Finite Group Databasebeta

← #36369822

Subgroups — #36369822

Order 22144.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#26012477C211yesmaximal, characteristic, complemented
2#26012470C211yesmaximal, characteristic, complemented
3#26012391C214yesmaximal, complemented
4#26012058C211yesmaximal, characteristic
5D173⋊M16—21no—
6#19149758—21no—
7#19149740—21no—
8#19149739C2212yes—
9#19149738—21no—
10#19149737C2212yes—
11C2×#14583826C2211yescharacteristic
12#19149675—21no—
13#19149666—21no—
14#19149650—21no—
15#19149650—21no—
16#19149648C412yescomplemented
17#19149648C2212yes—
18#19149638C411yescharacteristic, complemented
19C22×D4×C173C411yescharacteristic, complemented
20#14583836—41no—
21#14583835—22no—
22D173⋊C8—41no—
23C4⋊#10621924—41no—
24C22⋊#10621924—22no—
25C4×#10621924—41no—
26#14583826—41no—
27#14583826—41no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C22⋊D173—22no—
31C2×C22⋊D173—22no—
32C2×C22⋊D173C2×C411yescharacteristic
33C2×C22⋊D173C2311yescharacteristic
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C4⋊Dic173—21no—
37C22⋊Dic173—22no—
38C22⋊Dic173—22no—
39C22⋊Dic173C2×C412yes—
40C4×Dic173—21no—
41C24×C173C2×C412yes—
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173—41no—
45C2×D4×C173—41no—
46C22×C4×C173C2×C411yescharacteristic
47C2×#10621924—42no—
48#13063443—42no—
49C22×D173C22⋊C411yescharacteristic
50C22×D173C2×D411yescharacteristic
51Dic346—41no—
52D692—41no—
53C22⋊D173—41no—
54C22⋊D173—41no—
55C22⋊D173—82no—
56C22⋊D173—82no—
57C2×Dic173—22no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—21no—
61C4×D173—21no—
62C4×D173—21no—
63C4×D173—21no—
64C23×C173—22no—
65C23×C173—42no—
66C23×C173—82no—
67C23×C173C22×C411yescharacteristic
68D4×C173—42no—
69D4×C173—41no—
70D4×C173—41no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22⋊C411yescharacteristic
75C22×C173—21no—
76C22×C173—41no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173—84no—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—21no—
83C2×D173—21no—
84C2×D173—41no—
85C2×D173—41no—
86#10621924—82no—
87C4×C173—21no—
88C4×C173#9911yescharacteristic
89C4×C173#9911yescharacteristic
90Dic173—21no—
91Dic173—21no—
92Dic173—42no—
93C2×C173—21no—
94C2×C173—41no—
95C2×C173—82no—
96C2×C173C2×#9911yescharacteristic
97D173—41no—
98D173—41no—
99C173#212211yescharacteristic, complemented, Sylow
100#2122—1731nomaximal, Sylow
101#535—1731no—
102#413—1731no—
103#410—1731no—
104AΣL(1,𝔽16)—1734no—
105C2×#41—1731no—
106C22×D4#1062192411yescharacteristic, complemented
107C2×M16—3461no—
108#127—3461no—
109#124—3461no—
110#121—3461no—
111#121—3461no—
112#120—1732no—
113#120—1732no—
114#117—3461no—
115#117—1731no—
116#101—3461no—
117#100—1732no—
118#99—3461no—
119#99—1732no—
120C24C2×#1062192412yes—
121#41—6921no—
122#41—6921no—
123C2×Q8—1731no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—3462no—
129C2×D4—3462no—
130C2×D4—1731no—
131C2×D4—1731no—
132C2×D4—1731no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4C2×#1062192411yescharacteristic
136M16—3462no—
137M16—6921no—
138C2×C8—6921no—
139C4⋊C4—3461no—
140C4⋊C4—6921no—
141C22⋊C4—3462no—
142C22⋊C4—3462no—
143C22⋊C4—3462no—
144C22⋊C4—1732no—
145C42—3461no—
146C42—6921no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23—1731no—
151C23—1731no—
152C23C22×#1062192411yescharacteristic
153Q8—6921no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—13842no—
158D4—13842no—
159D4—6921no—
160D4—6921no—
161D4—6921no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—3461no—
165C2×C4—3461no—
166C2×C4—3461no—
167C2×C4—3461no—
168C2×C4—3462no—
169C2×C4—1731no—
170C2×C4—1731no—
171C2×C4—6922no—
172C2×C4C22⋊#1062192411yescharacteristic
173C2×C4C22⋊#1062192411yescharacteristic
174C8—6922no—
175C22—21no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—84no—
181C22—3461no—
182C22—3461no—
183C22—6921no—
184C22—6921no—
185C22C2×C22⋊#1062192411yescharacteristic
186C4—21no—
187C4—3461no—
188C4—3461no—
189C4—13842no—
190C4—6922no—
191C4#1914973711yescharacteristic
192C4#1914973711yescharacteristic
193C2—21no—
194C2—41no—
195C2—82no—
196C2—6921no—
197C2—6921no—
198C2C2×#1914973711yescharacteristic
subgroupG/H
centerC2C2×#19149737
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC22×D4×C173C4
socleC2×C173C2×#99
prime radical#26012058C2
cosocleC2×C22⋊D173C23
hypercenterC22×D4#10621924
hypocenterC173#2122
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26012477C2semidirect
2. #26012470C2semidirect
3. #26012391C2semidirect
16. #19149648C4semidirect
18. #19149638C4semidirect
19. C22×D4×C173C4semidirect
99. C173#2122semidirect
106. C22×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124772
#260124702
#260123912
#260120582
#2122173