Finite Group Databasebeta

← #37533015

Subgroups — #37533015

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#26041669C211yesmaximal, characteristic, complemented
3#26041668C211yesmaximal, characteristic, complemented
4#26041644C211yesmaximal, characteristic, complemented
5#26041020C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic, complemented
7#329×C181C211yesmaximal, characteristic, complemented
8C2×#15668438—21no—
9C2×#15668436—21no—
10C2×D1448—21no—
11C2×#15668434—21no—
12#19153813C2211yescharacteristic
13#19153808C2211yescharacteristic
14#19153804C2211yescharacteristic
15#19153800—21no—
16#19153798—21no—
17#19153793—21no—
18#19153792—21no—
19#19153789—21no—
20#19153788—21no—
21#19153738—21no—
22#19153738—21no—
23#19153727C2211yescharacteristic
24#124×C181C2211yescharacteristic
25#97×C181C2211yescharacteristic
26#97×C181C2211yescharacteristic
27C2×Dic362D411yescharacteristic
28C2×D724—21no—
29C2×D724—21no—
30C2×D724—21no—
31C2×D724D411yescharacteristic
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34C22×Dic181—21no—
35C2×C4×D181—21no—
36#15668438—42no—
37#15668436—42no—
38D1448—42no—
39#15668434—42no—
40C2×#13089678—21no—
41C4⋊Dic181—21no—
42C4⋊Dic181D411yescharacteristic
43C4⋊Dic181D411yescharacteristic
44#15668428—21no—
45#15668428—21no—
46#15668428—21no—
47C22⋊Dic181—21no—
48#15668426—21no—
49#15668426—21no—
50#15668426—21no—
51C2×Q8×C181D411yescharacteristic
52C2×D4×C181D411yescharacteristic
53C2×C8×C181—21no—
54C2×C8×C181—21no—
55C22⋊C4×C181—21no—
56C22⋊C4×C181—21no—
57C42×C181C2311yescharacteristic
58C22×D181—21no—
59C22×D181—41no—
60Dic362—41no—
61D724—41no—
62D724—41no—
63D724—41no—
64D724—41no—
65D724—41no—
66D724—42no—
67C22⋊D181—81no—
68C22⋊D181—81no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—81no—
73C4×D181—81no—
74#13089678—22no—
75C23×C181—21no—
76Q8×C181—41no—
77D4×C181—41no—
78C2×C4×C181—21no—
79C2×C4×C181C2×D411yescharacteristic
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C8×C181—22no—
83C8×C181—41no—
84C22×C181—41no—
85C22×C181—41no—
86C22×C181—41no—
87C22×C181#12011yescharacteristic
88C2×D181—41no—
89C2×D181—41no—
90C2×D181—41no—
91C2×D181—81no—
92C2×D181—81no—
93C2×D181—81no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—21no—
97C4×C181—41no—
98Dic181—41no—
99Dic181—41no—
100Dic181—41no—
101C2×C181—81no—
102C2×C181#45311yescharacteristic
103C2×C181#45311yescharacteristic
104C2×C181#45411yescharacteristic
105D181—81no—
106D181—161no—
107C181#165611yescharacteristic, complemented, Sylow
108#1656—1811nomaximal, Sylow
109#562—1811no—
110#496—1811no—
111#496—1811no—
112#490—1811no—
113#330—1811no—
114#329—1811no—
115#329D18111yescharacteristic, complemented
116C2×SD16—3621no—
117C2×SD16—3621no—
118C2×D8—3621no—
119C2×D8—3621no—
120#127—1811no—
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#97C2×D18111yescharacteristic
134#97C2×D18111yescharacteristic
135C2×Q8—1811no—
136C2×Q8D72411yescharacteristic
137C2×D4—3621no—
138C2×D4—3621no—
139C2×D4—3621no—
140C2×D4—3621no—
141C2×D4—3621no—
142C2×D4—1811no—
143C2×D4D72411yescharacteristic
144C22×C4—3621no—
145C22×C4—3621no—
146SD16—7242no—
147SD16—7242no—
148D8—7242no—
149D8—7242no—
150C2×C8—21no—
151C2×C8—21no—
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—3621no—
168C23—7241no—
169Q8—41no—
170Q8—7241no—
171D4—41no—
172D4—14481no—
173D4—14481no—
174D4—7241no—
175D4—7241no—
176D4—7241no—
177D4—7241no—
178D4—7242no—
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×D72411yescharacteristic
189C8—22no—
190C8—41no—
191C8—3622no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—14481no—
196C22—14481no—
197C22—14481no—
198C22—7241no—
199C22—7241no—
200C22—7241no—
201C22#1915378611yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—7241no—
207C4—7241no—
208C4—7241no—
209C2—81no—
210C2—28961no—
211C2—14481no—
212C2#2604144811yescharacteristic
213C2#2604144911yescharacteristic
214C2#2604145311yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#329×C181C2
socleC22×C181#120
hypercenter#329D181
hypocenterC181#1656
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #26042023C2semidirect
2. #26041669C2semidirect
3. #26041668C2semidirect
4. #26041644C2semidirect
5. #26041020C2semidirect
6. #26041018C2semidirect
7. #329×C181C2semidirect
107. C181#1656semidirect
115. #329D181semidirect

Maximal subgroups

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

subgroupindex
#260420232
#260416692
#260416682
#260416442
#260410202
#260410182
#329×C1812
#1656181