Finite Group Databasebeta

← #37535668

Subgroups — #37535668

Order 23168.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#26042076C211yesmaximal, characteristic, complemented
2#26041717C211yesmaximal, characteristic, complemented
3#26041714C211yesmaximal, characteristic, complemented
4#26041533C211yesmaximal, characteristic, complemented
5#26041088C211yesmaximal, characteristic, complemented
6#26041060C211yesmaximal, characteristic, complemented
7#360×C181C211yesmaximal, characteristic, complemented
8#41×D181—21no—
9#19153848—21no—
10#19153845—21no—
11#19153843—21no—
12#19153840—21no—
13#19153838C2211yescharacteristic
14#19153836—21no—
15C2×D1448C2211yescharacteristic, complemented
16C2×#15668432C2211yescharacteristic
17#19153791—21no—
18#19153761—21no—
19#19153759—21no—
20D2896—21no—
21#19153757—21no—
22#19153756C2211yescharacteristic, complemented
23#19153736—21no—
24#135×C181C2211yescharacteristic
25M32×C181C2211yescharacteristic, complemented
26#106×C181C2211yescharacteristic, complemented
27#15668450—21no—
28#15668449—21no—
29#15668449—41no—
30#15668448—21no—
31#15668448—41no—
32Q8×D181—21no—
33C2×D724—21no—
34C2×D724—41no—
35D4×D181—21no—
36D4×D181—41no—
37C2×C4×D181—41no—
38C2×C4×D181D411yescharacteristic
39Dic724D411yescharacteristic, complemented
40#15668439—21no—
41#15668438—21no—
42#15668437—21no—
43#15668436—21no—
44D1448—21no—
45D1448—21no—
46D1448D411yescharacteristic, complemented
47#15668434—21no—
48#15668432—21no—
49#15668432—21no—
50C2×#13089678D411yescharacteristic
51#15668426—41no—
52#15668424—21no—
53#41×C181—21no—
54Q16×C181D411yescharacteristic, complemented
55SD16×C181—21no—
56D8×C181D411yescharacteristic, complemented
57C2×C8×C181C2311yescharacteristic
58C4⋊C4×C181—21no—
59C16×C181—21no—
60C22×D181—41no—
61C22×D181—41no—
62C22×D181D811yescharacteristic
63Dic362—21no—
64Dic362—41no—
65D724—21no—
66D724—41no—
67D724—41no—
68D724—41no—
69D724—82no—
70C22⋊D181—41no—
71C22⋊D181—41no—
72C22⋊D181—41no—
73C2×Dic181—41no—
74C2×Dic181D811yescharacteristic
75C4×D181—21no—
76C4×D181—21no—
77C4×D181—41no—
78C4×D181—41no—
79C4×D181—41no—
80#13089678—21no—
81Q8×C181—21no—
82D4×C181—21no—
83D4×C181—41no—
84C2×C4×C181—41no—
85C2×C4×C181—41no—
86C2×C4×C181C2×D411yescharacteristic
87C8×C181C2×D411yescharacteristic
88C8×C181C2×D411yescharacteristic
89C22×C181—41no—
90C22×C181C2×D811yescharacteristic
91C2×D181—21no—
92C2×D181—41no—
93C2×D181—41no—
94C2×D181—82no—
95C2×D181—81no—
96C2×D181—81no—
97C2×D181—81no—
98C4×C181—41no—
99C4×C181—81no—
100C4×C181#12011yescharacteristic
101C4×C181AGL(1,ℤ/8)11yescharacteristic
102Dic181—21no—
103Dic181—41no—
104C2×C181—21no—
105C2×C181—81no—
106C2×C181#44711yescharacteristic
107D181—41no—
108D181—81no—
109D181—161no—
110C181#219011yescharacteristic, complemented, Sylow
111#2190—1811nomaximal, Sylow
112#575—1811no—
113#509—1811no—
114#509—1811no—
115#469—1811no—
116#360—1811no—
117#360D18111yescharacteristic, complemented
118#349—1811no—
119C2×#41—3621no—
120#137—3621no—
121#137—3621no—
122AGL(1,ℤ/8)—3621no—
123AGL(1,ℤ/8)—3621no—
124#135—3621no—
125#135—1811no—
126#135C2×D18111yescharacteristic
127C2×D8—1811no—
128C2×M16—1811no—
129#121—3621no—
130SD32—3621no—
131SD32—3621no—
132D16—3621no—
133D16—3621no—
134M32—1811no—
135M32C2×D18111yescharacteristic, complemented
136#106C2×D18111yescharacteristic, complemented
137#102—3621no—
138#41—21no—
139#41—3621no—
140#41—3621no—
141#41—3621no—
142#41—7241no—
143#41—7241no—
144C2×Q8—3621no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4—7241no—
148C2×D4—7241no—
149C22×C4—1811no—
150C22×C4—7241no—
151Q16—3621no—
152Q16—1811no—
153Q16D72411yescharacteristic, complemented
154SD16—21no—
155SD16—3621no—
156SD16—3621no—
157SD16—3621no—
158D8—3621no—
159D8—3621no—
160D8—3621no—
161D8—1811no—
162D8D72411yescharacteristic, complemented
163M16—3621no—
164M16—3621no—
165C2×C8—1811no—
166C2×C8C22×D18111yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—7241no—
169C16—21no—
170C16—3621no—
171C23—1811no—
172C23—7241no—
173C23—7241no—
174Q8—21no—
175Q8—3621no—
176Q8—7241no—
177D4—21no—
178D4—41no—
179D4—3621no—
180D4—14482no—
181D4—7241no—
182D4—7241no—
183D4—7241no—
184D4—7241no—
185D4—7241no—
186D4—7241no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3621no—
190C2×C4—3621no—
191C2×C4—1811no—
192C2×C4—7241no—
193C2×C4—7241no—
194C2×C4—7241no—
195C2×C4—7241no—
196C2×C4D4×D18111yescharacteristic
197C8—3621no—
198C8D4×D18111yescharacteristic
199C8C2×D72411yescharacteristic
200C22—41no—
201C22—3621no—
202C22—14481no—
203C22—14481no—
204C22—14482no—
205C22—14481no—
206C22—7241no—
207C22—7241no—
208C22D8×D18111yescharacteristic
209C4—41no—
210C4—81no—
211C4—3621no—
212C4—7241no—
213C4#1915378611yescharacteristic
214C4#1915384311yescharacteristic
215C2—21no—
216C2—81no—
217C2—28961no—
218C2—14481no—
219C2—7241no—
220C2#2604141411yescharacteristic
subgroupG/H
centerC2#26041414
commutator, cosocleC2×C8×C181C23
FrattiniC2×C8C22×D181
Fitting#360×C181C2
socleC2×C181#447
hypercenter#360D181
hypocenterC181#2190
LCSG ▷ C2×C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #26042076C2semidirect
2. #26041717C2semidirect
3. #26041714C2semidirect
4. #26041533C2semidirect
5. #26041088C2semidirect
6. #26041060C2semidirect
7. #360×C181C2semidirect
15. C2×D1448C22semidirect
22. #19153756C22semidirect
25. M32×C181C22semidirect
26. #106×C181C22semidirect
39. Dic724D4semidirect
46. D1448D4semidirect
54. Q16×C181D4semidirect
56. D8×C181D4semidirect
110. C181#2190semidirect
117. #360D181semidirect
135. M32C2×D181semidirect
136. #106C2×D181semidirect
153. Q16D724semidirect
162. D8D724semidirect

Maximal subgroups

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

subgroupindex
#260420762
#260417172
#260417142
#260415332
#260410882
#260410602
#360×C1812
#2190181