Finite Group Databasebeta

← #37532713

Subgroups — #37532713

Order 23168.

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

normal subgroups only

This group has 228 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#26041610C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041206C211yesmaximal, characteristic
5#26041015C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic
7#328×C181C211yesmaximal, characteristic, complemented
8#19153796—21no—
9#19153795C2211yescharacteristic
10#19153794—21no—
11#19153794—21no—
12#19153793—21no—
13#19153792—21no—
14C2×C4⋊Dic181C2211yescharacteristic
15C2×#15668426—21no—
16C2×#15668426C2211yescharacteristic
17C22⋊C4×D181—21no—
18#19153748—21no—
19#19153747—21no—
20#19153741—21no—
21#19153740—21no—
22C22⋊#13089678C2211yescharacteristic
23#19153723—21no—
24C22⋊Q8×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×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⋊Dic181—21no—
37C4⋊Dic181—21no—
38C4⋊Dic181D411yescharacteristic
39#15668428—21no—
40#15668428—21no—
41#15668428—21no—
42C22⋊Dic181—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—42no—
48C2×Q8×C181D411yescharacteristic
49C22×C4×C181—21no—
50C22×C4×C181C2311yescharacteristic
51C2×C8×C181—21no—
52C4⋊C4×C181—21no—
53C4⋊C4×C181D411yescharacteristic
54C22⋊C4×C181—21no—
55C22×D181—21no—
56C22×D181—21no—
57C22×D181—41no—
58C22×D181—41no—
59C22×D181SD1612yes—
60Dic362—41no—
61Dic362—81no—
62C2×Dic181—21no—
63C2×Dic181—21no—
64C2×Dic181—21no—
65C2×Dic181—41no—
66C2×Dic181—41no—
67C2×Dic181—41no—
68C2×Dic181—41no—
69C4×D181—81no—
70C4×D181—81no—
71#13089678—41no—
72C23×C181C2×D411yescharacteristic
73Q8×C181—41no—
74C2×C4×C181—21no—
75C2×C4×C181—21no—
76C2×C4×C181—41no—
77C2×C4×C181—41no—
78C2×C4×C181—41no—
79C2×C4×C181—41no—
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C8×C181—41no—
83C22×C181—21no—
84C22×C181—21no—
85C22×C181#12011yescharacteristic
86C22×C181C2×SD1611yescharacteristic
87C22×C181#13711yescharacteristic
88C2×D181—22no—
89C2×D181—41no—
90C2×D181—41no—
91C2×D181—42no—
92C2×D181—81no—
93C4×C181—21no—
94C4×C181—21no—
95C4×C181—41no—
96C4×C181—41no—
97C4×C181—81no—
98Dic181—41no—
99Dic181—41no—
100Dic181—41no—
101Dic181—81no—
102C2×C181—21no—
103C2×C181—21no—
104C2×C181#44811yescharacteristic
105C2×C181#45511yescharacteristic
106C2×C181#45811yescharacteristic
107D181—42no—
108C181#160811yescharacteristic, complemented, Sylow
109#1608—1811nomaximal, Sylow
110#543—1811no—
111#483—1811no—
112#483—1811no—
113#397—1811no—
114#328—1811no—
115#328D18111yescharacteristic, complemented
116#323—1811no—
117C22⋊Q8—3621no—
118C22⋊Q8—3621no—
119C22⋊Q8—3621no—
120C22⋊Q8—3621no—
121C22⋊Q8—3621no—
122C22⋊Q8—1811no—
123C22⋊Q8C2×D18111yescharacteristic
124C2×C4⋊C4—1811no—
125C2×C22⋊C4—3621no—
126C2×C22⋊C4—3621no—
127C2×C22⋊C4—1811no—
128#106—3621no—
129#106—3621no—
130#103—3621no—
131#103—3621no—
132C22⋊C8—1811no—
133C22⋊C8C2×D18111yescharacteristic
134#95—3621no—
135#95C2×D18111yescharacteristic
136C24—1811no—
137C2×Q8—3621no—
138C2×Q8—1811no—
139C2×Q8D72411yescharacteristic
140C22×C4—21no—
141C22×C4—3621no—
142C22×C4—3621no—
143C22×C4—3621no—
144C22×C4—1811no—
145C22×C4C22×D18111yescharacteristic
146C2×C8—21no—
147C2×C8—3621no—
148C4⋊C4—21no—
149C4⋊C4—3621no—
150C4⋊C4—3621no—
151C4⋊C4—3621no—
152C4⋊C4—3621no—
153C4⋊C4—3621no—
154C4⋊C4—3621no—
155C4⋊C4—1811no—
156C4⋊C4D72411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3621no—
159C22⋊C4—3621no—
160C22⋊C4—3621no—
161C22⋊C4—3621no—
162C22⋊C4—3621no—
163C22⋊C4—7242no—
164C23—3621no—
165C23—3621no—
166C23—1812no—
167C23—7241no—
168C23—7241no—
169C23D4×D18111yescharacteristic
170Q8—41no—
171Q8—14481no—
172Q8—7241no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3621no—
180C2×C4—3621no—
181C2×C4—3621no—
182C2×C4—14481no—
183C2×C4—14481no—
184C2×C4—7241no—
185C2×C4—7241no—
186C2×C4—7241no—
187C2×C4—7241no—
188C2×C4D4×D18111yescharacteristic
189C2×C4C2×D72411yescharacteristic
190C8—41no—
191C8—7241no—
192C22—21no—
193C22—21no—
194C22—3622no—
195C22—14481no—
196C22—7242no—
197C22—7241no—
198C22—7241no—
199C22#1915378611yescharacteristic
200C22SD16×D18111yescharacteristic
201C22#1915384811yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—14481no—
208C4—7241no—
209C4—7241no—
210C4—7241no—
211C2—21no—
212C2—21no—
213C2—7242no—
214C2#2604141911yescharacteristic
215C2#2604145811yescharacteristic
216C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#328×C181C2
socleC22×C181#120
prime radicalC23×D181D4
hypercenter#328D181
hypocenterC181#1608
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26041610C2semidirect
3. #26041607C2semidirect
5. #26041015C2semidirect
7. #328×C181C2semidirect
108. C181#1608semidirect
115. #328D181semidirect

Maximal subgroups

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

subgroupindex
#260419042
#260416102
#260416072
#260412062
#260410152
#260410022
#328×C1812
#1608181