Finite Group Databasebeta

← #36352868

Subgroups — #36352868

Order 22144.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012233C211yesmaximal, characteristic
2#26011883C211yesmaximal, characteristic, complemented
3#26011853C211yesmaximal, characteristic, complemented
4#26011231C211yesmaximal, characteristic, complemented
5#26011230C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic, complemented
7#489×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—21no—
9C2×#14583813—21no—
10#19149668—21no—
11#19149667C2211yescharacteristic
12#19149667C2211yescharacteristic
13#19149665—21no—
14#19149665—21no—
15#19149649—21no—
16#19149649—21no—
17#19149647—21no—
18#19149602—21no—
19#19149599—21no—
20#19149590C2211yescharacteristic, complemented
21#19149590C2211yescharacteristic, complemented
22C4⋊Q8×C173C2211yescharacteristic, complemented
23#124×C173C2211yescharacteristic
24#103×C173—21no—
25#102×C173—21no—
26#97×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C2×Dic346D411yescharacteristic
29C2×Dic346D411yescharacteristic
30C2×D692—21no—
31C2×D692D411yescharacteristic
32C2×D692D411yescharacteristic
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35#14583816—42no—
36#14583813—42no—
37C2×#13063436—21no—
38C2×#13063436—21no—
39C22⋊Dic173—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×Q8×C173—21no—
48C2×Q8×C173D411yescharacteristic
49C2×D4×C173D411yescharacteristic
50C2×C8×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173—21no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C42×C173C2311yescharacteristic
56C22×D173—21no—
57C22×D173—21no—
58C22×D173—81no—
59C22×D173—81no—
60C22×D173—81no—
61Dic346—41no—
62Dic346—41no—
63D692—41no—
64D692—41no—
65D692—81no—
66C22⋊D173—81no—
67C22⋊D173—81no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70#13063436—22no—
71#13063436—41no—
72C23×C173—21no—
73Q8×C173—41no—
74Q8×C173—41no—
75Q8×C173—41no—
76D4×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C8×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173#12011yescharacteristic
87C2×D173—41no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—84no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—21no—
97C4×C173—41no—
98C4×C173—81no—
99Dic173—41no—
100Dic173—41no—
101C2×C173—81no—
102C2×C173#45311yescharacteristic
103C2×C173#45511yescharacteristic
104C2×C173#45511yescharacteristic
105D173—81no—
106D173—81no—
107C173#165911yescharacteristic, complemented, Sylow
108#1659—1731nomaximal, Sylow
109H(3,𝔽4)—1731no—
110#497—1731no—
111#489—1731no—
112#489D17311yescharacteristic, complemented
113#329—1731no—
114#329—1731no—
115#329—1731no—
116C2×Q16—3461no—
117C2×SD16—3461no—
118C4⋊Q8C2×D17311yescharacteristic, complemented
119#124—3461no—
120#124—3461no—
121#124—3461no—
122#124—1731no—
123#124—1731no—
124#124C2×D17311yescharacteristic
125#120—3461no—
126#120—3461no—
127#120—3461no—
128#103—21no—
129#103—3461no—
130#102—21no—
131#102—3461no—
132#97—1731no—
133#97—1731no—
134#97C2×D17311yescharacteristic, complemented
135C24—3461no—
136C2×Q8—21no—
137C2×Q8—1731no—
138C2×Q8—1731no—
139C2×Q8C22⋊D17311yescharacteristic
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4—1731no—
144C2×D4—1731no—
145C2×D4C22⋊D17311yescharacteristic
146Q16—6922no—
147SD16—6922no—
148C2×C8—21no—
149C2×C8—3461no—
150C2×C8—3461no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C22⋊C4—3461no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C42—3461no—
163C42C22×D17311yescharacteristic
164C23—21no—
165C23—3461no—
166C23—3461no—
167C23—13841no—
168C23—13841no—
169C23—13841no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—6921no—
174Q8—6921no—
175D4—41no—
176D4—13841no—
177D4—13841no—
178D4—13841no—
179D4—6921no—
180D4—6921no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4D4×D17311yescharacteristic
186C2×C4D4×D17311yescharacteristic
187C2×C4C2×C22⋊D17311yescharacteristic
188C8—41no—
189C8—3462no—
190C8—6921no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—13844no—
195C22—6921no—
196C22—6921no—
197C22—6921no—
198C22—6921no—
199C22—6921no—
200C22—6921no—
201C22#1914964711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—6921no—
208C4—6921no—
209C2—81no—
210C2—13841no—
211C2—13841no—
212C2#2601165711yescharacteristic
213C2#2601166711yescharacteristic
214C2#2601166811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#489×C173C2
socleC22×C173#120
prime radical#26012233C2
hypercenter#489D173
hypocenterC173#1659
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #26011883C2semidirect
3. #26011853C2semidirect
4. #26011231C2semidirect
5. #26011230C2semidirect
6. #26011230C2semidirect
7. #489×C173C2semidirect
20. #19149590C22semidirect
21. #19149590C22semidirect
22. C4⋊Q8×C173C22semidirect
26. #97×C173C22semidirect
107. C173#1659semidirect
112. #489D173semidirect
118. C4⋊Q8C2×D173semidirect
134. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260118832
#260118532
#260112312
#260112302
#489×C1732
#1659173