Finite Group Databasebeta

← #37546595

Subgroups — #37546595

Order 23168.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#26042008C211yesmaximal, characteristic, complemented
2#26041990C211yesmaximal, characteristic
3#26041680C212yesmaximal, complemented
4#26041675C211yesmaximal, characteristic, complemented
5#26041588C212yesmaximal, complemented
6#26041582C212yesmaximal
7#26041560C211yesmaximal, characteristic
8#26041403C211yesmaximal, characteristic, complemented
9C4×#15668436C211yesmaximal, characteristic
10#26041364C212yesmaximal
11#477×C181C211yesmaximal, characteristic, complemented
12C2×#15668436C2211yescharacteristic
13#19153816—21no—
14#19153816C2211yescharacteristic
15#19153807—22no—
16#19153807C2212yes—
17#19153807C2212yes—
18#19153796—21no—
19#19153792—22no—
20Q8×Dic181C2212yes—
21C4×Dic362—21no—
22C4×Dic362C2211yescharacteristic
23C4×D724C2211yescharacteristic
24#19153778C2212yes—
25C4⋊C4×D181—21no—
26#19153750C2212yes—
27#19153747C2212yes—
28#19153747C2211yescharacteristic
29#19153747C2212yes—
30#19153745C2212yes—
31#19153740C2211yescharacteristic
32#19153738C2212yes—
33#19153736C2211yescharacteristic
34C4×#13089678C2211yescharacteristic
35C8×Dic181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic
37C4×Q8×C181C2211yescharacteristic
38#107×C181C2212yes—
39C4⋊C8×C181C2211yescharacteristic
40#103×C181C2212yes—
41C2×Dic362—21no—
42C2×Dic362D411yescharacteristic
43C2×D724C2311yescharacteristic
44C2×C4×D181—22no—
45C2×C4×D181—21no—
46#15668436Q814yescomplemented
47C2×#13089678C2311yescharacteristic
48C2×#13089678C2311yescharacteristic
49C4⋊Dic181—22no—
50C4⋊Dic181—22no—
51C4⋊Dic181—22no—
52C4⋊Dic181—21no—
53C4⋊Dic181D411yescharacteristic
54C4⋊Dic181C2312yes—
55C4⋊Dic181C2311yescharacteristic
56#15668428—22no—
57#15668428—22no—
58#15668428D412yes—
59#15668426—22no—
60#15668426—21no—
61C4×Dic181—22no—
62C4×Dic181C2312yes—
63C2×Q8×C181—21no—
64C2×Q8×C181C2311yescharacteristic
65C2×C8×C181C2312yes—
66C4⋊C4×C181—21no—
67C4⋊C4×C181—21no—
68C4⋊C4×C181C2312yes—
69C4⋊C4×C181C2311yescharacteristic
70C42×C181—21no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73Dic362—22no—
74Dic362—42no—
75D724—21no—
76D724C2×Q812yes—
77C2×Dic181—22no—
78C2×Dic181—21no—
79C2×Dic181C2×D412yes—
80C2×Dic181C2×D412yes—
81C4×D181—42no—
82C4×D181—82no—
83#13089678—21no—
84#13089678C2×Q812yes—
85Q8×C181—21no—
86Q8×C181—81no—
87Q8×C181C2×Q812yes—
88C2×C4×C181—21no—
89C2×C4×C181—21no—
90C2×C4×C181—22no—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2×D411yescharacteristic
93C2×C4×C181C2411yescharacteristic
94C8×C181—22no—
95C22×C181C22×D411yescharacteristic
96C2×D181—22no—
97C2×D181—41no—
98C2×D181—41no—
99C4×C181—22no—
100C4×C181—21no—
101C4×C181—41no—
102C4×C181—42no—
103C4×C181#13512yes—
104C4×C181C22×Q811yescharacteristic
105C4×C1812−1+411yescharacteristic
106Dic181—22no—
107Dic181—22no—
108Dic181—41no—
109Dic181—42no—
110C2×C181D4×Q811yescharacteristic
111C2×C181C2×#13511yescharacteristic
