Finite Group Databasebeta

← #37533671

Subgroups — #37533671

Order 23168.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#26042115C211yesmaximal, characteristic
2#26041263C211yesmaximal, characteristic, complemented
3#26041261C212yesmaximal, complemented
4#26041245C212yesmaximal, complemented
5#408×C181C211yesmaximal, characteristic, complemented
6#19153864—21no—
7#19153863—22no—
8#19153862—22no—
9#19153861—21no—
10#19153861C414yes—
11C2×#15668449—22no—
12C2×#15668449C2212yes—
13#41×D181—24no—
14C2×#15668433C2212yescomplemented
15#19153735—21no—
16#19153734—22no—
17#19153733—21no—
18#19153732—22no—
19#19153728—24no—
20C2×#41×C181C2211yescharacteristic
21C2×M16×C181C2211yescharacteristic, complemented
22C22×C8×C181C2211yescharacteristic, complemented
23C22⋊C8×C181—22no—
24#15668450—24no—
25#15668449—24no—
26#15668449—24no—
27#15668449—24no—
28#15668449—22no—
29#15668449—22no—
30#15668449—22no—
31#15668449—22no—
32#15668448—24no—
33#15668448—28no—
34C2×Dic362—22no—
35C2×Dic362D412yes—
36Q8×D181—24no—
37C2×D724—22no—
38C2×D724D412yes—
39C2×C22⋊D181—24no—
40C2×C22⋊D181C2×C414yes—
41D4×D181—24no—
42D4×D181—28no—
43C2×C4×D181—24no—
44C2×C4×D181D414yes—
45#15668433—22no—
46#15668433—22no—
47C2×#13089678—22no—
48#41×C181—22no—
49#41×C181—22no—
50C2×Q8×C181D411yescharacteristic
51C2×D4×C181C2×C412yes—
52C2×D4×C181D411yescharacteristic
53C22×C4×C181D412yes—
54C22×C4×C181C2311yescharacteristic
55M16×C181—22no—
56C2×C8×C181—21no—
57C2×C8×C181—21no—
58C2×C8×C181—22no—
59C22×D181—24no—
60C22×D181C22⋊C414yes—
61Dic362—24no—
62Dic362—22no—
63Dic362—22no—
64D724—24no—
65D724—22no—
66D724—22no—
67C22⋊D181—28no—
68C22⋊D181—24no—
69C22⋊D181—24no—
70C2×Dic181—24no—
71C2×Dic181C22⋊C414yes—
72C4×D181—28no—
73C4×D181—24no—
74C4×D181—24no—
75#13089678—42no—
76C23×C181C22⋊C412yes—
77C23×C181C22×C411yescharacteristic
78Q8×C181—22no—
79D4×C181—24no—
80D4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181C22⋊C412yes—
84C2×C4×C181C2×D412yes—
85C2×C4×C181C2×D412yes—
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2×D411yescharacteristic
88C8×C181—41no—
89C8×C181—41no—
90C22×C181—22no—
91C22×C181—41no—
92C22×C181—42no—
93C22×C181C2×C22⋊C412yes—
94C22×C181C2×C22⋊C411yescharacteristic
95C2×D181—24no—
96C2×D181—44no—
97C2×D181—84no—
98C4×C181—22no—
99C4×C181#12011yescharacteristic
100C4×C181#12011yescharacteristic
101C4×C181#12012yes—
102Dic181—24no—
103C2×C181—22no—
104C2×C181—21no—
105C2×C181—42no—
106C2×C181#37911yescharacteristic
107D181—44no—
108C181#179111yescharacteristic, complemented, Sylow
109#1791—1811nomaximal, Sylow
110#585—1811no—
111#413—1811no—
112#413—1812no—
113#408—1812no—
114#408D18111yescharacteristic, complemented
1152−1+4—3622no—
1162−1+4—3621no—
117H(5,𝔽2)—3622no—
118H(5,𝔽2)—3621no—
119H(5,𝔽2)—1814no—
120C2×#41—3624no—
121C2×#41—3622no—
122C2×#41—1812no—
123C2×#41C2×D18111yescharacteristic
124C2×M16—1812no—
125C2×M16C2×D18111yescharacteristic, complemented
126C22×C8C2×D18111yescharacteristic, complemented
127#101—3622no—
128#101—3621no—
129#100—3622no—
130#100—3621no—
131C22⋊C8—22no—
132C22⋊C8—3624no—
133#41—22no—
134#41—22no—
135#41—3624no—
136#41—3624no—
137#41—3624no—
138#41—3624no—
139#41—3628no—
140#41—3624no—
141#41—3622no—
142#41—3622no—
143#41—3622no—
144#41—3622no—
145C2×Q8—3622no—
146C2×Q8—3624no—
147C2×Q8—1812no—
148C2×Q8D72411yescharacteristic
149C2×D4—3622no—
150C2×D4—3624no—
151C2×D4—3628no—
152C2×D4—3624no—
153C2×D4—1814no—
154C2×D4—1812no—
155C2×D4C4×D18112yes—
156C2×D4D72411yescharacteristic
157C22×C4—3624no—
158C22×C4—1814no—
159C22×C4C22⋊D18112yes—
160C22×C4C22×D18111yescharacteristic
161M16—22no—
162M16—3622no—
163M16—3622no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—3622no—
168C23—3624no—
169C23—1814no—
170C23#1566842612yes—
171C23C2×C4×D18111yescharacteristic
172Q8—22no—
173Q8—3624no—
174Q8—3622no—
175Q8—3622no—
176D4—24no—
177D4—22no—
178D4—3624no—
179D4—3628no—
180D4—3624no—
181D4—3624no—
182D4—3622no—
183D4—3622no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3628no—
187C2×C4—3624no—
188C2×C4—3624no—
189C2×C4—3624no—
190C2×C4—1814no—
191C2×C4#1566842612yes—
192C2×C4D4×D18112yes—
193C2×C4D4×D18112yes—
194C2×C4C2×C22⋊D18111yescharacteristic
195C2×C4C2×D72411yescharacteristic
196C8—41no—
197C8—41no—
198C8—7242no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—3624no—
203C22—14484no—
204C22—7244no—
205C22C22⋊C4×D18112yes—
206C22C2×#1566842611yescharacteristic
207C4—22no—
208C4—3624no—
209C4#1915378411yescharacteristic
210C4#1915378411yescharacteristic
211C4#1915378612yes—
212C2—21no—
213C2—22no—
214C2—42no—
215C2—7244no—
216C2#2604113411yescharacteristic
subgroupG/H
centerC4#19153784
commutatorC23×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#408×C181C2
socleC2×C181#379
prime radical#26042115C2
cosocleC22×C4×C181C23
hypercenter#408D181
hypocenterC181#1791
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #26041263C2semidirect
3. #26041261C2semidirect
4. #26041245C2semidirect
5. #408×C181C2semidirect
14. C2×#15668433C22semidirect
21. C2×M16×C181C22semidirect
22. C22×C8×C181C22semidirect
108. C181#1791semidirect
114. #408D181semidirect
125. C2×M16C2×D181semidirect
126. C22×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
#260412632
#260412612
#260412452
#408×C1812
#1791181