Finite Group Databasebeta

← #37532744

Subgroups — #37532744

Order 23168.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#26041904C211yesmaximal, characteristic
2#26041614C211yesmaximal, characteristic, complemented
3#26041214C211yesmaximal, characteristic
4#26041017C211yesmaximal, characteristic, complemented
5#26041016C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic
7#484×C181C211yesmaximal, characteristic, complemented
8#19153796—21no—
9#19153795C2211yescharacteristic
10#19153794—21no—
11#19153794—21no—
12#19153793—21no—
13#19153792—21no—
14C2×#15668428—21no—
15C2×#15668426—21no—
16C2×#15668426C2211yescharacteristic
17C22⋊C4×D181—21no—
18#19153749—21no—
19#19153739—21no—
20C22⋊#13089678C2211yescharacteristic
21#19153723C2211yescharacteristic
22C22⋊Q8×C181C2211yescharacteristic
23C2×C4⋊C4×C181C2211yescharacteristic
24#107×C181—21no—
25#103×C181—21no—
26C22⋊C8×C181C2211yescharacteristic
27C23×D181D411yescharacteristic
28C2×Dic362—21no—
29C2×Dic362D411yescharacteristic
30C22×Dic181—21no—
31C22×Dic181D411yescharacteristic
32C2×C4×D181—21no—
33C2×C4×D181—21no—
34C2×#13089678—21no—
35C4⋊Dic181—21no—
36C4⋊Dic181D411yescharacteristic
37#15668428—21no—
38#15668428—21no—
39#15668428—21no—
40#15668428—42no—
41C22⋊Dic181—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—42no—
47C2×Q8×C181D411yescharacteristic
48C22×C4×C181—21no—
49C22×C4×C181C2311yescharacteristic
50C2×C8×C181—21no—
51C4⋊C4×C181—21no—
52C4⋊C4×C181—21no—
53C4⋊C4×C181—21no—
54C4⋊C4×C181D411yescharacteristic
55C22⋊C4×C181—21no—
56C22×D181—21no—
57C22×D181—21no—
58C22×D181—41no—
59C22×D181—41no—
60C22×D181Q1612yes—
61Dic362—41no—
62Dic362—81no—
63C2×Dic181—22no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181—41no—
68C2×Dic181—41no—
69C2×Dic181—41no—
70C4×D181—81no—
71C4×D181—81no—
72#13089678—41no—
73C23×C181C2×D411yescharacteristic
74Q8×C181—41no—
75C2×C4×C181—21no—
76C2×C4×C181—21no—
77C2×C4×C181—41no—
78C2×C4×C181—41no—
79C2×C4×C181—41no—
80C2×C4×C181—41no—
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C8×C181—41no—
84C22×C181—21no—
85C22×C181—21no—
86C22×C181#12011yescharacteristic
87C22×C181C2×Q1611yescharacteristic
88C22×C181#13711yescharacteristic
89C2×D181—22no—
90C2×D181—41no—
91C2×D181—41no—
92C2×D181—42no—
93C2×D181—81no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—41no—
97C4×C181—41no—
98C4×C181—81no—
99Dic181—42no—
100Dic181—41no—
101Dic181—41no—
102Dic181—41no—
103C2×C181—21no—
104C2×C181—21no—
105C2×C181#45111yescharacteristic
106C2×C181#45311yescharacteristic
107C2×C181#45811yescharacteristic
108D181—42no—
109C181#161411yescharacteristic, complemented, Sylow
110#1614—1811nomaximal, Sylow
111#543—1811no—
112#484—1811no—
113#484D18111yescharacteristic, complemented
114#399—1811no—
115#328—1811no—
116#328—1811no—
117#323—1811no—
118C22⋊Q8—3621no—
119C22⋊Q8—3621no—
120C22⋊Q8—3621no—
121C22⋊Q8—3621no—
122C22⋊Q8—3621no—
123C22⋊Q8—1811no—
124C22⋊Q8C2×D18111yescharacteristic
125C2×C4⋊C4—3621no—
126C2×C4⋊C4C2×D18111yescharacteristic
127C2×C22⋊C4—3621no—
128C2×C22⋊C4—3621no—
129C2×C22⋊C4—1811no—
130#107—21no—
131#107—3621no—
132#103—21no—
133#103—3621no—
134C22⋊C8—1811no—
135C22⋊C8C2×D18111yescharacteristic
136#95—1811no—
137C24—1811no—
138C2×Q8—3621no—
139C2×Q8—1811no—
140C2×Q8C22⋊D18111yescharacteristic
141C22×C4—21no—
142C22×C4—3621no—
143C22×C4—3621no—
144C22×C4—3621no—
145C22×C4—1811no—
146C22×C4C22×D18111yescharacteristic
147C2×C8—21no—
148C2×C8—3621no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—3621no—
153C4⋊C4—3621no—
154C4⋊C4—3621no—
155C4⋊C4—3621no—
156C4⋊C4—1811no—
157C4⋊C4—7242no—
158C4⋊C4C22⋊D18111yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3621no—
161C22⋊C4—3621no—
162C22⋊C4—3621no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C22⋊C4—7242no—
166C23—3621no—
167C23—3621no—
168C23—1812no—
169C23—7241no—
170C23—7241no—
171C23D4×D18111yescharacteristic
172Q8—41no—
173Q8—14481no—
174Q8—7241no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—3622no—
182C2×C4—3621no—
183C2×C4—3621no—
184C2×C4—3621no—
185C2×C4—14481no—
186C2×C4—14481no—
187C2×C4—7241no—
188C2×C4—7241no—
189C2×C4—7241no—
190C2×C4D4×D18111yescharacteristic
191C2×C4C2×C22⋊D18111yescharacteristic
192C8—41no—
193C8—7241no—
194C22—21no—
195C22—21no—
196C22—3622no—
197C22—14481no—
198C22—7242no—
199C22—7241no—
200C22—7241no—
201C22#1915378411yescharacteristic
202C22Q16×D18111yescharacteristic
203C22#1915384811yescharacteristic
204C4—21no—
205C4—21no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—7242no—
210C4—7241no—
211C4—7241no—
212C4—7241no—
213C2—21no—
214C2—21no—
215C2—7242no—
216C2#2604143311yescharacteristic
217C2#2604144511yescharacteristic
218C2#2604146811yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#484×C181C2
socleC22×C181#120
prime radicalC23×D181D4
hypercenter#484D181
hypocenterC181#1614
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #26041614C2semidirect
4. #26041017C2semidirect
5. #26041016C2semidirect
7. #484×C181C2semidirect
109. C181#1614semidirect
113. #484D181semidirect

Maximal subgroups

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

subgroupindex
#260419042
#260416142
#260412142
#260410172
#260410162
#260410022
#484×C1812
#1614181