Finite Group Databasebeta

← C22×#89783 #14217635

Subgroups — C22×#89783 #14217635

Order 1792.

3350 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C23×#1136C211yesmaximal, characteristic, complemented
3C24×C7⋊C8C211yesmaximal, characteristic, complemented
4C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6C2×#89783C2124yesmaximal, direct product factor
7C22×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8C22×#6377C211yesmaximal, characteristic, complemented
9C23×C7×D4C411yescharacteristic
10C24×C4×C7C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C2×C4×C7×D4C4112yes—
13C2×C4×C7×D4C22112yes—
14C22×C7×C4⋊C4C411yescharacteristic
15C22×C7×C22⋊C4C411yescharacteristic
16C22×C7×C22⋊C4C411yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C22×#1136—28no—
19C22×#1136C22112yescomplemented
20C23×C7⋊C8—28no—
21C23×C7⋊C8C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23C23×C7⋊C8C2211yescharacteristic
24C23×C7⋊C8C2211yescharacteristic
25C23×C7⋊C8C22112yescomplemented
26C2×C22⋊(C7⋊C8)C22112yescomplemented
27C2×C22⋊(C7⋊C8)C22112yescomplemented
28#89783C22164yesdirect product factor
29C2×C4⋊(C7⋊C8)C22112yescomplemented
30C2×#6377C22112yescomplemented
31C22×#435—71nomaximal, Sylow
32C25×C7C2×C411yescharacteristic
33C25×C7C2×C411yescharacteristic
34C22×C7×D4—28no—
35C22×C7×D4C2×C4112yes—
36C23×C4×C7—24no—
37C23×C4×C7—24no—
38C23×C4×C7—24no—
39C23×C4×C7—24no—
40C23×C4×C7C2×C411yescharacteristic
41C23×C4×C7C2×C411yescharacteristic
42C23×C4×C7C2×C411yescharacteristic
43C23×C4×C7C2×C411yescharacteristic
44C23×C4×C7C2×C416yes—
45C23×C4×C7C2×C416yes—
46C23×C4×C7C2316yes—
47C23×C4×C7C2316yes—
48C23×C4×C7C2311yescharacteristic
49C4×C7×D4C2×C4148yes—
50C4×C7×D4C23116yes—
51C2×C7×C4⋊C4C2×C4112yes—
52C2×C7×C22⋊C4C2×C4112yes—
53C2×C7×C22⋊C4C2×C4112yes—
54C2×C42×C7C2×C416yes—
55C2×C42×C7C2316yes—
56C2×#1136—248no—
57C2×#1136C23116yescomplemented
58C22×C7⋊C8—24no—
59C22×C7⋊C8—24no—
60C22×C7⋊C8—24no—
61C22×C7⋊C8—248no—
62C22×C7⋊C8D418yescomplemented
63C22×C7⋊C8C2316yes—
64C22×C7⋊C8C2316yes—
65C22×C7⋊C8C2316yes—
66C22×C7⋊C8C2316yes—
67C22×C7⋊C8C23116yescomplemented
68C22⋊(C7⋊C8)C23116yescomplemented
69C22⋊(C7⋊C8)C23116yescomplemented
70C4⋊(C7⋊C8)C23116yescomplemented
71#6377C23116yescomplemented
72C23×M16—71no—
73C24×C8—71no—
74C22×C4×D4D711yescharacteristic
75C2×#435—724no—
76C22×C4⋊C8—71no—
77C22×C22⋊C8—71no—
78C22×C22⋊C8—71no—
79C22×#97—71no—
80C24×C7—24no—
81C24×C7—24no—
82C24×C7—24no—
83C24×C7—24no—
84C24×C7—24no—
85C24×C7M1618yescomplemented
86C24×C7C22×C416yes—
87C24×C7C22×C416yes—
88C24×C7C22×C411yescharacteristic
89C2×C7×D4—248no—
90C2×C7×D4C22×C4116yes—
91C22×C4×C7—24no—
92C22×C4×C7—24no—
93C22×C4×C7—24no—
94C22×C4×C7—24no—
95C22×C4×C7—224no—
96C22×C4×C7—224no—
97C22×C4×C7—212no—
98C22×C4×C7—212no—
99C22×C4×C7—212no—
100C22×C4×C7—212no—
101C22×C4×C7C22×C416yes—
102C22×C4×C7C22×C416yes—
103C22×C4×C7C22×C416yes—
104C22×C4×C7C22×C416yes—
105C22×C4×C7C22×C4112yes—
106C22×C4×C7C22×C4112yes—
107C22×C4×C7C22×C413yes—
108C22×C4×C7C2×D414yes—
109C22×C4×C7#4114yes—
110C22×C4×C7C2414yes—
111C22×C4×C7C2414yes—
112C22×C4×C7C2413yes—
113C7×C4⋊C4C22×C4116yes—
114C7×C22⋊C4C22×C4116yes—
115C7×C22⋊C4C22×C4116yes—
116C42×C7C22×C4112yes—
117C42×C7C2414yes—
118#1136—232no—
119C2×C7⋊C8—212no—