112C2×C181#57711yescharacteristic
113D181—42no—
114C181#338111yescharacteristic, complemented, Sylow
115#3381—1811nomaximal, Sylow
116#557—1811no—
117#554—1811no—
118#499—1812no—
119#498—1811no—
120#479—1812no—
121#478—1812no—
122#477D18111yescharacteristic, complemented
123#474—1811no—
124C8×Q8—1811no—
125C4×SD16—1811no—
126C4×SD16—1812no—
127C2×SD16—1811no—
128C4⋊Q8—3621no—
129C4⋊Q8—1811no—
130C4⋊Q8C2×D18111yescharacteristic
131#125—3622no—
132#125—1812no—
133#125—1812no—
134C22⋊Q8—3621no—
135C22⋊Q8—3622no—
136C4×Q8—3621no—
137C4×Q8—1812no—
138C4×Q8—1811no—
139C4×Q8C2×D18111yescharacteristic
140C4×D4—1812no—
141C4×D4—1811no—
142C2×C4⋊C4—3621no—
143#107—1812no—
144#107C2×D18112yes—
145#106—1812no—
146#106—1811no—
147#106—1812no—
148C4⋊C8—1812no—
149C4⋊C8C2×D18111yescharacteristic
150#103—1811no—
151#103C2×D18112yes—
152#102—1812no—
153#102—1811no—
154C4×C8—1812no—
155C4×C8—1811no—
156C2×Q8—21no—
157C2×Q8—3621no—
158C2×Q8—1811no—
159C2×Q8C22×D18111yescharacteristic
160C2×D4—1811no—
161C22×C4—3622no—
162C22×C4—3621no—
163SD16—1814no—
164C2×C8—1811no—
165C2×C8—1811no—
166C2×C8C22×D18112yes—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—3622no—
170C4⋊C4—3622no—
171C4⋊C4—3622no—
172C4⋊C4—3622no—
173C4⋊C4—3622no—
174C4⋊C4—3621no—
175C4⋊C4—1812no—
176C4⋊C4—1811no—
177C4⋊C4—1811no—
178C4⋊C4—1812no—
179C4⋊C4C22×D18111yescharacteristic
180C4⋊C4C22×D18112yes—
181C22⋊C4—3622no—
182C22⋊C4—3621no—
183C42—21no—
184C42—3622no—
185C42—1812no—
186C42C22×D18111yescharacteristic
187C23—3621no—
188Q8—21no—
189Q8—81no—
190Q8—3622no—
191Q8—7242no—
192Q8Q8×D18112yes—
193D4—3621no—
194D4—1812no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—3622no—
199C2×C4—3621no—
200C2×C4—14482no—
201C2×C4—1812no—
202C2×C4—1812no—
203C2×C4—7242no—
204C2×C4D4×D18111yescharacteristic
205C2×C4D4×D18111yescharacteristic
206C2×C4C23×D18111yescharacteristic
207C8—22no—
208C8—3621no—
209C8—1812no—
210C22—3622no—
211C22—7241no—
212C22—7241no—
213C22C2×D4×D18111yescharacteristic
214C4—22no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—3622no—
219C4—3622no—
220C4—7241no—
221C4—7242no—
222C4#1915383912yes—
223C4C2×Q8×D18111yescharacteristic
224C4#1915386311yescharacteristic
225C2—7242no—
226C2#2604194211yescharacteristic
227C2C2×#1915383911yescharacteristic
228C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#477×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#477D181
hypocenterC181#3381
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #26042008C2semidirect
3. #26041680C2semidirect
4. #26041675C2semidirect
5. #26041588C2semidirect
8. #26041403C2semidirect
11. #477×C181C2semidirect
46. #15668436Q8semidirect
114. C181#3381semidirect
122. #477D181semidirect

Maximal subgroups

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

subgroupindex
#260420082
#260419902
#260416802
#260416752
#260415882
#260415822
#260415602
#260414032
C4×#156684362
#260413642
#477×C1812
#3381181