Finite Group Databasebeta

← #36365010

Subgroups — #36365010

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#26012103C211yesmaximal, characteristic, complemented
2#26012096C211yesmaximal, characteristic
3#26011799C212yesmaximal, complemented
4#26011792C212yesmaximal, complemented
5#26011737C212yesmaximal, complemented
6#26011720C212yesmaximal, complemented
7#26011697C212yesmaximal, complemented
8#26011692C212yesmaximal, complemented
9#424×C173C211yesmaximal, characteristic, complemented
10C2×#14583815C2212yes—
11C2×#14583815C2212yes—
12#19149679C2211yescharacteristic
13#19149676C2211yescharacteristic
14#19149672—22no—
15#19149671C2211yescharacteristic
16#19149671C2211yescharacteristic
17#19149669—22no—
18#19149664—22no—
19#19149658C2212yes—
20#19149653C2212yes—
21#19149652—22no—
22C4×Dic346C2212yes—
23C4×D692C2212yes—
24#19149613C2212yes—
25#19149611C2212yes—
26#19149604C2212yes—
27#19149604C2212yes—
28#19149601C2212yes—
29#19149601C2212yes—
30C2×M16×C173C2211yescharacteristic
31C22×C8×C173C2211yescharacteristic
32#117×C173C2211yescharacteristic
33C4⋊C8×C173C2212yes—
34C4⋊C8×C173C2212yes—
35C2×Dic346C2312yes—
36C2×D692—22no—
37C2×D692C2312yes—
38C2×C22⋊D173—22no—
39C2×C4×D173—22no—
40#14583815—42no—
41#14583815—42no—
42C4⋊Dic173—22no—
43C4⋊Dic173C2312yes—
44C4⋊Dic173C2312yes—
45#14583805—22no—
46#14583805—22no—
47#14583805—22no—
48#14583805—22no—
49C22⋊Dic173—22no—
50#14583803—22no—
51#14583803—22no—
52C4×Dic173—22no—
53C22×C4×C173C2311yescharacteristic
54M16×C173—41no—
55C2×C8×C173—41no—
56C2×C8×C173C2312yes—
57C2×C8×C173C2312yes—
58C4⋊C4×C173D412yes—
59C22⋊C4×C173D412yes—
60C42×C173C2311yescharacteristic
61C42×C173C2311yescharacteristic
62C22×D173—22no—
63Dic346—22no—
64Dic346—22no—
65D692—22no—
66D692—22no—
67D692—82no—
68C22⋊D173—82no—
69C2×Dic173—22no—
70C2×Dic173—22no—
71C2×Dic173—22no—
72C4×D173—82no—
73C23×C173C2×D411yescharacteristic
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173C2×D412yes—
77C2×C4×C173C2×D412yes—
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2411yescharacteristic
80C8×C173—22no—
81C8×C173—22no—
82C22×C173—21no—
83C22×C173—22no—
84C22×C173C22×D411yescharacteristic
85C2×D173—42no—
86C2×D173—42no—
87C2×D173—42no—
88C4×C173—21no—
89C4×C173—22no—
90C4×C173—22no—
91C4×C173H(5,𝔽2)11yescharacteristic
92C4×C1732−1+411yescharacteristic
93Dic173—42no—
94Dic173—42no—
95Dic173—42no—
96C2×C173—41no—
97C2×C173#53711yescharacteristic
98C2×C173#57712yes—
99D173—82no—
100C173#326011yescharacteristic, complemented, Sylow
101#3260—1731nomaximal, Sylow
102#541—1731no—
103#540—1731no—
104#479—1732no—
105#478—1732no—
106#468—1732no—
107#465—1732no—
108#461—1732no—
109#460—1732no—
110#424D17311yescharacteristic, complemented
111C2×SD16—1732no—
112C2×SD16—1732no—
113C2×M16C2×D17311yescharacteristic
114C22×C8C2×D17311yescharacteristic
115C4⋊Q8—1731no—
116#127—1731no—
117#126—3462no—
118#125—3462no—
119#125—1731no—
120#125—1731no—
121#123—3462no—
122C22⋊Q8—1732no—
123#121—3462no—
124#121—1732no—
125C4×Q8—1732no—
126C4×D4—1732no—
127#117C2×D17311yescharacteristic
128#107—1732no—
129#106—1732no—
130C4⋊C8C2×D17312yes—
131C4⋊C8C2×D17312yes—
132#103—1732no—
133#103—1732no—
134#102—1732no—
135#102—1732no—
136C2×Q8—1732no—
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—1732no—
140C22×C4—3462no—
141C22×C4C22×D17311yescharacteristic
142SD16—6922no—
143SD16—6922no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D17312yes—
147C2×C8C22×D17312yes—
148C4⋊C4—3462no—
149C4⋊C4—3462no—
150C4⋊C4—3462no—
151C4⋊C4—3462no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4—1732no—
155C4⋊C4D69212yes—
156C22⋊C4—3462no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C22⋊C4D69212yes—
160C42—3462no—
161C42C22×D17311yescharacteristic
162C42C22×D17311yescharacteristic
163C23—3462no—
164C23C2×D69211yescharacteristic
165Q8—3462no—
166Q8—3462no—
167D4—3462no—
168D4—3462no—
169D4—13842no—
170D4—13842no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—13842no—
177C2×C4C2×D69212yes—
178C2×C4C2×D69212yes—
179C2×C4C2×D69211yescharacteristic
180C2×C4C23×D17311yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—6922no—
186C22—6922no—
187C22—6922no—
188C22C22×D69211yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—6922no—
193C4—6922no—
194C4—6922no—
195C4#1914972411yescharacteristic
196C4#1914972711yescharacteristic
197C2—41no—
198C2—13842no—
199C2#2601207211yescharacteristic
200C2#2601230312yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#424×C173C2
socleC22×C173C22×D4
prime radical#26012096C2
hypercenter#424D173
hypocenterC173#3260
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. #26012103C2semidirect
3. #26011799C2semidirect
4. #26011792C2semidirect
5. #26011737C2semidirect
6. #26011720C2semidirect
7. #26011697C2semidirect
8. #26011692C2semidirect
9. #424×C173C2semidirect
100. C173#3260semidirect
110. #424D173semidirect

Maximal subgroups

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

subgroupindex
#260121032
#260120962
#260117992
#260117922
#260117372
#260117202
#260116972
#260116922
#424×C1732
#3260173