120C2×C7⋊C8—212no—
121C2×C7⋊C8—212no—
122C2×C7⋊C8—232no—
123C2×C7⋊C8C2×D4124yescomplemented
124C2×C7⋊C8C2414yes—
125C2×C7⋊C8C2414yes—
126C2×C7⋊C8C2414yes—
127C2×C7⋊C8C2414yes—
128C23×D4Dic711yescharacteristic
129C24×C4C2×D711yescharacteristic
130C24×C4C2×D711yescharacteristic
131C22×M16—712no—
132C22×M16—148no—
133C23×C8—71no—
134C23×C8—712no—
135C23×C8—71no—
136C23×C8—71no—
137C23×C8—71no—
138C23×C8—148no—
139C2×C4×D4Dic7112yes—
140C2×C4×D4C2×D7112yes—
141C22×C4⋊C4Dic711yescharacteristic
142C22×C22⋊C4Dic711yescharacteristic
143C22×C22⋊C4Dic711yescharacteristic
144C22×C42C2×D711yescharacteristic
145#435—764no—
146C2×C4⋊C8—712no—
147C2×C22⋊C8—712no—
148C2×C22⋊C8—712no—
149C2×#97—712no—
150C23×C7—212no—
151C23×C7—212no—
152C23×C7—232no—
153C23×C7—212no—
154C23×C7—212no—
155C23×C7—212no—
156C23×C7—416no—
157C23×C7C4×D414yes—
158C23×C7C2×M1614yes—
159C23×C7C2×M16124yescomplemented
160C23×C7#13114yes—
161C23×C7C23×C414yes—
162C23×C7C23×C414yes—
163C23×C7C23×C413yes—
164C7×D4—232no—
165C2×C4×C7—212no—
166C2×C4×C7—212no—
167C2×C4×C7—212no—
168C2×C4×C7—212no—
169C2×C4×C7—224no—
170C2×C4×C7—224no—
171C2×C4×C7—24no—
172C2×C4×C7—24no—
173C2×C4×C7—24no—
174C2×C4×C7—24no—
175C2×C4×C7C4×D4112yes—
176C2×C4×C7C23×C414yes—
177C2×C4×C7C23×C414yes—
178C2×C4×C7C23×C414yes—
179C2×C4×C7C23×C414yes—
180C2×C4×C7C23×C413yes—
181C2×C4×C7C22×D416yes—
182C2×C4×C7C2×#4116yes—
183C2×C4×C7C2511yescharacteristic
184C7⋊C8—24no—
185C7⋊C8—24no—
186C7⋊C8—24no—
187C7⋊C8C22×D418yescomplemented
188C25C2×Dic711yescharacteristic
189C25C2×Dic711yescharacteristic
190C22×D4—28no—
191C22×D4C2×Dic7112yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4C2×Dic711yescharacteristic
197C23×C4C2×Dic711yescharacteristic
198C23×C4C2×Dic711yescharacteristic
199C23×C4C2×Dic711yescharacteristic
200C23×C4C2×Dic716yes—
201C23×C4C2×Dic716yes—
202C23×C4C22×D716yes—
203C23×C4C22×D716yes—
204C23×C4C22×D711yescharacteristic
205C2×M16—716no—
206C2×M16—1448no—
207C22×C8—78no—
208C22×C8—76no—
209C22×C8—76no—
210C22×C8—716no—
211C22×C8—76no—
212C22×C8—76no—
213C22×C8—1448no—
214C22×C8—144no—
215C22×C8—144no—
216C22×C8—144no—
217C4×D4C2×Dic7148yes—
218C4×D4C22×D7116yes—
219C2×C4⋊C4C2×Dic7112yes—
220C2×C22⋊C4C2×Dic7112yes—
221C2×C22⋊C4C2×Dic7112yes—
222C2×C42C2×Dic716yes—
223C2×C42C22×D716yes—
224C4⋊C8—716no—
225C22⋊C8—716no—
226C22⋊C8—716no—
227#97—716no—
228C22×C7—24no—
229C22×C7—248no—
230C22×C7—24no—
231C22×C7—24no—
232C22×C7—24no—
233C22×C7—24no—
234C22×C7—424no—
235C22×C7#435116yescomplemented
236C22×C7C2×C4×D416yes—
237C22×C7C22×M1616yes—
238C22×C7C22×M1618yescomplemented
239C22×C7C2×#13116yes—
240C22×C7C24×C411yescharacteristic
241C4×C7—24no—
242C4×C7—24no—
243C4×C7—24no—
244C4×C7—24no—
245C4×C7C2×C4×D416yes—
246C4×C7C23×D411yescharacteristic
247C4×C7C22×#4111yescharacteristic
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24#113618yescomplemented
254C24C22×Dic716yes—
255C24C22×Dic716yes—
256C24C22×Dic711yescharacteristic
257C2×D4—248no—
258C2×D4C22×Dic7116yes—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—224no—
264C22×C4—224no—
265C22×C4—212no—
266C22×C4—212no—
267C22×C4—212no—
268C22×C4—212no—
269C22×C4D4×D714yes—
270C22×C4#115914yes—
271C22×C4C22×Dic716yes—
272C22×C4C22×Dic716yes—
273C22×C4C22×Dic716yes—
