#486×C1732
#3141173