Finite Group Databasebeta

← #37542201

Subgroups — #37542201

Order 23168.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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
2#26042045C212yesmaximal, complemented
3#26041262C214yesmaximal, complemented
4#26041261C212yesmaximal, complemented
5C2×#19153735C211yesmaximal, characteristic, complemented
6#101×D181C214yesmaximal, complemented
7C2×#101×C181C211yesmaximal, characteristic, complemented
8#19153864—24no—
9#19153864—24no—
10C2×#15668450C412yes—
11C2×#15668450C2212yes—
12C2×#15668449C414yes—
13C2×#15668449C2211yescharacteristic
14C2×#15668449C2211yescharacteristic
15C2×Q8×D181C412yes—
16C2×Q8×D181C2212yes—
17#19153824—24no—
18#19153822—24no—
19C2×#15668433C2212yes—
20M16×D181C2218yes—
21#19153735C2214yes—
22#19153734C2214yes—
23#19153732C2214yes—
24C22×Q8×C181C2211yescharacteristic
25C2×M16×C181C2212yes—
26#101×C181C2214yes—
27#15668450—28no—
28#15668450—28no—
29#15668449—28no—
30#15668449—28no—
31#15668449—22no—
32#15668449—22no—
33#15668449D414yes—
34#15668449D414yes—
35C2×Dic362C2×C414yes—
36C2×Dic362C2×C411yescharacteristic
37C2×Dic362C2311yescharacteristic
38Q8×D181—28no—
39Q8×D181—28no—
40C2×D724C2×C414yes—
41C2×D724C2×C411yescharacteristic
42C2×D724C2311yescharacteristic
43C2×C22⋊D181C2×C412yes—
44C2×C22⋊D181C2×C412yes—
45C2×C4×D181C2×C414yes—
46C2×C4×D181C2×C412yes—
47C2×C4×D181C2×C414yes—
48C2×C4×D181C2312yes—
49#15668433—22no—
50#15668433C2314yes—
51#15668432—28no—
52C2×#13089678—22no—
53C8×D181—28no—
54C2×Q8×C181—22no—
55C2×Q8×C181—22no—
56C2×Q8×C181C2×C412yes—
57C2×Q8×C181C2312yes—
58C22×C4×C181C2×C412yes—
59C22×C4×C181C2311yescharacteristic
60M16×C181—22no—
61M16×C181C2314yes—
62C2×C8×C181—22no—
63C22×D181C22×C412yes—
64C22×D181C22×C412yes—
65Dic362—22no—
66Dic362—28no—
67Dic362C22⋊C414yes—
68D724—28no—
69D724—22no—
70D724C22⋊C414yes—
71C22⋊D181—22no—
72C22⋊D181—22no—
73C22⋊D181C22⋊C414yes—
74C22⋊D181C22⋊C414yes—
75C2×Dic181C22×C412yes—
76C2×Dic181C22×C412yes—
77C4×D181—28no—
78C4×D181—24no—
79C4×D181—28no—
80C4×D181C2×D418yes—
81#13089678—24no—
82C23×C181C22×C411yescharacteristic
83Q8×C181—24no—
84Q8×C181—24no—
85C2×C4×C181—22no—
86C2×C4×C181—22no—
87C2×C4×C181C22×C414yes—
88C2×C4×C181C22×C411yescharacteristic
89C2×C4×C181C2×D412yes—
90C2×C4×C181C2×D412yes—
91C2×C4×C181C2411yescharacteristic
92C8×C181—24no—
93C22×C181—41no—
94C22×C181C2×C22⋊C411yescharacteristic
95C22×C181C2×C22⋊C411yescharacteristic
96C22×C181C23×C411yescharacteristic
97C2×D181—22no—
98C2×D181—42no—
99C2×D181—42no—
100C2×D181C2×C22⋊C414yes—
101C4×C181—24no—
102C4×C181C2×C22⋊C412yes—
103C4×C181C22×D412yes—
104Dic181—22no—
105Dic181C2×C22⋊C414yes—
106C2×C181—21no—
107C2×C181—21no—
108C2×C181—21no—
109C2×C181C22×C22⋊C411yescharacteristic
110D181—24no—
111D181—42no—
