Finite Group Databasebeta

← #37533012

Subgroups — #37533012

Order 23168.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042023C211yesmaximal, characteristic, complemented
2#26041666C211yesmaximal, characteristic, complemented
3#26041643C211yesmaximal, characteristic, complemented
4#26041020C211yesmaximal, characteristic, complemented
5#26041019C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic, complemented
7#496×C181C211yesmaximal, characteristic, complemented
8C2×#15668437—21no—
9C2×#15668434—21no—
10#19153808C2211yescharacteristic
11#19153804C2211yescharacteristic
12#19153800—21no—
13#19153798—21no—
14#19153793—21no—
15#19153792—21no—
16#19153789—21no—
17#19153788—21no—
18#19153737—21no—
19#19153737—21no—
20#19153727C2211yescharacteristic
21#19153727C2211yescharacteristic
22C2×SD16×C181—21no—
23C2×D8×C181—21no—
24#127×C181C2211yescharacteristic
25#124×C181C2211yescharacteristic
26#97×C181C2211yescharacteristic
27C2×Dic362D411yescharacteristic
28C2×D724D411yescharacteristic
29C2×C22⋊D181—21no—
30C2×C22⋊D181—21no—
31C22×Dic181—21no—
32C2×C4×D181—21no—
33#15668437—42no—
34#15668434—42no—
35C2×#13089678—21no—
36C2×#13089678—21no—
37C4⋊Dic181—21no—
38C4⋊Dic181D411yescharacteristic
39C4⋊Dic181D411yescharacteristic
40#15668428—21no—
41#15668428—21no—
42#15668428—21no—
43C22⋊Dic181—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47C2×Q8×C181D411yescharacteristic
48C2×D4×C181—21no—
49C2×D4×C181—21no—
50C2×D4×C181—21no—
51C2×D4×C181D411yescharacteristic
52SD16×C181—42no—
53D8×C181—42no—
54C2×C8×C181—21no—
55C22⋊C4×C181—21no—
56C22⋊C4×C181—21no—
57C42×C181C2311yescharacteristic
58C22×D181—21no—
59Dic362—41no—
60D724—41no—
61C22⋊D181—81no—
62C22⋊D181—81no—
63C2×Dic181—21no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—81no—
67C4×D181—81no—
68#13089678—22no—
69#13089678—41no—
70C23×C181—21no—
71C23×C181—41no—
72Q8×C181—41no—
73D4×C181—41no—
74D4×C181—41no—
75D4×C181—41no—
76D4×C181—42no—
77D4×C181—41no—
78D4×C181—41no—
79C2×C4×C181—21no—
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C8×C181—22no—
84C22×C181—41no—
85C22×C181—41no—
86C22×C181—41no—
87C22×C181—81no—
88C22×C181—81no—
89C22×C181—81no—
90C22×C181#12011yescharacteristic
91C2×D181—41no—
92C2×D181—41no—
93C2×D181—41no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—21no—
97C4×C181—41no—
98Dic181—41no—
99Dic181—41no—
100Dic181—41no—
101C2×C181—81no—
102C2×C181—161no—
103C2×C181#45311yescharacteristic
104C2×C181#45311yescharacteristic
105C2×C181#45411yescharacteristic
106D181—81no—
107C181#165611yescharacteristic, complemented, Sylow
108#1656—1811nomaximal, Sylow
109#562—1811no—
110#496—1811no—
111#496D18111yescharacteristic, complemented
112#490—1811no—
113#330—1811no—
114#329—1811no—
115#329—1811no—
116C2×SD16—21no—
117C2×SD16—3621no—
118C2×D8—21no—
119C2×D8—3621no—
120#127C2×D18111yescharacteristic
121#125—1811no—
122#124—1811no—
123#124C2×D18111yescharacteristic
124#123—3621no—
125#123—3621no—
126C22⋊Q8—3621no—
127C22⋊Q8—3621no—
128#121—3621no—
129#121—3621no—
130#102—3621no—
131#102—3621no—
132#97—1811no—
133#97—1811no—
134#97C2×D18111yescharacteristic
135C2×Q8—1811no—
136C2×Q8C22⋊D18111yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3621no—
141C2×D4—3621no—
142C2×D4—1811no—
143C2×D4C22⋊D18111yescharacteristic
144C22×C4—3621no—
145C22×C4—3621no—
146SD16—42no—
147SD16—7242no—
148D8—42no—
149D8—7242no—
150C2×C8—21no—
151C2×C8—3621no—
152C2×C8—3621no—
153C4⋊C4—3621no—
154C4⋊C4—3621no—
155C4⋊C4—3621no—
156C4⋊C4—3621no—
157C4⋊C4—1811no—
158C4⋊C4—1811no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—3621no—
162C22⋊C4—3621no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C42C22×D18111yescharacteristic
166C23—21no—
167C23—41no—
168C23—3621no—
169Q8—41no—
170Q8—7241no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—14481no—
178D4—14481no—
179D4—7241no—
180C2×C4—21no—
181C2×C4—3621no—
182C2×C4—3621no—
183C2×C4—3621no—
184C2×C4—14481no—
185C2×C4—14481no—
186C2×C4D4×D18111yescharacteristic
187C2×C4D4×D18111yescharacteristic
188C2×C4C2×C22⋊D18111yescharacteristic
189C8—22no—
190C8—3622no—
191C8—7241no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—7241no—
199C22—7241no—
200C22—7241no—
201C22#1915378411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—7241no—
207C4—7241no—
208C4—7241no—
209C2—81no—
210C2—161no—
211C2—14481no—
212C2#2604144511yescharacteristic
213C2#2604144611yescharacteristic
214C2#2604145011yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#496×C181C2
socleC22×C181#120
hypercenter#496D181
hypocenterC181#1656
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26042023C2semidirect
2. #26041666C2semidirect
3. #26041643C2semidirect
4. #26041020C2semidirect
5. #26041019C2semidirect
6. #26041018C2semidirect
7. #496×C181C2semidirect
107. C181#1656semidirect
111. #496D181semidirect

Maximal subgroups

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

subgroupindex
#260420232
#260416662
#260416432
#260410202
#260410192
#260410182
#496×C1812
#1656181