Finite Group Databasebeta

← #36365025

Subgroups — #36365025

Order 22144.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012114C212yesmaximal, complemented
2#26011784C212yesmaximal, complemented
3#26011770C212yesmaximal, complemented
4#26011739C212yesmaximal, complemented
5#26011727C211yesmaximal, characteristic
6#26011715C212yesmaximal, complemented
7#26011709C212yesmaximal, complemented
8#26011517C211yesmaximal, characteristic, complemented
9#467×C173C211yesmaximal, characteristic, complemented
10C2×#14583813C2212yes—
11C2×D1384C2211yescharacteristic
12C2×#14583810C2211yescharacteristic
13#19149678C2212yes—
14#19149674—22no—
15#19149668C2212yes—
16#19149665—22no—
17#19149661—22no—
18#19149659—22no—
19#19149653C2212yes—
20Q8×Dic173C2212yes—
21#19149641C2212yes—
22#19149638C2211yescharacteristic
23#19149613C2211yescharacteristic
24#19149612C2212yes—
25#19149608C2212yes—
26#19149608C2212yes—
27#19149603C2212yes—
28#19149601C2212yes—
29#19149599C2212yes—
30C2×Q16×C173C2211yescharacteristic
31C22×C8×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#107×C173C2211yescharacteristic
34#103×C173C2212yes—
35C2×Dic346—22no—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39#14583813—42no—
40D1384—41no—
41#14583810—41no—
42C2×#13063436C2312yes—
43C4⋊Dic173—22no—
44C4⋊Dic173D411yescharacteristic
45C4⋊Dic173D411yescharacteristic
46C4⋊Dic173C2312yes—
47#14583805—22no—
48C22⋊Dic173D412yes—
49#14583803—22no—
50#14583803—22no—
51#14583803—22no—
52#14583803—22no—
53C4×Dic173—22no—
54C4×Dic173C2312yes—
55C2×Q8×C173C2312yes—
56C22×C4×C173C2311yescharacteristic
57Q16×C173—41no—
58C2×C8×C173—41no—
59C2×C8×C173C2311yescharacteristic
60C2×C8×C173C2311yescharacteristic
61C4⋊C4×C173—22no—
62C4⋊C4×C173C2312yes—
63C22⋊C4×C173—22no—
64C22×D173—22no—
65Dic346—82no—
66D692—22no—
67D692—22no—
68C22⋊D173—82no—
69C2×Dic173—22no—
70C2×Dic173C2×D412yes—
71C2×Dic173C2×D412yes—
72C4×D173—82no—
73#13063436—22no—
74C23×C173C2×D411yescharacteristic
75Q8×C173—22no—
76Q8×C173—22no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—22no—
80C2×C4×C173—22no—
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2411yescharacteristic
83C8×C173—21no—
84C8×C173—21no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173C22×D411yescharacteristic
89C2×D173—42no—
90C2×D173—42no—
91C2×D173—42no—
92C4×C173—21no—
93C4×C173—42no—
94C4×C173—42no—
95C4×C173H(5,𝔽2)11yescharacteristic
96C4×C1732−1+411yescharacteristic
97Dic173—22no—
98Dic173—22no—
99Dic173—42no—
100C2×C173—41no—
101C2×C173#53711yescharacteristic
102C2×C173#57611yescharacteristic
103C2×C173#57811yescharacteristic
104D173—82no—
105C173#326211yescharacteristic, complemented, Sylow
106#3262—1731nomaximal, Sylow
107#542—1732no—
108#477—1732no—
109#474—1732no—
110#468—1732no—
111#467D17311yescharacteristic, complemented
112#466—1731no—
113#464—1732no—
114#463—1732no—
115#424—1731no—
116C2×Q16C2×D17311yescharacteristic
117C2×SD16—1732no—
118C2×D8—1731no—
119C2×M16—1731no—
120C22×C8C2×D17311yescharacteristic
121C4⋊Q8—1732no—
122#126—3462no—
123#124—3462no—
124#124—1732no—
125#123—3462no—
126C22⋊Q8—3462no—
127C22⋊Q8C2×D17312yes—
128#121—1732no—
129C4×Q8—1732no—
130C4×D4—1732no—
131#117—1731no—
132#107—1731no—
133#107—1732no—
134#107C2×D17311yescharacteristic
135C4⋊C8—1732no—
136C4⋊C8—1732no—
137#103—1732no—
138#103C2×D17312yes—
139#102—1732no—
140#102—1732no—
141C2×Q8—3462no—
142C2×Q8C22×D17312yes—
143C2×D4—3462no—
144C2×D4—1732no—
145C22×C4—3462no—
146C22×C4C22×D17311yescharacteristic
147Q16—41no—
148SD16—6922no—
149D8—6921no—
150M16—6921no—
151C2×C8—41no—
152C2×C8—1732no—
153C2×C8C22×D17311yescharacteristic
154C2×C8C22×D17311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—3462no—
157C4⋊C4—3462no—
158C4⋊C4—1732no—
159C4⋊C4—1731no—
160C4⋊C4—1731no—
161C4⋊C4C22×D17312yes—
162C22⋊C4—22no—
163C22⋊C4—3462no—
164C22⋊C4—3462no—
165C22⋊C4—3462no—
166C22⋊C4—3462no—
167C22⋊C4—1732no—
168C42—3462no—
169C42—1732no—
170C23—3462no—
171C23D4×D17311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—13842no—
175D4—3462no—
176D4—3462no—
177D4—13842no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—3462no—
183C2×C4—13842no—
184C2×C4—1732no—
185C2×C4—1732no—
186C2×C4D4×D17311yescharacteristic
187C2×C4C23×D17311yescharacteristic
188C8—21no—
189C8—21no—
190C8—3462no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6922no—
195C22—6922no—
196C22—6922no—
197C22C2×D4×D17311yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3462no—
202C4—3462no—
203C4—6922no—
204C4#1914972411yescharacteristic
205C4#1914972611yescharacteristic
206C2—41no—
207C2—13842no—
208C2#2601206811yescharacteristic
209C2#2601229311yescharacteristic
210C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#26011727C2
hypercenter#467D173
hypocenterC173#3262
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26011784C2semidirect
3. #26011770C2semidirect
4. #26011739C2semidirect
6. #26011715C2semidirect
7. #26011709C2semidirect
8. #26011517C2semidirect
9. #467×C173C2semidirect
105. C173#3262semidirect
111. #467D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260117842
#260117702
#260117392
#260117272
#260117152
#260117092
#260115172
#467×C1732
#3262173