Finite Group Databasebeta

← #36353953

Subgroups — #36353953

Order 22144.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#19149677C211yesmaximal, characteristic
2#26011968C211yesmaximal, characteristic
3#26011500C211yesmaximal, characteristic
4C2×#19149597C211yesmaximal, characteristic
5#26011252C211yesmaximal, characteristic
6#26011250C211yesmaximal, characteristic
7#419×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic
9C2×#14583810C2211yescharacteristic
10#19149677—22no—
11#19149677C414yescomplemented
12C4×Dic346—22no—
13#19149644—22no—
14#19149638C2211yescharacteristic
15#19149635—22no—
16C2×#14583805—22no—
17C2×C4×Dic173C2211yescharacteristic
18#19149603—22no—
19#19149597—22no—
20C22×Q8×C173C2211yescharacteristic
21C2×M16×C173C2211yescharacteristic
22#117×C173C2211yescharacteristic
23#103×C173—22no—
24C2×Dic346—21no—
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346C2×C412yes—
29C2×Dic346D412yes—
30C22×Dic173—21no—
31C22×Dic173D412yes—
32#14583810—21no—
33#14583810—21no—
34C2×#13063436—21no—
35C4⋊Dic173D412yes—
36#14583805—24no—
37#14583805—28no—
38C22⋊Dic173—21no—
39C4×Dic173—24no—
40C4×Dic173—21no—
41C4×Dic173—21no—
42C4×Dic173C2×C412yes—
43C2×Q8×C173—21no—
44C2×Q8×C173—21no—
45C2×Q8×C173—21no—
46C2×Q8×C173—21no—
47C2×Q8×C173—21no—
48C2×Q8×C173C2×C412yes—
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51M16×C173—21no—
52M16×C173—21no—
53C2×C8×C173—21no—
54C4⋊C4×C173D412yes—
55C22⋊C4×C173—21no—
56C42×C173—21no—
57Dic346—22no—
58Dic346—22no—
59Dic346—22no—
60Dic346—22no—
61C2×Dic173—24no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173C22⋊C414yes—
67#13063436—41no—
68C23×C173C2×D411yescharacteristic
69Q8×C173—22no—
70Q8×C173—22no—
71Q8×C173—41no—
72Q8×C173—41no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173C22×C411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173#4111yescharacteristic
82C2×C4×C173#4111yescharacteristic
83C8×C173—41no—
84C22×C173—22no—
85C22×C173C2×C22⋊C411yescharacteristic
86C22×C173#12011yescharacteristic
87C22×C173#12311yescharacteristic
88C4×C173—42no—
89C4×C173—42no—
90C4×C173C4×D411yescharacteristic
91C4×C173C4×D411yescharacteristic
92C4×C173#12111yescharacteristic
93C4×C173#12111yescharacteristic
94Dic173—24no—
95Dic173—24no—
96Dic173—42no—
97C2×C173—21no—
98C2×C173—21no—
99C2×C173#38611yescharacteristic
100C2×C173#45612yes—
101C173#188611yescharacteristic, complemented, Sylow
102#1886—1731nomaximal, Sylow
103C2×C4⋊Q8—1731no—
104#519—1731no—
105#419—1731no—
106#419D17311yescharacteristic
107C2×#101—1731no—
108#339—1731no—
109#339—1731no—
110C22×Q8—1731no—
111C22×Q8C2×D17311yescharacteristic
112C2×M16—1731no—
113C2×M16C2×D17311yescharacteristic
114C4⋊Q8—3462no—
115C4⋊Q8—1734no—
116C4×Q8—3462no—
117C4×Q8—3462no—
118#117—3462no—
119#117—1731no—
120#117C2×D17311yescharacteristic
121C2×C4⋊C4—3462no—
122C2×C42—1731no—
123#103—22no—
124#103—3462no—
125#101—3462no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—3461no—
132C2×Q8—3461no—
133C2×Q8—3461no—
134C2×Q8—3461no—
135C2×Q8—1732no—
136C2×Q8—1732no—
137C2×Q8C4×D17312yes—
138C22×C4—21no—
139C22×C4—3461no—
140C22×C4—1732no—
141C22×C4C22×D17311yescharacteristic
142M16—21no—
143M16—21no—
144M16—3461no—
145M16—3461no—
146C2×C8—21no—
147C2×C8—3461no—
148C4⋊C4—3464no—
149C4⋊C4—3468no—
150C4⋊C4—1732no—
151C4⋊C4C22⋊D17312yes—
152C22⋊C4—21no—
153C22⋊C4—3461no—
154C42—21no—
155C42—3464no—
156C42—3461no—
157C42—3461no—
158C42—1732no—
159C23D4×D17311yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—3462no—
165Q8—3462no—
166Q8—3462no—
167Q8—3462no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—3464no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3462no—
177C2×C4—1734no—
178C2×C4C2×C4×D17311yescharacteristic
179C2×C4D4×D17311yescharacteristic
180C2×C4D4×D17311yescharacteristic
181C2×C4C2×C22⋊D17311yescharacteristic
182C2×C4#1458382511yescharacteristic
183C2×C4#1458382611yescharacteristic
184C8—41no—
185C8—6921no—
186C22—22no—
187C22C22⋊C4×D17311yescharacteristic
188C22#1914964711yescharacteristic
189C22#1914966111yescharacteristic
190C4—42no—
191C4—42no—
192C4—3464no—
193C4—3464no—
194C4—6922no—
195C4#1914963911yescharacteristic
196C4C4×C22⋊D17311yescharacteristic
197C4#1914965011yescharacteristic
198C4#1914965211yescharacteristic
199C2—21no—
200C2—21no—
201C2#2601136711yescharacteristic
202C2#2601167212yes—
subgroupG/H
centerC22#19149647
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#419×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
cosocleC22×C4×C173C23
hypercenter#419D173
hypocenterC173#1886
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #19149677C4semidirect
101. C173#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#191496772
#260119682
#260115002
C2×#191495972
#260112522
#260112502
#419×C1732
#1886173