Finite Group Databasebeta

← #36354686

Subgroups — #36354686

Order 22144.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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#26012050C211yesmaximal, characteristic, complemented
2C2×#19149648C211yesmaximal, characteristic, complemented
3C2×#19149594C211yesmaximal, characteristic, complemented
4C2×#19149593C211yesmaximal, characteristic, complemented
5#26011347C211yesmaximal, characteristic, complemented
6#26011257C211yesmaximal, characteristic, complemented
7C2×#120×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173—21no—
9C22×C22⋊D173C2211yescharacteristic
10#19149662—21no—
11#19149660—21no—
12#19149651—22no—
13#19149648—22no—
14#19149648—22no—
15#19149648—21no—
16#19149647—21no—
17C2×C22⋊Dic173—21no—
18C2×C22⋊Dic173—22no—
19C2×C22⋊Dic173C2211yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic
21C2×C22⋊Dic173C2211yescharacteristic
22#19149594—22no—
23#19149593—22no—
24C25×C173C2211yescharacteristic
25C22×D4×C173—21no—
26C22×D4×C173C2211yescharacteristic
27#120×C173—22no—
28#120×C173—22no—
29C2×C22⋊C4×C173—21no—
30C2×C22⋊C4×C173C2211yescharacteristic
31C23×D173D411yescharacteristic, complemented
32C2×C22⋊D173—24no—
33C2×C22⋊D173—22no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C2×C22⋊D173—21no—
39C2×C22⋊D173—21no—
40C2×C22⋊D173—44no—
41C22×Dic173—21no—
42C22×Dic173—21no—
43C22×Dic173D411yescharacteristic, complemented
44C22×Dic173D412yescomplemented
45#14583805—22no—
46C22⋊Dic173—24no—
47C22⋊Dic173—22no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173—21no—
52C22⋊Dic173—21no—
53C22⋊Dic173—44no—
54C22⋊Dic173D412yescomplemented
55#14583803—22no—
56C24×C173—21no—
57C24×C173—21no—
58C24×C173—21no—
59C24×C173—21no—
60C24×C173—21no—
61C24×C173—21no—
62C24×C173—44no—
63C24×C173D411yescharacteristic, complemented
64C24×C173D412yescomplemented
65C24×C173C2311yescharacteristic
66C2×D4×C173—24no—
67C2×D4×C173—21no—
68C2×D4×C173—21no—
69C2×D4×C173—21no—
70C2×D4×C173—21no—
71C2×D4×C173—21no—
72C2×D4×C173—21no—
73C2×D4×C173—44no—
74C22×C4×C173—21no—
75C22×C4×C173D411yescharacteristic, complemented
76C22⋊C4×C173—24no—
77C22⋊C4×C173—21no—
78C22⋊C4×C173D412yescomplemented
79C22×D173—21no—
80C22×D173—21no—
81C22×D173—21no—
82C22×D173—42no—
83C22⋊D173—48no—
84C22⋊D173—41no—
85C22⋊D173—41no—
86C22⋊D173—41no—
87C22⋊D173—41no—
88C22⋊D173—82no—
89C2×Dic173—22no—
90C2×Dic173—21no—
91C2×Dic173—21no—
92C2×Dic173—22no—
93C2×Dic173—21no—
94C2×Dic173—42no—
95C2×Dic173—42no—
96C2×Dic173—42no—
97C2×Dic173—41no—
98C23×C173—24no—
99C23×C173—24no—
100C23×C173—24no—
101C23×C173—22no—
102C23×C173—22no—
103C23×C173—22no—
104C23×C173—21no—
105C23×C173—21no—
106C23×C173—21no—
107C23×C173—44no—
108C23×C173—42no—
109C23×C173—42no—
110C23×C173—42no—
111C23×C173—42no—
112C23×C173—44no—
113C23×C173—44no—
114C23×C173—44no—
115C23×C173—42no—
116C23×C173—41no—
117C23×C173—41no—
118C23×C173—41no—
119C23×C173—41no—
120C23×C173C2×D411yescharacteristic
121C23×C173C2×D411yescharacteristic
122C23×C173C2×D411yescharacteristic
123C23×C173C2×D411yescharacteristic
124C23×C173C2×D411yescharacteristic
125C23×C173C2×D411yescharacteristic
126C23×C173#4111yescharacteristic
127D4×C173—48no—
128D4×C173—41no—
129D4×C173—41no—
130D4×C173—41no—
131D4×C173—41no—
132C2×C4×C173—21no—
133C2×C4×C173—22no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—42no—
137C22×C173—22no—
138C22×C173—22no—
139C22×C173—22no—
140C22×C173—24no—
141C22×C173—24no—
142C22×C173—24no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173—42no—
148C22×C173—44no—
149C22×C173—42no—
150C22×C173—44no—
151C22×C173—42no—
152C22×C173—44no—
153C22×C173—42no—
154C22×C173—42no—
155C22×C173—44no—
156C22×C173—41no—
157C22×C173—42no—
158C22×C173—81no—
159C22×C173#12011yescharacteristic
160C22×C173#12011yescharacteristic
161C22×C173#12011yescharacteristic
162C22×C173#12111yescharacteristic
163C22×C173#12111yescharacteristic
164C22×C173#12111yescharacteristic
165C22×C173#12711yescharacteristic
166C2×D173—42no—
167C2×D173—41no—