112C181#288911yescharacteristic, complemented, Sylow
113#2889—1811nomaximal, Sylow
114C2×2−1+4—1811no—
115#568—1812no—
116#413—1812no—
117#413—1814no—
118C2×#101—1811no—
119C2×#101—1814no—
120C2×#101D18111yescharacteristic, complemented
1212−1+4—3624no—
1222−1+4—3624no—
123C2×#41—1812no—
124C2×#41—1812no—
125C2×#41—1814no—
126C2×#41—1811no—
127C2×#41—1811no—
128C22×Q8—1812no—
129C22×Q8—1812no—
130C22×Q8C2×D18111yescharacteristic
131#131—3624no—
132#131—3624no—
133C2×M16—1812no—
134C2×M16—1818no—
135C2×M16C2×D18112yes—
136#101—1814no—
137#101—1814no—
138#101C2×D18114yes—
139#100—1814no—
140#41—3628no—
141#41—3628no—
142#41—3628no—
143#41—3628no—
144#41—3622no—
145#41—3622no—
146#41—1814no—
147#41—1814no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—3628no—
151C2×Q8—3628no—
152C2×Q8—1814no—
153C2×Q8—1811no—
154C2×Q8—1811no—
155C2×Q8C4×D18112yes—
156C2×Q8C22×D18112yes—
157C2×D4—1812no—
158C2×D4—1812no—
159C2×D4—1814no—
160C2×D4—1811no—
161C2×D4—1811no—
162C22×C4—1814no—
163C22×C4—1814no—
164C22×C4—1812no—
165C22×C4—1812no—
166C22×C4C4×D18112yes—
167C22×C4C22×D18111yescharacteristic
168M16—22no—
169M16—3628no—
170M16—3622no—
171M16—1814no—
172M16C22×D18114yes—
173C2×C8—22no—
174C2×C8—3628no—
175C2×C8—3622no—
176C23—1812no—
177C23—1812no—
178C23C2×C4×D18111yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—3628no—
182Q8—3622no—
183Q8—1814no—
184D4—3628no—
185D4—3622no—
186D4—3622no—
187D4—3622no—
188D4—1814no—
189D4—1814no—
190D4—1814no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—3628no—
194C2×C4—3628no—
195C2×C4—3624no—
196C2×C4—1812no—
197C2×C4—1812no—
198C2×C4—1818no—
199C2×C4C2×C4×D18114yes—
200C2×C4C2×C4×D18111yescharacteristic
201C2×C4D4×D18112yes—
202C2×C4D4×D18112yes—
203C2×C4C23×D18111yescharacteristic
204C8—24no—
205C8—3624no—
206C22—41no—
207C22—3622no—
208C22—1814no—
209C22—7242no—
210C22—7242no—
211C22C22⋊C4×D18111yescharacteristic
212C22C22⋊C4×D18111yescharacteristic
213C22C22×C4×D18111yescharacteristic
214C4—24no—
215C4—3622no—
216C4—1814no—
217C4C22⋊C4×D18112yes—
218C4C2×D4×D18112yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—3624no—
223C2—7242no—
224C2C2×C22⋊C4×D18111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D181
commutatorC22×C181C23×C4
FrattiniC2×C4C23×D181
FittingC2×#101×C181C2
socleC2×C181C22×C22⋊C4
prime radicalC2×#19153864C2
cosocleC2×C4×C181C24
hypercenterC2×#101D181
hypocenterC181#2889
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #26042045C2semidirect
3. #26041262C2semidirect
4. #26041261C2semidirect
5. C2×#19153735C2semidirect
6. #101×D181C2semidirect
7. C2×#101×C181C2semidirect
112. C181#2889semidirect
120. C2×#101D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
#260420452
#260412622
#260412612
C2×#191537352
#101×D1812
C2×#101×C1812
#2889181