Finite Group Databasebeta

← #37534970

Subgroups — #37534970

Order 23168.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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#26041838C211yesmaximal, characteristic
2C2×#19153731C211yesmaximal, characteristic, complemented
3C2×#19153730C211yesmaximal, characteristic, complemented
4#26041158C211yesmaximal, characteristic, complemented
5#26041152C211yesmaximal, characteristic, complemented
6#26041045C211yesmaximal, characteristic, complemented
7#386×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2211yescharacteristic
9#19153799—21no—
10#19153797—21no—
11#19153788—21no—
12#19153788—21no—
13#19153785—21no—
14#19153784—21no—
15C2×#15668428—21no—
16C2×C22⋊Dic181C2211yescharacteristic
17C2×C22⋊Dic181C2211yescharacteristic
18C2×C22⋊Dic181C2211yescharacteristic
19C2×#15668426—21no—
20#19153731—22no—
21#19153730—22no—
22#19153723—21no—
23#19153723—21no—
24C22×D4×C181C2211yescharacteristic
25C23×C4×C181C2211yescharacteristic
26C2×C22⋊C4×C181—21no—
27C2×C22⋊C4×C181C2211yescharacteristic
28#95×C181—21no—
29C23×D181D411yescharacteristic
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—21no—
37C2×C22⋊D181—21no—
38C22×Dic181—21no—
39C22×Dic181—21no—
40C22×Dic181D411yescharacteristic
41C22×Dic181D411yescharacteristic
42C22×Dic181D411yescharacteristic
43#15668428—21no—
44#15668428—21no—
45#15668428—42no—
46C22⋊Dic181—21no—
47C22⋊Dic181—21no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181—21no—
52C22⋊Dic181—21no—
53C22⋊Dic181D412yescomplemented
54#15668426—21no—
55#15668426—21no—
56#15668426—42no—
57C24×C181D411yescharacteristic
58C24×C181C2311yescharacteristic
59C2×D4×C181—21no—
60C2×D4×C181—21no—
61C2×D4×C181—21no—
62C2×D4×C181—21no—
63C2×D4×C181—21no—
64C2×D4×C181—21no—
65C22×C4×C181—21no—
66C22×C4×C181—21no—
67C22×C4×C181—21no—
68C22×C4×C181—21no—
69C22×C4×C181—42no—
70C22×C4×C181D411yescharacteristic
71C22⋊C4×C181—21no—
72C22⋊C4×C181—21no—
73C22⋊C4×C181—42no—
74C22×D181—21no—
75C22×D181—21no—
76C22×D181—21no—
77C22×D181—41no—
78C22×D181—41no—
79C22⋊D181—41no—
80C22⋊D181—41no—
81C22⋊D181—41no—
82C22⋊D181—41no—
83C22⋊D181—81no—
84C22⋊D181—81no—
85C2×Dic181—21no—
86C2×Dic181—21no—
87C2×Dic181—21no—
88C2×Dic181—22no—
89C2×Dic181—21no—
90C2×Dic181—21no—
91C2×Dic181—41no—
92C2×Dic181—41no—
93C2×Dic181—41no—
94C2×Dic181—41no—
95C2×Dic181—41no—
96C2×Dic181—41no—
97C2×Dic181—41no—
98C23×C181—21no—
99C23×C181—21no—
100C23×C181—21no—
101C23×C181—41no—
102C23×C181—41no—
103C23×C181—41no—
104C23×C181—41no—
105C23×C181C2×D411yescharacteristic
106C23×C181C2×D411yescharacteristic
107C23×C181C2×D411yescharacteristic
108C23×C181C2×D411yescharacteristic
109C23×C181#4111yescharacteristic
110C23×C181#4111yescharacteristic
111C23×C181#4111yescharacteristic
112D4×C181—41no—
113D4×C181—41no—
114D4×C181—41no—
115D4×C181—41no—
116C2×C4×C181—22no—
117C2×C4×C181—22no—
118C2×C4×C181—22no—
119C2×C4×C181—21no—
120C2×C4×C181—21no—
121C2×C4×C181—21no—
122C2×C4×C181—41no—
123C2×C4×C181—41no—
124C2×C4×C181—41no—
125C2×C4×C181—41no—
126C2×C4×C181—41no—
127C2×C4×C181—41no—
128C2×C4×C181—41no—
129C22×C181—21no—
130C22×C181—21no—
131C22×C181—21no—
132C22×C181—21no—
133C22×C181—21no—
134C22×C181—21no—
135C22×C181—41no—
136C22×C181—41no—
137C22×C181—41no—
138C22×C181—41no—
139C22×C181—42no—
140C22×C181—41no—
141C22×C181—41no—
142C22×C181—41no—
143C22×C181—81no—
144C22×C181#12011yescharacteristic
145C22×C181#12111yescharacteristic
146C22×C181#12111yescharacteristic
147C22×C181#12111yescharacteristic
148C22×C181#12311yescharacteristic
149C22×C181#12311yescharacteristic
150C22×C181#12411yescharacteristic
151C2×D181—42no—
152C2×D181—41no—
153C2×D181—41no—
154C2×D181—41no—
155C2×D181—81no—
156C4×C181—41no—
157C4×C181—42no—