274C22×C4C22×Dic716yes—
275C22×C4C22×Dic7112yes—
276C22×C4C22×Dic7112yes—
277C22×C4C22×Dic713yes—
278C22×C4C23×D714yes—
279C22×C4C23×D714yes—
280C22×C4C23×D713yes—
281M16—1432no—
282C2×C8—724no—
283C2×C8—74no—
284C2×C8—74no—
285C2×C8—74no—
286C2×C8—74no—
287C2×C8—1432no—
288C2×C8—1412no—
289C2×C8—1412no—
290C2×C8—1412no—
291C4⋊C4C22×Dic7116yes—
292C22⋊C4C22×Dic7116yes—
293C22⋊C4C22×Dic7116yes—
294C42C22×Dic7112yes—
295C42C23×D714yes—
296C2×C7—28no—
297C2×C7—44no—
298C2×C7C2×#435112yescomplemented
299C2×C7C22×C4×D411yescharacteristic
300C2×C7C23×M1611yescharacteristic
301C2×C7C22×#13111yescharacteristic
302C23—212no—
303C23—212no—
304C23—232no—
305C23—212no—
306C23—212no—
307C23—212no—
308C23—416no—
309C23C2×#113614yes—
310C23C2×#1136124yescomplemented
311C23D4×Dic714yes—
312C23#651114yes—
313C23C23×Dic714yes—
314C23C23×Dic714yes—
315C23C23×Dic713yes—
316D4—232no—
317C2×C4—212no—
318C2×C4—212no—
319C2×C4—212no—
320C2×C4—212no—
321C2×C4—224no—
322C2×C4—224no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4D4×Dic7112yes—
328C2×C4C2×D4×D716yes—
329C2×C4C2×#115916yes—
330C2×C4C23×Dic714yes—
331C2×C4C23×Dic714yes—
332C2×C4C23×Dic714yes—
333C2×C4C23×Dic714yes—
334C2×C4C23×Dic713yes—
335C2×C4C24×D711yescharacteristic
336C8—78no—
337C8—144no—
338C8—144no—
339C8—144no—
340C7C22×#43511yescharacteristic, complemented, Sylow
341C22—24no—
342C22—24no—
343C22—248no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—424no—
348C22#89783116yesdirect product factor
349C22C22×#113616yes—
350C22C22×#113618yescomplemented
351C22C2×D4×Dic716yes—
352C22C2×#651116yes—
353C22C24×Dic711yescharacteristic
354C4—24no—
355C4—24no—
356C4—24no—
357C4—24no—
358C4C2×D4×Dic716yes—
359C4C22×D4×D711yescharacteristic
360C4C22×#115911yescharacteristic
361C2—28no—
362C2—44no—
363C2C2×#89783112yesdirect product factor
364C2C23×#113611yescharacteristic
365C2C22×D4×Dic711yescharacteristic
366C2C22×#651111yescharacteristic
subgroupG/H
centerC23×C4C22×D7
commutatorC22×C7C24×C4
FrattiniC2×C4C24×D7
FittingC22×C4×C7×D4C2
socleC24×C7C22×C4
prime radicalC23×C7×D4C4
cosocleC2×C4×C7C25
hypercenterC22×C4×D4D7
hypocenterC7C22×#435
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C23×#1136C2semidirect
3. C24×C7⋊C8C2semidirect
4. C22×C22⋊(C7⋊C8)C2semidirect
5. C22×C22⋊(C7⋊C8)C2semidirect
6. C2×#89783C2direct (N × H)
7. C22×C4⋊(C7⋊C8)C2semidirect
8. C22×#6377C2semidirect
19. C22×#1136C22semidirect
25. C23×C7⋊C8C22semidirect
26. C2×C22⋊(C7⋊C8)C22semidirect
27. C2×C22⋊(C7⋊C8)C22semidirect
28. #89783C22direct (N × H)
29. C2×C4⋊(C7⋊C8)C22semidirect
30. C2×#6377C22semidirect
57. C2×#1136C23semidirect
62. C22×C7⋊C8D4semidirect
67. C22×C7⋊C8C23semidirect
68. C22⋊(C7⋊C8)C23semidirect
69. C22⋊(C7⋊C8)C23semidirect
70. C4⋊(C7⋊C8)C23semidirect
71. #6377C23semidirect
85. C24×C7M16semidirect
123. C2×C7⋊C8C2×D4semidirect
159. C23×C7C2×M16semidirect
187. C7⋊C8C22×D4semidirect
235. C22×C7#435semidirect
238. C22×C7C22×M16semidirect
253. C24#1136semidirect
298. C2×C7C2×#435semidirect
310. C23C2×#1136semidirect
340. C7C22×#435semidirect
348. C22#89783direct (N × H)
350. C22C22×#1136semidirect
363. C2C2×#89783direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C23×#11362
C24×C7⋊C82
C22×C22⋊(C7⋊C8)2
C2×#897832
C22×C4⋊(C7⋊C8)2
C22×#63772
C22×#4357