Finite Group Databasebeta

← #37544804

Subgroups — #37544804

Order 23168.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041891C211yesmaximal, characteristic, complemented
2#26041674C211yesmaximal, characteristic, complemented
3#26041658C211yesmaximal, characteristic, complemented
4#26041647C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041498C212yesmaximal, complemented
7#26041486C212yesmaximal, complemented
8#26041441C212yesmaximal
9#26041435C212yesmaximal, complemented
10#26041340C211yesmaximal, characteristic, complemented
11#536×C181C211yesmaximal, characteristic
12C2×#15668439C2212yes—
13C2×#15668439C2212yes—
14C2×#15668439C2212yes—
15C2×#15668437C2212yes—
16#19153816C2211yescharacteristic
17#19153809—22no—
18#19153808C2211yescharacteristic
19#19153806—22no—
20#19153795C2212yes—
21C4×Dic362C2212yes—
22C4⋊#13089678C2212yes—
23#19153740C2212yes—
24#19153740C2212yes—
25#19153740C2212yes—
26#19153737C2212yes—
27C22⋊#13089678C2212yes—
28#19153727C2211yescharacteristic
29C4×#13089678C2211yescharacteristic
30C2×#41×C181C2211yescharacteristic
31C22×Q8×C181C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33C4⋊Q8×C181C2211yescharacteristic
34#124×C181C2211yescharacteristic
35#124×C181C2211yescharacteristic
36#123×C181—22no—
37C22⋊Q8×C181—22no—
38#117×C181C2211yescharacteristic
39C2×Dic362C2312yes—
40#15668439—42no—
41#15668439—42no—
42#15668439—42no—
43#15668437—42no—
44C2×#13089678C2312yes—
45C2×#13089678C2312yes—
46C4⋊Dic181—22no—
47C4⋊Dic181C2312yes—
48#15668428—22no—
49#15668428—22no—
50#15668428—22no—
51C22⋊Dic181—22no—
52C4×Dic181—22no—
53#41×C181—42no—
54C2×Q8×C181—21no—
55C2×Q8×C181—42no—
56C2×Q8×C181C2311yescharacteristic
57C2×Q8×C181C2311yescharacteristic
58C2×Q8×C181C2311yescharacteristic
59C2×D4×C181—21no—
60C2×D4×C181C2311yescharacteristic
61C22×C4×C181—21no—
62C22×C4×C181—21no—
63C22×C4×C181C2311yescharacteristic
64C4⋊C4×C181—22no—
65C4⋊C4×C181—22no—
66C4⋊C4×C181D412yes—
67C22⋊C4×C181—22no—
68C22⋊C4×C181—22no—
69C22⋊C4×C181D412yes—
70C42×C181C2311yescharacteristic
71C42×C181C2311yescharacteristic
72Dic362—22no—
73Dic362—22no—
74C2×Dic181—22no—
75C2×Dic181—22no—
76#13089678—22no—
77#13089678—22no—
78C23×C181—21no—
79C23×C181C2×D411yescharacteristic
80Q8×C181—22no—
81Q8×C181—22no—
82Q8×C181—22no—
83Q8×C181—42no—
84D4×C181—22no—
85D4×C181—42no—
86C2×C4×C181—22no—
87C2×C4×C181—21no—
88C2×C4×C181—21no—
89C2×C4×C181—21no—
90C2×C4×C181—42no—
91C2×C4×C181—42no—
92C2×C4×C181C2×D412yes—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2×D411yescharacteristic
95C2×C4×C181C2411yescharacteristic
96C22×C181—21no—
97C22×C181—22no—
98C22×C181—41no—
99C22×C181—42no—
100C22×C181C22×D411yescharacteristic
101C4×C181—21no—
102C4×C181—22no—
103C4×C181—22no—
104C4×C181—41no—
105C4×C181—41no—
106C4×C181—41no—
107C4×C181H(5,𝔽2)12yes—
108Dic181—42no—
109Dic181—42no—
110C2×C181—41no—
111C2×C181—81no—
112C2×C181#53511yescharacteristic
113C2×C181#57812yes—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541—1811no—
117#536D18111yescharacteristic
118#497—1811no—
119#494—1811no—
120#491—1811no—
121#488—1811no—
122#464—1812no—
123#462—1812no—
124#452—1812no—
125#451—1812no—
126#433—1811no—
127C2×#41C2×D18111yescharacteristic
128C22×Q8C2×D18111yescharacteristic
129C2×Q16—1812no—
130C2×Q16—1812no—
131C2×Q16—1812no—
132C2×SD16—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8C2×D18111yescharacteristic
135C4⋊Q8C2×D18111yescharacteristic
136#126—3622no—
137#125—3622no—
138#125—1811no—
139#124C2×D18111yescharacteristic
140#124C2×D18111yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—1812no—
144C4×Q8—1812no—
145#117C2×D18111yescharacteristic
146C4⋊C8—1812no—
147#103—1812no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151C22⋊C8—1812no—
152#97—1811no—
153C4×C8—1811no—
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1812no—
158C2×Q8C22×D18111yescharacteristic
159C2×Q8C22×D18111yescharacteristic
160C2×Q8C22×D18111yescharacteristic
161C2×D4—21no—
162C2×D4C22×D18111yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D18111yescharacteristic
166Q16—7242no—
167Q16—7242no—
168Q16—7242no—
169SD16—7242no—
170C2×C8—1812no—
171C2×C8—1812no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—3622no—
175C4⋊C4—3622no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4—1812no—
179C4⋊C4C22⋊D18112yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—3622no—
183C22⋊C4C22⋊D18112yes—
184C42—3622no—
185C42C22×D18111yescharacteristic
186C42C22×D18111yescharacteristic
187C23—21no—
188C23C2×C22⋊D18111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3622no—
194Q8—3622no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4C2×C22⋊D18112yes—
206C2×C4C2×C22⋊D18112yes—
207C2×C4C2×C22⋊D18111yescharacteristic
208C2×C4C23×D18111yescharacteristic
209C8—3622no—
210C8—3622no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D18111yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—7242no—
223C4—7242no—
224C4#1915386112yes—
225C2—41no—
226C2—81no—
227C2#2604184611yescharacteristic
228C2#2604209412yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC22×C181C22×D4
prime radicalC2×#41×C181C22
hypercenter#536D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26041891C2semidirect
2. #26041674C2semidirect
3. #26041658C2semidirect
4. #26041647C2semidirect
5. #26041633C2semidirect
6. #26041498C2semidirect
7. #26041486C2semidirect
9. #26041435C2semidirect
10. #26041340C2semidirect
114. C181#3223semidirect

Maximal subgroups

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

subgroupindex
#260418912
#260416742
#260416582
#260416472
#260416332
#260414982
#260414862
#260414412
#260414352
#260413402
#536×C1812
#3223181