Finite Group Databasebeta

← #37546608

Subgroups — #37546608

Order 23168.

324 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 133 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#26042007C211yesmaximal, characteristic
2#26041682C211yesmaximal, characteristic
3#26041678C212yesmaximal
4#26041586C212yesmaximal
5#26041583C212yesmaximal
6#26041564C211yesmaximal, characteristic
7#26041561C211yesmaximal, characteristic
8Q8×#13089678C211yesmaximal, characteristic
9C4×#15668439C211yesmaximal, characteristic
10C4×#15668439C212yesmaximal
11#557×C181C211yesmaximal, characteristic
12C2×#15668439C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153814—21no—
15#19153808C2212yes—
16#19153806—22no—
17C4×Dic362C2211yescharacteristic
18C4×Dic362C2212yes—
19#19153781—21no—
20#19153749C2211yescharacteristic
21#19153749C2212yes—
22#19153749C2212yes—
23#19153747C2212yes—
24#19153747C2212yes—
25C4⋊#13089678C2211yescharacteristic
26C4⋊#13089678C2212yes—
27#19153740C2212yes—
28#19153740C2211yescharacteristic
29#19153739C2212yes—
30#19153739C2211yescharacteristic
31C4×#13089678C2212yes—
32C4×#13089678C2211yescharacteristic
33C4⋊Q8×C181—21no—
34C4⋊Q8×C181C2211yescharacteristic
35#125×C181—22no—
36#125×C181C2212yes—
37C4×Q8×C181—21no—
38C4×Q8×C181C2211yescharacteristic
39C4×Q8×C181C2212yes—
40C4×Q8×C181C2211yescharacteristic
41C2×Dic362—21no—
42C2×Dic362C2311yescharacteristic
43#15668439Q814yescomplemented
44C2×#13089678C2312yes—
45C2×#13089678C2311yescharacteristic
46C2×#13089678C2311yescharacteristic
47C4⋊Dic181—21no—
48C4⋊Dic181C2311yescharacteristic
49C4⋊Dic181C2312yes—
50#15668428—22no—
51#15668428—21no—
52#15668428—22no—
53#15668428—22no—
54C4×Dic181—22no—
55C4×Dic181—21no—
56C2×Q8×C181—21no—
57C2×Q8×C181D411yescharacteristic
58C2×Q8×C181C2311yescharacteristic
59C4⋊C4×C181—22no—
60C4⋊C4×C181—21no—
61C4⋊C4×C181—22no—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—22no—
64C4⋊C4×C181D411yescharacteristic
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2312yes—
68C42×C181—22no—
69C42×C181—21no—
70C42×C181C2312yes—
71C42×C181C2311yescharacteristic
72Dic362—21no—
73Dic362—42no—
74Dic362C2×Q812yes—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—22no—
78#13089678—22no—
79#13089678—21no—
80#13089678C2×Q812yes—
81Q8×C181—22no—
82Q8×C181—21no—
83Q8×C181—42no—
84Q8×C181C2×Q812yes—
85C2×C4×C181—21no—
86C2×C4×C181—21no—
87C2×C4×C181—22no—
88C2×C4×C181C2×D412yes—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2×D412yes—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2411yescharacteristic
93C22×C181C22×D411yescharacteristic
94C4×C181—22no—
95C4×C181—22no—
96C4×C181—22no—
97C4×C181—21no—
98C4×C181—42no—
99C4×C181—41no—
100C4×C181#13512yes—
101C4×C181C22×Q811yescharacteristic
102C4×C1812−1+411yescharacteristic
103Dic181—22no—
104Dic181—42no—
105Dic181—41no—
106C2×C181D4×Q811yescharacteristic
107C2×C181C2×#13511yescharacteristic
108C2×C181#57811yescharacteristic
109C181#338311yescharacteristic, complemented, Sylow
110#3383—1811nomaximal, Sylow
111#557—1811no—
112#557D18111yescharacteristic
113#500—1811no—
114#499—1812no—
115#479—1812no—
116#479—1812no—
117#475—1811no—
118#475—1811no—
119C8×Q8—1811no—
120C4×Q16—1812no—
121C4×Q16—1811no—
122C2×Q16—1811no—
123C4⋊Q8—21no—
124C4⋊Q8—3621no—
125C4⋊Q8—1811no—
126C4⋊Q8C2×D18111yescharacteristic
127#125—22no—
128#125—3622no—
129#125—1812no—
130#125C2×D18112yes—
131C4×Q8—21no—
132C4×Q8—3621no—
133C4×Q8—1811no—
134C4×Q8—1812no—
135C4×Q8C2×D18111yescharacteristic
136C4×Q8C2×D18112yes—
137C4×Q8C2×D18111yescharacteristic
138#107—1811no—
139#107—1812no—
140#107—1812no—
141#106—1812no—
142#106—1812no—
143C4⋊C8—1811no—
144C4⋊C8—1812no—
145#103—1812no—
146#103—1811no—
147#103—1812no—
148#103—1811no—
149C4×C8—1812no—
150C4×C8—1811no—
151C2×Q8—21no—
152C2×Q8—3621no—
153C2×Q8—1811no—
154C2×Q8C22⋊D18111yescharacteristic
155C2×Q8C22×D18111yescharacteristic
156Q16—1814no—
157C2×C8—1812no—
158C2×C8—1811no—
159C2×C8—1811no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—3622no—
166C4⋊C4—3621no—
167C4⋊C4—3622no—
168C4⋊C4—3621no—
169C4⋊C4—3622no—
170C4⋊C4—1811no—
171C4⋊C4—1812no—
172C4⋊C4C22⋊D18111yescharacteristic
173C4⋊C4C22⋊D18112yes—
174C4⋊C4C22×D18111yescharacteristic
175C4⋊C4C22×D18112yes—
176C42—22no—
177C42—21no—
178C42—3622no—
179C42—3621no—
180C42C22×D18112yes—
181C42C22×D18111yescharacteristic
182Q8—22no—
183Q8—21no—
184Q8—42no—
185Q8—3621no—
186Q8—1812no—
187Q8—7242no—
188Q8Q8×D18112yes—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—3621no—
193C2×C4—3621no—
194C2×C4—3622no—
195C2×C4C2×C22⋊D18112yes—
196C2×C4C2×C22⋊D18111yescharacteristic
197C2×C4C2×C22⋊D18112yes—
198C2×C4C2×C22⋊D18111yescharacteristic
199C2×C4C23×D18111yescharacteristic
200C8—3622no—
201C8—3621no—
202C8—1812no—
203C22C22×C22⋊D18111yescharacteristic
204C4—22no—
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—3622no—
211C4—7242no—
212C4—7241no—
213C4#1915383612yes—
214C4C2×Q8×D18111yescharacteristic
215C4#1915386411yescharacteristic
216C2Q8×C22⋊D18111yescharacteristic
217C2C2×#1915383611yescharacteristic
218C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#557×C181C2
socle, prime radicalC22×C181C22×D4
hypercenter#557D181
hypocenterC181#3383
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
43. #15668439Q8semidirect
109. C181#3383semidirect

Maximal subgroups

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

subgroupindex
#260420072
#260416822
#260416782
#260415862
#260415832
#260415642
#260415612
Q8×#130896782
C4×#156684392
#557×C1812
#3383181