Finite Group Databasebeta

← #37532640

Subgroups — #37532640

Order 23168.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#26041837C211yesmaximal, characteristic, complemented
2#26041415C211yesmaximal, characteristic, complemented
3#26041412C211yesmaximal, characteristic, complemented
4#26041189C211yesmaximal, characteristic, complemented
5#26041012C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#327×C181C211yesmaximal, characteristic, complemented
8C22×D724—21no—
9C22×D724C2211yescharacteristic, complemented
10C2×D4×D181—21no—
11C2×D1448—21no—
12C2×#15668434—21no—
13#19153801—21no—
14#19153798—21no—
15#19153790C2211yescharacteristic
16#19153788—21no—
17#19153786—21no—
18#19153784—21no—
19C2×#15668426—21no—
20C2×#15668426C2211yescharacteristic
21#19153738—21no—
22#19153736—21no—
23C22⋊#13089678C2211yescharacteristic, complemented
24#121×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic, complemented
26#95×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C23×D181—21no—
29C23×D181D411yescharacteristic
30C2×D724—21no—
31C2×D724—21no—
32C2×D724—21no—
33C2×D724—42no—
34C2×D724—42no—
35C2×D724—42no—
36C2×D724—42no—
37C2×D724D411yescharacteristic, complemented
38C2×C22⋊D181—21no—
39C2×C22⋊D181—21no—
40C2×C22⋊D181—21no—
41D4×D181—42no—
42D4×D181—42no—
43C22×Dic181D411yescharacteristic
44C2×C4×D181—21no—
45D1448—42no—
46#15668434—42no—
47C2×#13089678—21no—
48C4⋊Dic181D411yescharacteristic, complemented
49#15668428—21no—
50C22⋊Dic181—21no—
51#15668426—21no—
52#15668426—21no—
53#15668426—21no—
54#15668426—21no—
55#15668426—42no—
56C2×D4×C181—21no—
57C2×D4×C181D411yescharacteristic, complemented
58C22×C4×C181—21no—
59C22×C4×C181C2311yescharacteristic
60C2×C8×C181—21no—
61C4⋊C4×C181D411yescharacteristic, complemented
62C22⋊C4×C181—21no—
63C22×D181—21no—
64C22×D181—21no—
65C22×D181—22no—
66C22×D181—21no—
67C22×D181—42no—
68C22×D181—41no—
69C22×D181—41no—
70C22×D181—41no—
71C22×D181—41no—
72C22×D181—41no—
73C22×D181—41no—
74C22×D181—41no—
75C22×D181—41no—
76C22×D181—81no—
77C22×D181—81no—
78C22×D181D812yescomplemented
79D724—22no—
80D724—22no—
81D724—41no—
82D724—41no—
83D724—41no—
84D724—42no—
85D724—48no—
86C22⋊D181—42no—
87C22⋊D181—41no—
88C22⋊D181—41no—
89C22⋊D181—81no—
90C2×Dic181—21no—
91C2×Dic181—21no—
92C2×Dic181—41no—
93C4×D181—42no—
94#13089678—41no—
95C23×C181—21no—
96C23×C181C2×D411yescharacteristic
97D4×C181—41no—
98D4×C181—41no—
99D4×C181—41no—
100C2×C4×C181—22no—
101C2×C4×C181—21no—
102C2×C4×C181—41no—
103C2×C4×C181—41no—
104C2×C4×C181—41no—
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C8×C181—41no—
108C22×C181—21no—
109C22×C181—21no—
110C22×C181—41no—
111C22×C181—41no—
112C22×C181—41no—
113C22×C181#12011yescharacteristic
114C22×C181C2×D811yescharacteristic
115C22×C181AGL(1,ℤ/8)11yescharacteristic
116C2×D181—22no—
117C2×D181—41no—
118C2×D181—41no—
119C2×D181—42no—
120C2×D181—42no—
121C2×D181—42no—
122C2×D181—41no—
123C2×D181—41no—
124C2×D181—42no—
125C2×D181—41no—
126C2×D181—81no—
127C2×D181—84no—
128C2×D181—81no—
129C2×D181—81no—
130C2×D181—81no—
131C2×D181—81no—
132C4×C181—21no—
133C4×C181—21no—
134C4×C181—42no—
135C4×C181—41no—
136Dic181—41no—
137Dic181—41no—
138C2×C181—21no—
139C2×C181—21no—
140C2×C181—81no—
141C2×C181#44711yescharacteristic
142C2×C181#45311yescharacteristic
143C2×C181B(4,𝔽2)11yescharacteristic
144D181—42no—
145D181—82no—
146D181—81no—
147C181#159511yescharacteristic, complemented, Sylow
148#1595—1811nomaximal, Sylow
