Finite Group Databasebeta

← #37543254

Subgroups — #37543254

Order 23168.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153846C211yesmaximal, characteristic
2C22×Dic724C211yesmaximal, characteristic, complemented
3#26041850C211yesmaximal, characteristic
4#26041756C211yesmaximal, characteristic, complemented
5#26041500C214yesmaximal, complemented
6#26041488C214yesmaximal, complemented
7#26041305C211yesmaximal, characteristic, complemented
8#26041287C211yesmaximal, characteristic, complemented
9#416×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C22×Dic362C2211yescharacteristic
12#19153846—24no—
13C2×Dic724—24no—
14C2×Dic724C2214yes—
15C2×#15668439C2212yes—
16C2×#15668437C2212yes—
17C2×#15668433C2211yescharacteristic
18#19153814C2212yes—
19#19153803C2212yes—
20#19153797—22no—
21#19153795—22no—
22C4×Dic362—22no—
23#19153781C2214yes—
24#19153775C2211yescharacteristic
25#19153772—22no—
26#19153745C2214yes—
27#19153739C2214yes—
28C2×#41×C181C2211yescharacteristic
29C22×C8×C181C2211yescharacteristic
30#117×C181C2211yescharacteristic
31#103×C181C2212yes—
32#102×C181C2212yes—
33C2×Dic362—21no—
34C2×Dic362—21no—
35C2×Dic362—22no—
36C2×Dic362—22no—
37C2×Dic362—22no—
38C2×Dic362D414yes—
39C2×Dic362C2312yes—
40C2×Dic362C2312yes—
41C22×Dic181—21no—
42C22×Dic181—21no—
43Dic724—28no—
44#15668439—24no—
45#15668437—24no—
46#15668433—22no—
47C2×#13089678C2312yes—
48C4⋊Dic181D412yes—
49#15668428—24no—
50#15668428—24no—
51C22⋊Dic181—24no—
52C22⋊Dic181D412yes—
53C4×Dic181—24no—
54C4×Dic181C2312yes—
55#41×C181—22no—
56#41×C181—22no—
57C2×Q8×C181C2311yescharacteristic
58C2×D4×C181—21no—
59C2×D4×C181C2311yescharacteristic
60C22×C4×C181—21no—
61C22×C4×C181C2311yescharacteristic
62C2×C8×C181—22no—
63C2×C8×C181C2312yes—
64C4⋊C4×C181C2312yes—
65C22⋊C4×C181—21no—
66C42×C181—21no—
67Dic362—24no—
68Dic362—22no—
69Dic362—22no—
70Dic362—22no—
71Dic362—42no—
72Dic362C2×D414yes—
73C2×Dic181—24no—
74C2×Dic181—22no—
75C2×Dic181—22no—
76C2×Dic181—42no—
77C2×Dic181C2×D414yes—
78#13089678—22no—
79C23×C181—21no—
80C23×C181C2×D411yescharacteristic
81Q8×C181—21no—
82Q8×C181—21no—
83D4×C181—21no—
84D4×C181—21no—
85D4×C181—42no—
86C2×C4×C181—22no—
87C2×C4×C181—21no—
88C2×C4×C181—42no—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2×D412yes—
91C2×C4×C181#4112yes—
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181—22no—
95C22×C181—41no—
96C22×C181—42no—
97C22×C181#12112yes—
98C22×C181C22×D411yescharacteristic
99C4×C181—41no—
100C4×C181—42no—
101C4×C181#12112yes—
102C4×C181C22×D411yescharacteristic
103C4×C181C2×#4111yescharacteristic
104Dic181—24no—
105Dic181—24no—
106Dic181—42no—
107C2×C181—22no—
108C2×C181—81no—
109C2×C181C2×#12111yescharacteristic
110C2×C181#57812yes—
111C181#304411yescharacteristic, complemented, Sylow
112#3044—1811nomaximal, Sylow
113C2×#137—1811no—
114C22×Q16—1811no—
115#536—1811no—
116#519—1811no—
117#464—1814no—
118#462—1814no—
119#424—1811no—
120#419—1811no—
121#416D18111yescharacteristic
122C2×#41C2×D18111yescharacteristic
123C22×Q8—1811no—
124C22×Q8—1811no—
125#137—3624no—
126C2×Q16—3624no—
127C2×Q16—1812no—
128C2×Q16—1814no—
129C2×SD16—1812no—
130C2×M16—1811no—
131C22×C8C2×D18111yescharacteristic
132C4⋊Q8—1812no—
133#124—1812no—
134#123—3622no—
135C22⋊Q8—3622no—
136C4×Q8—3622no—
137C4×Q8—1814no—
138#117—3622no—
139#117—1811no—
140#117C2×D18111yescharacteristic
141C4⋊C8—1814no—
142#103—1814no—
143#103C2×D18112yes—
144#102C2×D18112yes—
145#41—22no—
146#41—22no—
147C2×Q8—3621no—
148C2×Q8—3621no—
149C2×Q8—3622no—
150C2×Q8—3622no—
151C2×Q8—3622no—
152C2×Q8—1814no—
153C2×Q8—1812no—
154C2×Q8—1812no—
155C2×Q8C22×D18111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D18111yescharacteristic
158C22×C4—21no—
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4C22×D18111yescharacteristic
162Q16—3628no—
163Q16—3624no—
164SD16—3624no—
165M16—3622no—
166C2×C8—22no—
167C2×C8—1812no—
168C2×C8C22×D18112yes—
169C4⋊C4—3624no—
170C4⋊C4—3624no—
171C4⋊C4—1812no—
172C4⋊C4C22×D18112yes—
173C22⋊C4—21no—
174C22⋊C4—3624no—
175C22⋊C4—1812no—
176C42—21no—
177C42—3624no—
178C42—1812no—
179C23—21no—
180C23D4×D18111yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—3624no—
184Q8—3622no—
185Q8—3622no—
186Q8—3622no—
187Q8—1814no—
188Q8—7242no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—3624no—
196C2×C4—3622no—
197C2×C4—3622no—
198C2×C4—1814no—
199C2×C4—7242no—
200C2×C4D4×D18111yescharacteristic
201C2×C4D4×D18112yes—
202C2×C4#1566844912yes—
203C2×C4C23×D18111yescharacteristic
204C8—22no—
205C8—3622no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1915378912yes—
210C22C2×D4×D18111yescharacteristic
211C4—41no—
212C4—42no—
213C4—3624no—
214C4—3624no—
215C4—7242no—
216C4#1915378912yes—
217C4C2×D4×D18111yescharacteristic
218C4C2×#1566844911yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1915378911yescharacteristic
222C2#2604209312yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#416×C181C2
socleC22×C181C22×D4
prime radicalC2×#41×C181C22
hypercenter#416D181
hypocenterC181#3044
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic724C2semidirect
4. #26041756C2semidirect
5. #26041500C2semidirect
6. #26041488C2semidirect
7. #26041305C2semidirect
8. #26041287C2semidirect
111. C181#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#191538462
C22×Dic7242
#260418502
#260417562
#260415002
#260414882
#260413052
#260412872
#416×C1812
#3044181