Finite Group Databasebeta

← #36365024

Subgroups — #36365024

Order 22144.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012112C212yesmaximal, complemented
2#26011787C212yesmaximal, complemented
3#26011772C212yesmaximal, complemented
4#26011737C212yesmaximal, complemented
5#26011732C211yesmaximal, characteristic, complemented
6#26011727C211yesmaximal, characteristic
7#26011714C212yesmaximal, complemented
8#26011707C212yesmaximal, complemented
9#424×C173C211yesmaximal, characteristic, complemented
10C2×Dic692C2211yescharacteristic
11C2×#14583815C2212yes—
12C2×D1384C2211yescharacteristic
13#19149679C2212yes—
14#19149674—22no—
15#19149667C2212yes—
16#19149665—22no—
17#19149661—22no—
18#19149658C2212yes—
19#19149657—22no—
20#19149653C2212yes—
21C4×Dic346C2212yes—
22C4×D692C2212yes—
23#19149613C2211yescharacteristic
24#19149613C2211yescharacteristic
25#19149613C2212yes—
26#19149604C2212yes—
27#19149604C2212yes—
28#19149601C2212yes—
29#19149601C2212yes—
30C2×M16×C173C2211yescharacteristic
31C22×C8×C173C2211yescharacteristic
32#117×C173C2211yescharacteristic
33C4⋊C8×C173C2212yes—
34C4⋊C8×C173C2212yes—
35C2×Dic346—22no—
36C2×Dic346C2312yes—
37C2×D692C2312yes—
38C2×C22⋊D173—22no—
39C2×C4×D173—22no—
40Dic692—41no—
41#14583815—42no—
42D1384—41no—
43C4⋊Dic173—22no—
44C4⋊Dic173C2312yes—
45C4⋊Dic173C2312yes—
46#14583805—22no—
47#14583805—22no—
48C22⋊Dic173—22no—
49#14583803—22no—
50#14583803—22no—
51#14583803—22no—
52#14583803—22no—
53C4×Dic173—22no—
54C22×C4×C173C2311yescharacteristic
55M16×C173—41no—
56C2×C8×C173—41no—
57C2×C8×C173C2311yescharacteristic
58C2×C8×C173C2311yescharacteristic
59C2×C8×C173C2312yes—
60C4⋊C4×C173D411yescharacteristic
61C4⋊C4×C173D411yescharacteristic
62C22⋊C4×C173D412yes—
63C42×C173C2312yes—
64C22×D173—22no—
65Dic346—22no—
66Dic346—22no—
67Dic346—82no—
68D692—22no—
69D692—22no—
70C22⋊D173—82no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C4×D173—82no—
75C23×C173C2×D411yescharacteristic
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2411yescharacteristic
82C8×C173—21no—
83C8×C173—22no—
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—22no—
94C4×C173—22no—
95C4×C173H(5,𝔽2)11yescharacteristic
96C4×C1732−1+411yescharacteristic
97Dic173—42no—
98Dic173—42no—
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#467—1731no—
112#466—1731no—
113#464—1732no—
114#463—1732no—
115#424D17311yescharacteristic, complemented
116C2×Q16—1731no—
117C2×SD16—1732no—
118C2×D8—1731no—
119C2×M16C2×D17311yescharacteristic
120C22×C8C2×D17311yescharacteristic
121C4⋊Q8—1732no—
122#126—3462no—
123#124—3462no—
124#124—1732no—
125#123—3462no—
126C22⋊Q8—3462no—
127C22⋊Q8—1732no—
128#121—1732no—
129C4×Q8—1732no—
130C4×D4—1732no—
131#117C2×D17311yescharacteristic
132#107—1731no—
133#107—1731no—
134#107—1732no—
135C4⋊C8C2×D17312yes—
136C4⋊C8C2×D17312yes—
137#103—1732no—
138#103—1732no—
139#102—1732no—
140#102—1732no—
141C2×Q8—3462no—
142C2×Q8—1732no—
143C2×D4—3462no—
144C2×D4—1732no—
145C22×C4—3462no—
146C22×C4C22×D17311yescharacteristic
147Q16—6921no—
148SD16—6922no—
149D8—6921no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D17311yescharacteristic
153C2×C8C22×D17312yes—
154C2×C8C22×D17311yescharacteristic
155C4⋊C4—3462no—
156C4⋊C4—3462no—
157C4⋊C4—3462no—
158C4⋊C4—1732no—
159C4⋊C4—1732no—
160C4⋊C4D69211yescharacteristic
161C4⋊C4D69211yescharacteristic
162C22⋊C4—3462no—
163C22⋊C4—3462no—
164C22⋊C4—3462no—
165C22⋊C4—3462no—
166C22⋊C4—3462no—
167C22⋊C4D69212yes—
168C42—3462no—
169C42C22×D17312yes—
170C23—3462no—
171C23C2×D69211yescharacteristic
172Q8—3462no—
173Q8—3462no—
174Q8—13842no—
175D4—3462no—
176D4—3462no—
177D4—13842no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—3462no—
181C2×C4—3462no—
182C2×C4—3462no—
183C2×C4—13842no—
184C2×C4C2×D69212yes—
185C2×C4C2×D69212yes—
186C2×C4C2×D69211yescharacteristic
187C2×C4C23×D17311yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6922no—
195C22—6922no—
196C22—6922no—
197C22C22×D69211yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—6922no—
202C4—6922no—
203C4—6922no—
204C4#1914972411yescharacteristic
205C4#1914972711yescharacteristic
206C2—41no—
207C2—13842no—
208C2#2601207211yescharacteristic
209C2#2601229611yescharacteristic
210C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#424×C173C2
socleC22×C173C22×D4
prime radical#26011727C2
hypercenter#424D173
hypocenterC173#3262
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011787C2semidirect
3. #26011772C2semidirect
4. #26011737C2semidirect
5. #26011732C2semidirect
7. #26011714C2semidirect
8. #26011707C2semidirect
9. #424×C173C2semidirect
105. C173#3262semidirect
115. #424D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260117872
#260117722
#260117372
#260117322
#260117272
#260117142
#260117072
#424×C1732
#3262173