149#534—1811no—
150#447—1811no—
151#447—1811no—
152#392—1811no—
153#327—1811no—
154#327D18111yescharacteristic, complemented
155#323—1811no—
156C22×D4—3621no—
157C22×D4—3621no—
158C22×D4—1811no—
159C2×D8—3621no—
160C2×D8—3621no—
161#123—3621no—
162#123—3621no—
163#121—3621no—
164#121—1811no—
165#121C2×D18111yescharacteristic
166#120—3621no—
167#120—3621no—
168C2×C22⋊C4—3621no—
169C2×C22⋊C4—1811no—
170#102—3621no—
171#102—3621no—
172C22⋊C8—1811no—
173C22⋊C8C2×D18111yescharacteristic, complemented
174#95C2×D18111yescharacteristic, complemented
175C24—3621no—
176C24—3621no—
177C24—1811no—
178C2×D4—21no—
179C2×D4—3621no—
180C2×D4—3621no—
181C2×D4—3621no—
182C2×D4—3621no—
183C2×D4—3621no—
184C2×D4—3621no—
185C2×D4—1811no—
186C2×D4—7242no—
187C2×D4—7242no—
188C2×D4—7242no—
189C2×D4—7242no—
190C2×D4—7242no—
191C2×D4—7242no—
192C2×D4D72411yescharacteristic, complemented
193C22×C4—21no—
194C22×C4—3621no—
195C22×C4—1811no—
196C22×C4C22×D18111yescharacteristic
197D8—7242no—
198D8—7242no—
199C2×C8—21no—
200C2×C8—3621no—
201C4⋊C4—3621no—
202C4⋊C4—1811no—
203C4⋊C4D72411yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C22⋊C4—3621no—
208C22⋊C4—3621no—
209C22⋊C4—3621no—
210C22⋊C4—7242no—
211C23—21no—
212C23—3622no—
213C23—3621no—
214C23—3621no—
215C23—3621no—
216C23—14481no—
217C23—14481no—
218C23—1812no—
219C23—7241no—
220C23—7241no—
221C23—7241no—
222C23—7241no—
223C23—7241no—
224C23—7242no—
225C23—7241no—
226C23—7241no—
227C23—7241no—
228C23D4×D18111yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—3622no—
233D4—3622no—
234D4—14481no—
235D4—7241no—
236D4—7241no—
237D4—7241no—
238D4—7242no—
239D4—7248no—
240D4—7242no—
241D4—7241no—
242D4—7241no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—3621no—
249C2×C4—3621no—
250C2×C4—7242no—
251C2×C4—7241no—
252C2×C4D4×D18111yescharacteristic
253C2×C4C2×D72411yescharacteristic
254C8—41no—
255C8—7241no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3622no—
262C22—14481no—
263C22—14481no—
264C22—14481no—
265C22—14484no—
266C22—14481no—
267C22—14481no—
268C22—7242no—
269C22—7242no—
270C22—7242no—
271C22—7241no—
272C22—7241no—
273C22—7242no—
274C22—7241no—
275C22—7241no—
276C22—7241no—
277C22#1915378611yescharacteristic
278C22D8×D18111yescharacteristic
279C22#1915384311yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—41no—
284C4—7241no—
285C4—7241no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—14482no—
290C2—14481no—
291C2—7242no—
292C2#2604141411yescharacteristic
293C2#2604144811yescharacteristic
294C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#327×C181C2
socleC22×C181#120
hypercenter#327D181
hypocenterC181#1595
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #26041837C2semidirect
2. #26041415C2semidirect
3. #26041412C2semidirect
4. #26041189C2semidirect
5. #26041012C2semidirect
6. #26041002C2semidirect
7. #327×C181C2semidirect
9. C22×D724C22semidirect
23. C22⋊#13089678C22semidirect
25. C22⋊C8×C181C22semidirect
26. #95×C181C22semidirect
37. C2×D724D4semidirect
48. C4⋊Dic181D4semidirect
57. C2×D4×C181D4semidirect
61. C4⋊C4×C181D4semidirect
78. C22×D181D8semidirect
147. C181#1595semidirect
154. #327D181semidirect
173. C22⋊C8C2×D181semidirect
174. #95C2×D181semidirect
192. C2×D4D724semidirect
203. C4⋊C4D724semidirect

Maximal subgroups

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

subgroupindex
#260418372
#260414152
#260414122
#260411892
#260410122
#260410022
#327×C1812
#1595181