Finite Group Databasebeta

← #36365029

Subgroups — #36365029

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#26011780C212yesmaximal, complemented
3#26011774C212yesmaximal, complemented
4#26011744C211yesmaximal, characteristic
5#26011722C212yesmaximal, complemented
6#26011715C212yesmaximal, complemented
7#26011709C212yesmaximal, complemented
8#26011519C211yesmaximal, characteristic, complemented
9#470×C173C211yesmaximal, characteristic, complemented
10C2×#14583813C2212yes—
11C2×D1384C2211yescharacteristic
12C22×#13063436C2211yescharacteristic
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#19149611C2211yescharacteristic
24#19149610C2212yes—
25#19149608C2212yes—
26#19149608C2212yes—
27#19149603C2212yes—
28#19149601C2212yes—
29#19149599C2212yes—
30C2×Q16×C173C2211yescharacteristic
31C2×M16×C173C2211yescharacteristic
32C22⋊Q8×C173C2212yes—
33#106×C173C2211yescharacteristic
34#103×C173C2212yes—
35C2×Dic346—22no—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39#14583813—42no—
40D1384—41no—
41C2×#13063436—41no—
42C2×#13063436C2312yes—
43C4⋊Dic173—22no—
44C4⋊Dic173D412yes—
45C4⋊Dic173C2312yes—
46#14583805—22no—
47C22⋊Dic173D411yescharacteristic
48C22⋊Dic173D411yescharacteristic
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—
58M16×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—22no—
87C22×C173C22×D411yescharacteristic
88C2×D173—42no—
89C2×D173—42no—
90C2×D173—42no—
91C4×C173—21no—
92C4×C173—42no—
93C4×C173—42no—
94C4×C173H(5,𝔽2)11yescharacteristic
95C4×C1732−1+411yescharacteristic
96Dic173—22no—
97Dic173—22no—
98Dic173—42no—
99C2×C173—41no—
100C2×C173#53711yescharacteristic
101C2×C173#57712yes—
102D173—82no—
103C173#326311yescharacteristic, complemented, Sylow
104#3263—1731nomaximal, Sylow
105#542—1732no—
106#476—1732no—
107#475—1732no—
108#470D17311yescharacteristic, complemented
109#469—1731no—
110#465—1732no—
111#464—1732no—
112#463—1732no—
113#424—1731no—
114C2×Q16C2×D17311yescharacteristic
115C2×SD16—1732no—
116C2×D8—1731no—
117C2×M16C2×D17311yescharacteristic
118C22×C8—1731no—
119C4⋊Q8—1732no—
120#126—3462no—
121#124—3462no—
122#124—1732no—
123#123—3462no—
124C22⋊Q8—3462no—
125C22⋊Q8C2×D17312yes—
126#121—1732no—
127C4×Q8—1732no—
128C4×D4—1732no—
129#117—1731no—
130#106—1731no—
131#106—1732no—
132#106C2×D17311yescharacteristic
133C4⋊C8—1732no—
134C4⋊C8—1732no—
135#103—1732no—
136#103C2×D17312yes—
137#102—1732no—
138#102—1732no—
139C2×Q8—3462no—
140C2×Q8C22×D17312yes—
141C2×D4—3462no—
142C2×D4—1732no—
143C22×C4—3462no—
144C22×C4C22×D17311yescharacteristic
145Q16—41no—
146SD16—6922no—
147D8—6921no—
148M16—41no—
149C2×C8—1732no—
150C2×C8—6921no—
151C2×C8C22×D17311yescharacteristic
152C2×C8C22×D17311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—3462no—
155C4⋊C4—3462no—
156C4⋊C4—1732no—
157C4⋊C4—1732no—
158C4⋊C4C22×D17312yes—
159C22⋊C4—22no—
160C22⋊C4—3462no—
161C22⋊C4—3462no—
162C22⋊C4—3462no—
163C22⋊C4—3462no—
164C22⋊C4—1731no—
165C22⋊C4—1731no—
166C42—3462no—
167C42—1732no—
168C23—3462no—
169C23D4×D17311yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—13842no—
173D4—3462no—
174D4—3462no—
175D4—13842no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—3462no—
181C2×C4—13842no—
182C2×C4—1732no—
183C2×C4—1732no—
184C2×C4D4×D17311yescharacteristic
185C2×C4C23×D17311yescharacteristic
186C8—21no—
187C8—21no—
188C8—3462no—
189C22—22no—
190C22—21no—
191C22—6922no—
192C22—6922no—
193C22—6922no—
194C22C2×D4×D17311yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—3462no—
199C4—3462no—
200C4—6922no—
201C4#1914972411yescharacteristic
202C4#1914972611yescharacteristic
203C2—41no—
204C2—13842no—
205C2#2601206811yescharacteristic
206C2#2601230012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
prime radical#26011744C2
hypercenter#470D173
hypocenterC173#3263
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26011780C2semidirect
3. #26011774C2semidirect
5. #26011722C2semidirect
6. #26011715C2semidirect
7. #26011709C2semidirect
8. #26011519C2semidirect
9. #470×C173C2semidirect
103. C173#3263semidirect
108. #470D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260117802
#260117742
#260117442
#260117222
#260117152
#260117092
#260115192
#470×C1732
#3263173