Finite Group Databasebeta

← #37546596

Subgroups — #37546596

Order 23168.

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

normal subgroups only

This group has 338 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, complemented
2#26041678C212yesmaximal, complemented
3#26041675C211yesmaximal, characteristic, complemented
4#26041583C212yesmaximal, complemented
5#26041579C212yesmaximal
6#26041571C211yesmaximal, characteristic, complemented
7#26041559C211yesmaximal, characteristic
8Q8×#13089678C211yesmaximal, characteristic, complemented
9C4×#15668437C211yesmaximal, characteristic
10C4×#15668437C212yesmaximal
11#554×C181C211yesmaximal, characteristic
12C2×#15668437C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153814—21no—
15#19153808C2212yes—
16#19153806—22no—
17C4×Dic362C2211yescharacteristic
18C4×Dic362C2212yes—
19#19153781—21no—
20#19153749C2212yes—
21#19153749C2212yes—
22#19153747C2212yes—
23#19153747C2211yescharacteristic
24#19153747C2212yes—
25C4⋊#13089678C2211yescharacteristic
26C4⋊#13089678C2212yes—
27#19153739C2212yes—
28#19153739C2211yescharacteristic
29#19153737C2212yes—
30#19153737C2211yescharacteristic
31C4×#13089678C2211yescharacteristic
32C4×#13089678C2212yes—
33C4⋊Q8×C181C2211yescharacteristic
34#125×C181C2212yes—
35C22⋊Q8×C181—21no—
36C22⋊Q8×C181—22no—
37C4×Q8×C181C2211yescharacteristic
38C4×D4×C181C2211yescharacteristic
39C4×D4×C181C2212yes—
40C2×C4⋊C4×C181—21no—
41C2×Dic362—21no—
42C2×Dic362C2311yescharacteristic
43#15668437Q814yescomplemented
44C2×#13089678C2311yescharacteristic
45C2×#13089678C2312yes—
46C2×#13089678C2311yescharacteristic
47C4⋊Dic181—21no—
48C4⋊Dic181C2312yes—
49C4⋊Dic181C2311yescharacteristic
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×D4×C181C2311yescharacteristic
59C22×C4×C181—21no—
60C22×C4×C181—22no—
61C4⋊C4×C181—22no—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—22no—
64C4⋊C4×C181D411yescharacteristic
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2312yes—
68C22⋊C4×C181—21no—
69C22⋊C4×C181—22no—
70C42×C181C2312yes—
71C42×C181C2311yescharacteristic
72Dic362—21no—
73Dic362—42no—
74Dic362C2×Q812yes—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—22no—
78#13089678—21no—
79#13089678—22no—
80#13089678C2×Q812yes—
81C23×C181—21no—
82Q8×C181—22no—
83Q8×C181—81no—
84D4×C181—21no—
85D4×C181C2×Q812yes—
86C2×C4×C181—21no—
87C2×C4×C181—22no—
88C2×C4×C181—42no—
89C2×C4×C181—82no—
90C2×C4×C181C2×D412yes—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2×D412yes—
93C2×C4×C181C2×D411yescharacteristic
94C2×C4×C181C2411yescharacteristic
95C22×C181—22no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181C22×D411yescharacteristic
99C4×C181—22no—
100C4×C181—22no—
101C4×C181—21no—
102C4×C181—41no—
103C4×C181—42no—
104C4×C181#13512yes—
105C4×C181C22×Q811yescharacteristic
106C4×C1812−1+411yescharacteristic
107Dic181—22no—
108Dic181—42no—
109Dic181—41no—
110C2×C181—42no—
111C2×C181D4×Q811yescharacteristic
112C2×C181C2×#13511yescharacteristic
113C2×C181#57711yescharacteristic
114C181#338111yescharacteristic, complemented, Sylow
115#3381—1811nomaximal, Sylow
116#557—1811no—
117#554D18111yescharacteristic
118#499—1812no—
119#498—1811no—
120#479—1812no—
121#478—1812no—
122#477—1811no—
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#125C2×D18112yes—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C4×Q8—3621no—
137C4×Q8—1811no—
138C4×Q8—1812no—
139C4×Q8C2×D18111yescharacteristic
140C4×D4C2×D18111yescharacteristic
141C4×D4C2×D18112yes—
142C2×C4⋊C4—21no—
143#107—1812no—
144#107—1812no—
145#106—1812no—
146#106—1811no—
147#106—1812no—
148C4⋊C8—1811no—
149C4⋊C8—1812no—
150#103—1812no—
151#103—1811no—
152#102—1812no—
153#102—1811no—
154C4×C8—1811no—
155C4×C8—1812no—
156C2×Q8—21no—
157C2×Q8—3621no—
158C2×Q8—1811no—
159C2×Q8C22⋊D18111yescharacteristic
160C2×D4C22×D18111yescharacteristic
161C22×C4—21no—
162C22×C4—22no—
163SD16—1814no—
164C2×C8—1811no—
165C2×C8—1812no—
166C2×C8—1811no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—3622no—
171C4⋊C4—3621no—
172C4⋊C4—3622no—
173C4⋊C4—3621no—
174C4⋊C4—3622no—
175C4⋊C4—1812no—
176C4⋊C4—1811no—
177C4⋊C4C22⋊D18111yescharacteristic
178C4⋊C4C22⋊D18112yes—
179C4⋊C4C22×D18112yes—
180C4⋊C4C22×D18111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—22no—
183C42—3622no—
184C42—3621no—
185C42C22×D18112yes—
186C42C22×D18111yescharacteristic
187C23—21no—
188Q8—22no—
189Q8—81no—
190Q8—3621no—
191Q8—1812no—
192Q8—7242no—
193D4—21no—
194D4Q8×D18112yes—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—3621no—
200C2×C4—3621no—
201C2×C4—3622no—
202C2×C4C2×C22⋊D18112yes—
203C2×C4C2×C22⋊D18112yes—
204C2×C4C2×C22⋊D18111yescharacteristic
205C2×C4C2×C22⋊D18111yescharacteristic
206C2×C4C23×D18111yescharacteristic
207C8—3621no—
208C8—3622no—
209C8—1812no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C22×C22⋊D18111yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—41no—
218C4—42no—
219C4—3622no—
220C4—7242no—
221C4—7241no—
222C4#1915383612yes—
223C4C2×Q8×D18111yescharacteristic
224C4#1915386411yescharacteristic
225C2—42no—
226C2Q8×C22⋊D18111yescharacteristic
227C2C2×#1915383611yescharacteristic
228C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#554×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#554D181
hypocenterC181#3381
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #26042007C2semidirect
2. #26041678C2semidirect
3. #26041675C2semidirect
4. #26041583C2semidirect
6. #26041571C2semidirect
8. Q8×#13089678C2semidirect
43. #15668437Q8semidirect
114. C181#3381semidirect

Maximal subgroups

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

subgroupindex
#260420072
#260416782
#260416752
#260415832
#260415792
#260415712
#260415592
Q8×#130896782
C4×#156684372
#554×C1812
#3381181