168C2×D173—42no—
169C2×D173—81no—
170C4×C173—42no—
171C4×C173—41no—
172Dic173—42no—
173Dic173—41no—
174Dic173—42no—
175Dic173—81no—
176C2×C173—21no—
177C2×C173—21no—
178C2×C173—21no—
179C2×C173—21no—
180C2×C173—21no—
181C2×C173—21no—
182C2×C173—44no—
183C2×C173—81no—
184C2×C173#39211yescharacteristic
185C2×C173B(4,𝔽2)12yes—
186D173—81no—
187C173#202111yescharacteristic, complemented, Sylow
188#2021—1731nomaximal, Sylow
189#534—1731no—
190C2×#120—1731no—
191C2×#120D17311yescharacteristic, complemented
192C2×#99—1731no—
193C2×#99—1731no—
194#379—1731no—
195#342—1731no—
196C25C2×D17311yescharacteristic
197C22×D4—21no—
198C22×D4—3461no—
199C22×D4—1731no—
200C22×D4C2×D17311yescharacteristic
201#123—3461no—
202#123—3461no—
203#121—3462no—
204#120—22no—
205#120—22no—
206#120—3461no—
207#120—3462no—
208#120—3461no—
209#120—3462no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—3461no—
212C2×C22⋊C4—3462no—
213C2×C22⋊C4—1731no—
214C2×C22⋊C4—1731no—
215C2×C22⋊C4—1731no—
216C2×C22⋊C4C2×D17311yescharacteristic
217#99—3462no—
218#99—3462no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—44no—
226C24—1731no—
227C24C22⋊D17312yescomplemented
228C24C22⋊D17311yescharacteristic, complemented
229C24C22×D17311yescharacteristic
230C2×D4—24no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—44no—
238C2×D4—3462no—
239C2×D4—3464no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—6924no—
247C22×C4—21no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4—1732no—
252C22×C4C22⋊D17311yescharacteristic, complemented
253C4⋊C4—3462no—
254C22⋊C4—24no—
255C22⋊C4—21no—
256C22⋊C4—3462no—
257C22⋊C4—3464no—
258C22⋊C4—3462no—
259C22⋊C4—3461no—
260C22⋊C4—3461no—
261C22⋊C4—3461no—
262C22⋊C4—3461no—
263C22⋊C4—3461no—
264C22⋊C4—1732no—
265C22⋊C4—6924no—
266C22⋊C4C22⋊D17312yescomplemented
267C23—24no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—3461no—
290C23—3461no—
291C23—3461no—
292C23—6922no—
293C23D4×D17311yescharacteristic
294C23D4×D17311yescharacteristic
295C23D4×D17311yescharacteristic
296C23C2×C22⋊D17311yescharacteristic
297C23C2×C22⋊D17311yescharacteristic
298C23C2×C22⋊D17311yescharacteristic
299C23#1458382511yescharacteristic
300D4—48no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—13842no—
306D4—6928no—
307D4—6921no—
308D4—6921no—
309D4—6921no—
310D4—6921no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—3461no—
317C2×C4—3462no—
318C2×C4—3462no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—6922no—
322C2×C4—6921no—
323C2×C4—6922no—
324C2×C4—6922no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—41no—
345C22—42no—
346C22—81no—
347C22—13841no—
348C22—6922no—
349C22—6921no—
350C22—6922no—
351C22#1914964711yescharacteristic
352C22#1914964711yescharacteristic
353C22#1914964811yescharacteristic
354C22#1914965011yescharacteristic
355C22#1914965111yescharacteristic
356C22#1914965111yescharacteristic
357C22#1914967511yescharacteristic
358C4—42no—
359C4—41no—
360C4—13841no—
361C4—6922no—
362C4—6921no—
363C4—6922no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—44no—
371C2—81no—
372C2—13841no—
373C2#2601139911yescharacteristic
374C2#2601167712yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
FittingC2×#120×C173C2
socleC22×C173#120
hypercenterC2×#120D173
hypocenterC173#2021
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #26012050C2semidirect
2. C2×#19149648C2semidirect
3. C2×#19149594C2semidirect
4. C2×#19149593C2semidirect
5. #26011347C2semidirect
6. #26011257C2semidirect
7. C2×#120×C173C2semidirect
31. C23×D173D4semidirect
43. C22×Dic173D4semidirect
44. C22×Dic173D4semidirect
54. C22⋊Dic173D4semidirect
63. C24×C173D4semidirect
64. C24×C173D4semidirect
75. C22×C4×C173D4semidirect
78. C22⋊C4×C173D4semidirect
187. C173#2021semidirect
191. C2×#120D173semidirect
227. C24C22⋊D173semidirect
228. C24C22⋊D173semidirect
252. C22×C4C22⋊D173semidirect
266. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120502
C2×#191496482
C2×#191495942
C2×#191495932
#260113472
#260112572
C2×#120×C1732
#2021173