Finite Group Databasebeta

← #36364964

Subgroups — #36364964

Order 22144.

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

normal subgroups only

This group has 354 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#26012025C212yesmaximal, complemented
2#26011799C212yesmaximal, complemented
3#26011792C212yesmaximal, complemented
4#26011749C211yesmaximal, characteristic, complemented
5#26011744C211yesmaximal, characteristic
6#26011737C212yesmaximal, complemented
7#26011714C212yesmaximal, complemented
8#26011707C212yesmaximal, complemented
9#423×C173C211yesmaximal, characteristic, complemented
10C2×Dic692C2211yescharacteristic
11C2×#14583815C2212yes—
12C2×D1384C2211yescharacteristic
13#19149673—22no—
14#19149671C2212yes—
15#19149667C2212yes—
16#19149662—22no—
17#19149658C2212yes—
18#19149653C2212yes—
19C4×Dic346C2212yes—
20C4×D692C2212yes—
21C4×C22⋊D173—22no—
22#19149636—22no—
23#19149613C2212yes—
24#19149611C2211yescharacteristic
25#19149611C2211yescharacteristic
26#19149604C2212yes—
27#19149604C2212yes—
28#19149601C2212yes—
29#19149601C2212yes—
30C2×M16×C173C2211yescharacteristic
31C2×M16×C173C2211yescharacteristic
32C2×C42×C173C2211yescharacteristic
33C4⋊C8×C173C2212yes—
34C4⋊C8×C173C2212yes—
35C2×Dic346C2312yes—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39Dic692—41no—
40#14583815—42no—
41D1384—41no—
42C4⋊Dic173C2312yes—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583805—22no—
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×C173D412yes—
55C22×C4×C173C2311yescharacteristic
56M16×C173—41no—
57M16×C173—41no—
58C2×C8×C173C2311yescharacteristic
59C2×C8×C173C2312yes—
60C2×C8×C173C2311yescharacteristic
61C42×C173D411yescharacteristic
62C42×C173D411yescharacteristic
63C42×C173C2312yes—
64C22×D173—22no—
65Dic346—22no—
66Dic346—22no—
67D692—22no—
68D692—22no—
69C22⋊D173—44no—
70C2×Dic173—22no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C4×D173—44no—
74C23×C173C2×D411yescharacteristic
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173—42no—
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D412yes—
81C2×C4×C173C2411yescharacteristic
82C8×C173—21no—
83C8×C173—22no—
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—22no—
93C4×C173—22no—
94C4×C173H(5,𝔽2)11yescharacteristic
95C4×C1732−1+411yescharacteristic
96Dic173—42no—
97Dic173—42no—
98Dic173—42no—
99C2×C173—41no—
100C2×C173#53711yescharacteristic
101C2×C173#57512yes—
102D173—82no—
103C173#325611yescharacteristic, complemented, Sylow
104#3256—1731nomaximal, Sylow
105#529—1732no—
106#479—1732no—
107#478—1732no—
108#470—1731no—
109#469—1731no—
110#468—1732no—
111#464—1732no—
112#463—1732no—
113#423D17311yescharacteristic, complemented
114C2×Q16—1731no—
115C2×SD16—1732no—
116C2×D8—1731no—
117C2×M16C2×D17311yescharacteristic
118C2×M16C2×D17311yescharacteristic
119#126—3462no—
120#125—1732no—
121#124—1732no—
122#123—3462no—
123C22⋊Q8—1732no—
124#121—1732no—
125C4×Q8—1732no—
126C4×D4—3462no—
127C4×D4—1732no—
128#117—3462no—
129C2×C42C2×D17311yescharacteristic
130#107—1732no—
131#106—1731no—
132#106—1731no—
133C4⋊C8C2×D17312yes—
134C4⋊C8C2×D17312yes—
135#103—1732no—
136#103—1732no—
137#102—1732no—
138#102—1732no—
139C2×Q8—1732no—
140C2×D4—3462no—
141C2×D4—1732no—
142C22×C4—3462no—
143C22×C4D69212yes—
144C22×C4C22×D17311yescharacteristic
145Q16—6921no—
146SD16—6922no—
147D8—6921no—
148M16—41no—
149M16—41no—
150C2×C8C22×D17311yescharacteristic
151C2×C8C22×D17312yes—
152C2×C8C22×D17311yescharacteristic
153C4⋊C4—3462no—
154C4⋊C4—3462no—
155C4⋊C4—3462no—
156C4⋊C4—3462no—
157C4⋊C4—1732no—
158C4⋊C4—1732no—
159C22⋊C4—3462no—
160C22⋊C4—3462no—
161C22⋊C4—3462no—
162C22⋊C4—3462no—
163C22⋊C4—3462no—
164C42—3462no—
165C42D69211yescharacteristic
166C42D69211yescharacteristic
167C42C22×D17312yes—
168C23—3462no—
169C23C2×D69211yescharacteristic
170Q8—3462no—
171Q8—3462no—
172D4—3462no—
173D4—3462no—
174D4—6924no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—3462no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—6924no—
182C2×C4C2×D69212yes—
183C2×C4C2×D69211yescharacteristic
184C2×C4C2×D69212yes—
185C2×C4C23×D17311yescharacteristic
186C8—21no—
187C8—22no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—6922no—
192C22—6922no—
193C22—6922no—
194C22C22×D69211yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—6922no—
199C4—6922no—
200C4—6922no—
201C4#1914972411yescharacteristic
202C4#1914972711yescharacteristic
203C2—41no—
204C2—13842no—
205C2#2601207211yescharacteristic
206C2#2601229112yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#423×C173C2
socleC22×C173C22×D4
prime radical#26011744C2
hypercenter#423D173
hypocenterC173#3256
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #26012025C2semidirect
2. #26011799C2semidirect
3. #26011792C2semidirect
4. #26011749C2semidirect
6. #26011737C2semidirect
7. #26011714C2semidirect
8. #26011707C2semidirect
9. #423×C173C2semidirect
103. C173#3256semidirect
113. #423D173semidirect

Maximal subgroups

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

subgroupindex
#260120252
#260117992
#260117922
#260117492
#260117442
#260117372
#260117142
#260117072
#423×C1732
#3256173