Finite Group Databasebeta

← #36353521

Subgroups — #36353521

Order 22144.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19149727C211yesmaximal, characteristic
2C2×#19149607C212yesmaximal, complemented
3#26011490C212yesmaximal, complemented
4C2×#19149598C211yesmaximal, characteristic, complemented
5#391×C173C211yesmaximal, characteristic, complemented
6#19149727—22no—
7#19149727—24no—
8#19149727C414yescomplemented
9C2×#14583827—22no—
10C2×#14583826—22no—
11C2×#14583826C2212yes—
12C2×Q8×D173—22no—
13C2×#14583810C2212yes—
14#19149607—24no—
15#19149602—22no—
16#19149599—22no—
17#19149598—22no—
18C22×Q8×C173C2211yescharacteristic
19C2×C4⋊C4×C173C2211yescharacteristic
20C2×C42×C173C2211yescharacteristic
21#95×C173—22no—
22#14583827—28no—
23#14583827—28no—
24#14583826—28no—
25#14583826—28no—
26#14583826—22no—
27#14583826—22no—
28#14583826C2×C414yes—
29#14583826D414yes—
30C2×Dic346—22no—
31C2×Dic346D412yes—
32Q8×D173—28no—
33Q8×D173—28no—
34C2×D692—22no—
35C2×D692D412yes—
36C2×C22⋊D173—22no—
37C2×C4×D173—24no—
38C2×C4×D173—22no—
39#14583810—22no—
40#14583810—22no—
41C2×#13063436—22no—
42C2×Q8×C173—22no—
43C2×Q8×C173—22no—
44C2×Q8×C173C2×C412yes—
45C2×Q8×C173D412yes—
46C22×C4×C173—21no—
47C22×C4×C173—21no—
48C22×C4×C173D412yes—
49C22×C4×C173C2311yescharacteristic
50C4⋊C4×C173—22no—
51C42×C173—22no—
52C22×D173—22no—
53Dic346—28no—
54Dic346—22no—
55Dic346C22⋊C414yes—
56D692—28no—
57D692—22no—
58D692C22⋊C414yes—
59C22⋊D173—24no—
60C22⋊D173—24no—
61C2×Dic173—22no—
62C4×D173—216no—
63C4×D173—24no—
64C4×D173—24no—
65#13063436—42no—
66C23×C173C2×D411yescharacteristic
67Q8×C173—24no—
68Q8×C173—24no—
69C2×C4×C173—22no—
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—41no—
73C2×C4×C173—41no—
74C2×C4×C173—42no—
75C2×C4×C173—41no—
76C2×C4×C173C22⋊C414yes—
77C2×C4×C173C22×C411yescharacteristic
78C2×C4×C173C2×D412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173C2×D411yescharacteristic
81C22×C173—21no—
82C22×C173—21no—
83C22×C173C2×C22⋊C411yescharacteristic
84C22×C173#12011yescharacteristic
85C22×C173#12011yescharacteristic
86C2×D173—24no—
87C2×D173—42no—
88C2×D173—42no—
89C4×C173—24no—
90C4×C173—42no—
91C4×C173—81no—
92C4×C173C2×C22⋊C412yes—
93C4×C173#12012yes—
94Dic173—24no—
95C2×C173—21no—
96C2×C173—21no—
97C2×C173#37911yescharacteristic
98C2×C173#45411yescharacteristic
99C2×C173#45611yescharacteristic
100D173—44no—
101C173#179311yescharacteristic, complemented, Sylow
102#1793—1731nomaximal, Sylow
103C2×2−1+4—1731no—
104C2×#104—1732no—
105#417—1732no—
106C2×#101—1731no—
107#391D17311yescharacteristic, complemented
1082−1+4—3464no—
1092−1+4—3462no—
1102−1+4—1734no—
111C2×#41—3462no—
112C2×#41—3462no—
113C2×#41—1732no—
114C22×Q8—3462no—
115C22×Q8C2×D17311yescharacteristic
116C2×M16—1732no—
117C2×C4⋊C4C2×D17311yescharacteristic
118C2×C42C2×D17311yescharacteristic
119#104—3464no—
120#103—3462no—
121#102—3462no—
122#101—3462no—
123#95—22no—
124#41—3468no—
125#41—3468no—
126#41—3468no—
127#41—3462no—
128#41—3462no—
129#41—3468no—
130#41—1734no—
131#41—1734no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—3462no—
135C2×Q8—3468no—
136C2×Q8—3468no—
137C2×Q8—1732no—
138C2×Q8C4×D17312yes—
139C2×Q8D69212yes—
140C2×D4—3462no—
141C2×D4—3462no—
142C2×D4—1732no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3464no—
146C22×C4—3462no—
147C22×C4C22⋊D17312yes—
148C22×C4C22×D17311yescharacteristic
149M16—3462no—
150M16—3462no—
151C2×C8—3462no—
152C4⋊C4—22no—
153C42—22no—
154C23—3462no—
155C23C2×C22⋊D17311yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—3468no—
159Q8—3462no—
160Q8—1734no—
161D4—3468no—
162D4—3464no—
163D4—3464no—
164D4—3462no—
165D4—1734no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—34616no—
174C2×C4—3462no—
175C2×C4—3464no—
176C2×C4—3464no—
177C2×C4#1458380314yes—
178C2×C4C2×C4×D17311yescharacteristic
179C2×C4D4×D17312yes—
180C2×C4D4×D17312yes—
181C2×C4C2×D69211yescharacteristic
182C8—6922no—
183C22—21no—
184C22—21no—
185C22—3464no—
186C22—6922no—
187C22—6922no—
188C22C2×#1458380311yescharacteristic
189C22#1914964711yescharacteristic
190C22#1914964711yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—3464no—
195C4C22⋊C4×D17312yes—
196C4#1914964912yes—
197C2—21no—
198C2—21no—
199C2—6924no—
200C2#2601134611yescharacteristic
201C2#2601166311yescharacteristic
202C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#391×C173C2
socleC22×C173#120
prime radicalC2×#19149727C2
cosocleC22×C4×C173C23
hypercenter#391D173
hypocenterC173#1793
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#19149607C2semidirect
3. #26011490C2semidirect
4. C2×#19149598C2semidirect
5. #391×C173C2semidirect
8. #19149727C4semidirect
101. C173#1793semidirect
107. #391D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497272
C2×#191496072
#260114902
C2×#191495982
#391×C1732
#1793173