158C4×C181—81no—
159Dic181—42no—
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163Dic181—81no—
164C2×C181—21no—
165C2×C181—21no—
166C2×C181—21no—
167C2×C181—21no—
168C2×C181—21no—
169C2×C181—21no—
170C2×C181—81no—
171C2×C181#39411yescharacteristic
172C2×C181B(4,𝔽2)11yescharacteristic
173C2×C181#45811yescharacteristic
174D181—81no—
175C181#204411yescharacteristic, complemented, Sylow
176#2044—1811nomaximal, Sylow
177#534—1811no—
178C2×#99—1811no—
179C2×#99—1811no—
180#386—1811no—
181#386D18111yescharacteristic, complemented
182#385—1811no—
183#342—1811no—
184C22×D4—1811no—
185C22×D4C2×D18111yescharacteristic
186C23×C4C2×D18111yescharacteristic
187#123—3621no—
188#123—3621no—
189#121—3621no—
190#121—3621no—
191#120—3621no—
192#120—3621no—
193C2×C4⋊C4—3621no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—3621no—
196C2×C22⋊C4—1811no—
197C2×C22⋊C4—1811no—
198C2×C22⋊C4—1811no—
199C2×C22⋊C4C2×D18111yescharacteristic
200#99—3622no—
201#99—3622no—
202#95—21no—
203#95—3621no—
204#95—3621no—
205C24—1811no—
206C24C22⋊D18111yescharacteristic
207C24C22×D18111yescharacteristic
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—3621no—
215C2×D4—3621no—
216C2×D4—3621no—
217C2×D4—3621no—
218C2×D4—3621no—
219C2×D4—3621no—
220C2×D4—3621no—
221C2×D4—3621no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—3621no—
228C22×C4—3621no—
229C22×C4—1811no—
230C22×C4—1811no—
231C22×C4—1811no—
232C22×C4C22⋊D18111yescharacteristic
233C4⋊C4—3621no—
234C4⋊C4—3621no—
235C4⋊C4—7242no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—42no—
239C22⋊C4—3621no—
240C22⋊C4—3621no—
241C22⋊C4—3621no—
242C22⋊C4—3621no—
243C22⋊C4—3621no—
244C22⋊C4—3621no—
245C22⋊C4—3621no—
246C22⋊C4—3621no—
247C22⋊C4—3621no—
248C22⋊C4—1812no—
249C22⋊C4—7242no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—3621no—
258C23—3621no—
259C23—3621no—
260C23—7241no—
261C23—7241no—
262C23D4×D18111yescharacteristic
263C23D4×D18111yescharacteristic
264C23D4×D18111yescharacteristic
265C23C2×C22⋊D18111yescharacteristic
266C23#1566844811yescharacteristic
267C23#1566844911yescharacteristic
268C23#1566844911yescharacteristic
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—14481no—
274D4—14481no—
275D4—7241no—
276D4—7241no—
277D4—7241no—
278D4—7241no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3622no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—7241no—
299C2×C4—7241no—
300C2×C4—7241no—
301C2×C4—7241no—
302C2×C4—7241no—
303C2×C4—7241no—
304C2×C4—7241no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22—14481no—
321C22—7242no—
322C22—7241no—
323C22—7241no—
324C22—7241no—
325C22#1915378411yescharacteristic
326C22#1915378811yescharacteristic
327C22#1915378911yescharacteristic
328C22#1915378911yescharacteristic
329C22#1915379811yescharacteristic
330C22#1915379911yescharacteristic
331C22#1915380211yescharacteristic
332C4—41no—
333C4—42no—
334C4—81no—
335C4—14481no—
336C4—7242no—
337C4—7241no—
338C4—7241no—
339C4—7241no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—14481no—
348C2#2604119411yescharacteristic
349C2#2604146511yescharacteristic
350C2#2604146811yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#386×C181C2
socleC22×C181#120
prime radical#26041838C2
hypercenter#386D181
hypocenterC181#2044
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#19153731C2semidirect
3. C2×#19153730C2semidirect
4. #26041158C2semidirect
5. #26041152C2semidirect
6. #26041045C2semidirect
7. #386×C181C2semidirect
53. C22⋊Dic181D4semidirect
175. C181#2044semidirect
181. #386D181semidirect

Maximal subgroups

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

subgroupindex
#260418382
C2×#191537312
C2×#191537302
#260411582
#260411522
#260410452
#386×C1812
#2044181