Finite Group Databasebeta

← #36354099

Subgroups — #36354099

Order 22144.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#26012067C211yesmaximal, characteristic
2#26011968C211yesmaximal, characteristic, complemented
3#26011509C211yesmaximal, characteristic
4#26011507C211yesmaximal, characteristic
5C2×#19149597C211yesmaximal, characteristic, complemented
6#26011247C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C22×Dic346C2211yescharacteristic
10C2×#14583810C2211yescharacteristic
11#19149679C412yescomplemented
12#19149667C412yescomplemented
13#19149663—22no—
14#19149659—22no—
15C4×Dic346—22no—
16#19149644—22no—
17#19149638C2211yescharacteristic
18#19149635—22no—
19#19149607—22no—
20#19149605—22no—
21#19149597—22no—
22C2×M16×C173C2211yescharacteristic
23#117×C173C2211yescharacteristic
24#117×C173C2211yescharacteristic
25#14583826—21no—
26#14583826—21no—
27#14583826—21no—
28#14583826—21no—
29C2×Dic346—21no—
30C2×Dic346—21no—
31C2×Dic346—21no—
32C2×Dic346—21no—
33C2×Dic346C2×C411yescharacteristic
34C2×Dic346C2×C412yes—
35C2×Dic346D412yes—
36C2×D692C2×C411yescharacteristic
37C2×C22⋊D173—21no—
38C22×Dic173D412yes—
39C2×C4×D173—21no—
40#14583810—21no—
41#14583810—21no—
42C2×#13063436—21no—
43C4⋊Dic173—24no—
44C4⋊Dic173—21no—
45#14583805—24no—
46C22⋊Dic173—21no—
47#14583803—24no—
48C4×Dic173—24no—
49C4×Dic173D412yes—
50C22×C4×C173C2311yescharacteristic
51M16×C173—21no—
52M16×C173—21no—
53C2×C8×C173—21no—
54C4⋊C4×C173—21no—
55C4⋊C4×C173—21no—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—21no—
58C42×C173C2×C412yes—
59C42×C173D412yes—
60C22×D173—21no—
61Dic346—22no—
62Dic346—22no—
63Dic346—22no—
64Dic346—22no—
65Dic346—21no—
66Dic346—21no—
67D692—21no—
68D692—21no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—21no—
74C2×Dic173—22no—
75C2×Dic173C22⋊C414yes—
76C4×D173—41no—
77C4×D173—41no—
78#13063436—41no—
79C23×C173C2×D411yescharacteristic
80C2×C4×C173—22no—
81C2×C4×C173—22no—
82C2×C4×C173C22×C411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173#4111yescharacteristic
87C2×C4×C173#4111yescharacteristic
88C8×C173—41no—
89C22×C173—41no—
90C22×C173C2×C22⋊C411yescharacteristic
91C22×C173#12011yescharacteristic
92C22×C173#12311yescharacteristic
93C2×D173—42no—
94C2×D173—41no—
95C4×C173—42no—
96C4×C173—42no—
97C4×C173C4×D411yescharacteristic
98C4×C173C4×D411yescharacteristic
99C4×C173#12111yescharacteristic
100C4×C173#12111yescharacteristic
101Dic173—24no—
102Dic173—42no—
103Dic173—41no—
104C2×C173—21no—
105C2×C173—21no—
106C2×C173—21no—
107C2×C173#38611yescharacteristic
108D173—81no—
109C173#191211yescharacteristic, complemented, Sylow
110#1912—1731nomaximal, Sylow
111#536—1731no—
112#519—1731no—
113#421—1731no—
114#421—1731no—
115C2×#101—1731no—
116H(3,ℤ/4)—1731no—
117H(3,ℤ/4)D17311yescharacteristic, complemented
118C2×#41—1731no—
119C22×Q8—1731no—
120C2×M16—1731no—
121C2×M16C2×D17311yescharacteristic
122C4⋊Q8—1732no—
123#124—1732no—
124#123—3462no—
125C22⋊Q8—3462no—
126C4×Q8—3462no—
127C4×Q8—3462no—
128#117—3462no—
129#117—1731no—
130#117C2×D17311yescharacteristic
131#117C2×D17311yescharacteristic
132#104—3462no—
133#104—3462no—
134#101—3462no—
135#41—3461no—
136#41—3461no—
137#41—3461no—
138#41—3461no—
139C2×Q8—3461no—
140C2×Q8—3461no—
141C2×Q8—3461no—
142C2×Q8—3461no—
143C2×Q8—1731no—
144C2×Q8—1732no—
145C2×Q8—1732no—
146C2×D4—3461no—
147C2×D4—1731no—
148C22×C4—3461no—
149C22×C4—1732no—
150C22×C4C22×D17311yescharacteristic
151M16—21no—
152M16—21no—
153M16—3461no—
154M16—3461no—
155C2×C8—21no—
156C2×C8—3461no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—3464no—
160C4⋊C4—3464no—
161C4⋊C4—3461no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—3464no—
165C22⋊C4—3461no—
166C42—3464no—
167C42—1732no—
168C42C4×D17312yes—
169C42D69212yes—
170C23—3461no—
171C23D4×D17311yescharacteristic
172Q8—3462no—
173Q8—3462no—
174Q8—3462no—
175Q8—3462no—
176Q8—3461no—
177Q8—3461no—
178D4—3461no—
179D4—3461no—
180D4—6921no—
181D4—6921no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—3461no—
187C2×C4—3462no—
188C2×C4—1734no—
189C2×C4—6921no—
190C2×C4—6921no—
191C2×C4C2×C4×D17311yescharacteristic
192C2×C4D4×D17311yescharacteristic
193C2×C4D4×D17311yescharacteristic
194C2×C4C2×D69211yescharacteristic
195C2×C4#1458382611yescharacteristic
196C2×C4#1458382711yescharacteristic
197C8—41no—
198C8—6921no—
199C22—41no—
200C22—6922no—
201C22—6921no—
202C22C22⋊C4×D17311yescharacteristic
203C22#1914964911yescharacteristic
204C22#1914966411yescharacteristic
205C4—42no—
206C4—42no—
207C4—3464no—
208C4—6922no—
209C4—6921no—
210C4#1914964111yescharacteristic
211C4C4×D69211yescharacteristic
212C4#1914965211yescharacteristic
213C4#1914965411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—13841no—
218C2#2601136911yescharacteristic
subgroupG/H
centerC2#26011369
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingH(3,ℤ/4)×C173C2
socleC2×C173#386
prime radicalC2×#14583826C22
cosocleC22×C4×C173C23
hypercenterH(3,ℤ/4)D173
hypocenterC173#1912
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #26011968C2semidirect
5. C2×#19149597C2semidirect
6. #26011247C2semidirect
7. H(3,ℤ/4)×C173C2semidirect
11. #19149679C4semidirect
12. #19149667C4semidirect
109. C173#1912semidirect
117. H(3,ℤ/4)D173semidirect

Maximal subgroups

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

subgroupindex
#260120672
#260119682
#260115092
#260115072
C2×#191495972
#260112472
H(3,ℤ/4)×C1732
#1912173