Finite Group Databasebeta

← #37535709

Subgroups — #37535709

Order 23168.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#26041960C211yesmaximal, characteristic, complemented
2#26041469C211yesmaximal, characteristic, complemented
3#26041457C211yesmaximal, characteristic, complemented
4#26041073C211yesmaximal, characteristic, complemented
5#26041068C211yesmaximal, characteristic, complemented
6#26041066C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C181C211yesmaximal, characteristic
8#19153846—21no—
9#19153809—21no—
10#19153803C2211yescharacteristic
11#19153797—21no—
12#19153797—21no—
13#19153797C2211yescharacteristic
14#19153793—21no—
15D4×Dic181—21no—
16#19153772—21no—
17C2×C22⋊Dic181—21no—
18#19153743—21no—
19#19153733C2211yescharacteristic
20#19153731—21no—
21#19153731C2211yescharacteristic
22H(5,𝔽2)×C181C2211yescharacteristic
23#120×C181—21no—
24#120×C181C2211yescharacteristic
25#99×C181—21no—
26#99×C181C2211yescharacteristic
27C2×Dic362D411yescharacteristic, complemented
28C22×Dic181—21no—
29C22×Dic181—21no—
30C22×Dic181D411yescharacteristic, complemented
31#15668439—41no—
32#15668437—41no—
33#15668433—21no—
34C4⋊Dic181—21no—
35#15668428—21no—
36#15668428—21no—
37#15668428—21no—
38#15668428—41no—
39C22⋊Dic181—21no—
40C22⋊Dic181—21no—
41C22⋊Dic181—21no—
42C22⋊Dic181—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45C22⋊Dic181—41no—
46C22⋊Dic181D411yescharacteristic, complemented
47C4×Dic181—21no—
48C4×Dic181D411yescharacteristic, complemented
49C24×C181D411yescharacteristic
50#41×C181—21no—
51#41×C181—41no—
52C2×D4×C181—21no—
53C2×D4×C181—21no—
54C2×D4×C181—21no—
55C2×D4×C181—41no—
56C2×D4×C181—41no—
57C2×D4×C181C2311yescharacteristic
58C22⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—41no—
61C22⋊C4×C181D411yescharacteristic
62Dic362—41no—
63C2×Dic181—21no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181—42no—
68C2×Dic181—41no—
69C2×Dic181—41no—
70C2×Dic181—81no—
71#13089678—41no—
72C23×C181—21no—
73C23×C181—21no—
74C23×C181—21no—
75C23×C181—41no—
76C23×C181—42no—
77C23×C181C2×D411yescharacteristic
78C23×C181C2×D411yescharacteristic
79Q8×C181—21no—
80D4×C181—21no—
81D4×C181—41no—
82D4×C181—42no—
83D4×C181—41no—
84D4×C181—41no—
85D4×C181—41no—
86D4×C181—82no—
87C2×C4×C181—21no—
88C2×C4×C181—21no—
89C2×C4×C181—21no—
90C2×C4×C181—41no—
91C2×C4×C181—41no—
92C2×C4×C181C2×D411yescharacteristic
93C22×C181—22no—
94C22×C181—21no—
95C22×C181—21no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181—42no—
100C22×C181—41no—
101C22×C181—41no—
102C22×C181—81no—
103C22×C181—81no—
104C22×C181—81no—
105C22×C181#12011yescharacteristic
106C4×C181—21no—
107C4×C181—41no—
108C4×C181—41no—
109C4×C181—81no—
110Dic181—41no—
111Dic181—41no—
112Dic181—41no—
113Dic181—41no—
114Dic181—81no—
115C2×C181—21no—
116C2×C181—41no—
117C2×C181—41no—
118C2×C181—42no—
119C2×C181—81no—
120C2×C181B(4,𝔽2)11yescharacteristic
121C181#219711yescharacteristic, complemented, Sylow
122#2197—1811nomaximal, Sylow
123#551—1811no—
124#458—1811no—
125B(4,𝔽2)D18111yescharacteristic
126#455—1811no—
127#354—1811no—
128#352—1811no—
129AΣL(1,𝔽16)—1811no—
130H(5,𝔽2)C2×D18111yescharacteristic
131#137—3621no—
132#126—3621no—
133#124—1811no—
134#123—3621no—
135#123—3621no—
136#123—1811no—
137C22⋊Q8—3621no—
138#120—21no—
139#120C2×D18111yescharacteristic
140C4×D4—3621no—
141#117—3621no—
142C2×C22⋊C4—3621no—
143#104—3621no—
144#100—1811no—
145#99—21no—
146#99—3621no—
147#99—1811no—
148#99C2×D18111yescharacteristic
149C24C22⋊D18111yescharacteristic
150#41—21no—
151#41—41no—
152C2×Q8—1811no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C22×D18111yescharacteristic
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4—1811no—
162Q16—7241no—
163SD16—7241no—
164M16—3621no—
165C4⋊C4—3621no—
166C4⋊C4—3621no—
167C4⋊C4—3621no—
168C4⋊C4—3621no—
169C4⋊C4—7241no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—3621no—
174C22⋊C4—3621no—
175C22⋊C4—3621no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—1811no—
180C22⋊C4—7241no—
181C22⋊C4C22⋊D18111yescharacteristic
182C42—3621no—
183C42—1811no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23D4×D18111yescharacteristic
190C23C2×C22⋊D18111yescharacteristic
191Q8—21no—
192Q8—7241no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—41no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—3621no—
208C2×C4—3621no—
209C2×C4—14481no—
210C2×C4—7241no—
211C2×C4—7242no—
212C2×C4—7241no—
213C2×C4D4×D18111yescharacteristic
214C8—7241no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#1915378411yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—14481no—
233C4—7241no—
234C4—7241no—
235C4—7241no—
236C4—7241no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—81no—
242C2#2604146511yescharacteristic
subgroupG/H
centerC2#26041465
commutator, cosocleC2×D4×C181C23
FrattiniC2×D4C22×D181
Fitting, prime radicalB(4,𝔽2)×C181C2
socleC2×C181B(4,𝔽2)
hypercenterB(4,𝔽2)D181
hypocenterC181#2197
LCSG ▷ C2×D4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26041960C2semidirect
2. #26041469C2semidirect
3. #26041457C2semidirect
4. #26041073C2semidirect
5. #26041068C2semidirect
6. #26041066C2semidirect
27. C2×Dic362D4semidirect
30. C22×Dic181D4semidirect
46. C22⋊Dic181D4semidirect
48. C4×Dic181D4semidirect
121. C181#2197semidirect

Maximal subgroups

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

subgroupindex
#260419602
#260414692
#260414572
#260410732
#260410682
#260410662
B(4,𝔽2)×C1812
#2197181