Finite Group Databasebeta

← #37533679

Subgroups — #37533679

Order 23168.

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×#19153864C211yesmaximal, characteristic
2C2×#19153744C212yesmaximal, complemented
3#26041278C212yesmaximal, complemented
4C2×#19153735C211yesmaximal, characteristic, complemented
5#391×C181C211yesmaximal, characteristic, complemented
6#19153864—24no—
7#19153864—22no—
8#19153864C414yescomplemented
9C2×#15668450—22no—
10C2×#15668449—22no—
11C2×#15668449C2212yes—
12C2×Q8×D181—22no—
13C2×#15668433C2212yes—
14#19153744—24no—
15#19153739—22no—
16#19153736—22no—
17#19153735—22no—
18C22×Q8×C181C2211yescharacteristic
19C2×C4⋊C4×C181C2211yescharacteristic
20C2×C42×C181C2211yescharacteristic
21#95×C181—22no—
22#15668450—28no—
23#15668450—28no—
24#15668449—28no—
25#15668449—28no—
26#15668449—22no—
27#15668449—22no—
28#15668449C2×C414yes—
29#15668449D414yes—
30C2×Dic362—22no—
31C2×Dic362D412yes—
32Q8×D181—28no—
33Q8×D181—28no—
34C2×D724—22no—
35C2×D724D412yes—
36C2×C22⋊D181—22no—
37C2×C4×D181—24no—
38C2×C4×D181—22no—
39#15668433—22no—
40#15668433—22no—
41C2×#13089678—22no—
42C2×Q8×C181—22no—
43C2×Q8×C181—22no—
44C2×Q8×C181C2×C412yes—
45C2×Q8×C181D412yes—
46C22×C4×C181—21no—
47C22×C4×C181—21no—
48C22×C4×C181D412yes—
49C22×C4×C181C2311yescharacteristic
50C4⋊C4×C181—22no—
51C42×C181—22no—
52C22×D181—22no—
53Dic362—28no—
54Dic362—22no—
55Dic362C22⋊C414yes—
56D724—28no—
57D724—22no—
58D724C22⋊C414yes—
59C22⋊D181—24no—
60C22⋊D181—24no—
61C2×Dic181—22no—
62C4×D181—216no—
63C4×D181—24no—
64C4×D181—24no—
65#13089678—42no—
66C23×C181C2×D411yescharacteristic
67Q8×C181—24no—
68Q8×C181—24no—
69C2×C4×C181—22no—
70C2×C4×C181—22no—
71C2×C4×C181—22no—
72C2×C4×C181—41no—
73C2×C4×C181—41no—
74C2×C4×C181—42no—
75C2×C4×C181—41no—
76C2×C4×C181C22⋊C414yes—
77C2×C4×C181C22×C411yescharacteristic
78C2×C4×C181C2×D412yes—
79C2×C4×C181C2×D412yes—
80C2×C4×C181C2×D411yescharacteristic
81C22×C181—21no—
82C22×C181—21no—
83C22×C181C2×C22⋊C411yescharacteristic
84C22×C181#12011yescharacteristic
85C22×C181#12011yescharacteristic
86C2×D181—24no—
87C2×D181—42no—
88C2×D181—42no—
89C4×C181—24no—
90C4×C181—42no—
91C4×C181—81no—
92C4×C181C2×C22⋊C412yes—
93C4×C181#12012yes—
94Dic181—24no—
95C2×C181—21no—
96C2×C181—21no—
97C2×C181#37911yescharacteristic
98C2×C181#45411yescharacteristic
99C2×C181#45611yescharacteristic
100D181—44no—
101C181#179311yescharacteristic, complemented, Sylow
102#1793—1811nomaximal, Sylow
103C2×2−1+4—1811no—
104C2×#104—1812no—
105#417—1812no—
106C2×#101—1811no—
107#391D18111yescharacteristic, complemented
1082−1+4—3624no—
1092−1+4—3622no—
1102−1+4—1814no—
111C2×#41—3622no—
112C2×#41—3622no—
113C2×#41—1812no—
114C22×Q8—3622no—
115C22×Q8C2×D18111yescharacteristic
116C2×M16—1812no—
117C2×C4⋊C4C2×D18111yescharacteristic
118C2×C42C2×D18111yescharacteristic
119#104—3624no—
120#103—3622no—
121#102—3622no—
122#101—3622no—
123#95—22no—
124#41—3628no—
125#41—3628no—
126#41—3628no—
127#41—3628no—
128#41—3622no—
129#41—3622no—
130#41—1814no—
131#41—1814no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—3622no—
135C2×Q8—3628no—
136C2×Q8—3628no—
137C2×Q8—1812no—
138C2×Q8C4×D18112yes—
139C2×Q8D72412yes—
140C2×D4—3622no—
141C2×D4—3622no—
142C2×D4—1812no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3624no—
146C22×C4—3622no—
147C22×C4C22⋊D18112yes—
148C22×C4C22×D18111yescharacteristic
149M16—3622no—
150M16—3622no—
151C2×C8—3622no—
152C4⋊C4—22no—
153C42—22no—
154C23—3622no—
155C23C2×C22⋊D18111yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—3628no—
159Q8—3622no—
160Q8—1814no—
161D4—3628no—
162D4—3624no—
163D4—3624no—
164D4—3622no—
165D4—1814no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—36216no—
174C2×C4—3622no—
175C2×C4—3624no—
176C2×C4—3624no—
177C2×C4#1566842614yes—
178C2×C4C2×C4×D18111yescharacteristic
179C2×C4D4×D18112yes—
180C2×C4D4×D18112yes—
181C2×C4C2×D72411yescharacteristic
182C8—7242no—
183C22—21no—
184C22—21no—
185C22—3624no—
186C22—7242no—
187C22—7242no—
188C22C2×#1566842611yescharacteristic
189C22#1915378411yescharacteristic
190C22#1915378411yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—3624no—
195C4C22⋊C4×D18112yes—
196C4#1915378612yes—
197C2—21no—
198C2—21no—
199C2—7244no—
200C2#2604113411yescharacteristic
201C2#2604145111yescharacteristic
202C2#2604146111yescharacteristic
subgroupG/H
centerC22#19153784
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#391×C181C2
socleC22×C181#120
prime radicalC2×#19153864C2
cosocleC22×C4×C181C23
hypercenter#391D181
hypocenterC181#1793
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#19153744C2semidirect
3. #26041278C2semidirect
4. C2×#19153735C2semidirect
5. #391×C181C2semidirect
8. #19153864C4semidirect
101. C181#1793semidirect
107. #391D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
C2×#191537442
#260412782
C2×#191537352
#391×C1812
#1793181