Finite Group Databasebeta

← #13281004

Subgroups — #13281004

Order 1792.

2150 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×#89709C211yesmaximal, characteristic, complemented
3C2×#89708C211yesmaximal, characteristic, complemented
4#11729885C212yesmaximal
5#11729881C214yesmaximal, complemented
6#11729323C212yesmaximal, complemented
7#11729300C214yesmaximal, complemented
8C23×C7×D4C2211yescharacteristic
9C24×C4×C7C2212yes—
10C2×C4×C7×D4—24no—
11C2×C4×C7×D4—24no—
12C2×C4×C7×D4C418yescomplemented
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C42×C7C2211yescharacteristic
16C22×C22⋊Dic7C2218yes—
17C22×C4⋊Dic7C2212yes—
18C2×C7⋊#95C2212yes—
19C2×C7⋊#95C2212yes—
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2218yes—
24#89709—24no—
25#89708—24no—
26#14345—71nomaximal, Sylow
27C25×C7C2312yes—
28C22×C7×D4—28no—
29C22×C7×D4—22no—
30C22×C7×D4—22no—
31C22×C7×D4C2×C414yes—
32C23×C4×C7—24no—
33C23×C4×C7—24no—
34C23×C4×C7—48no—
35C23×C4×C7C2×C418yes—
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2311yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C23×Dic7C2314yes—
41C23×Dic7C2314yes—
42C4×C7×D4—232no—
43C2×C7×C4⋊C4—22no—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4C2×C414yes—
46C2×C7×C22⋊C4—24no—
47C2×C7×C22⋊C4—24no—
48C2×C7×C22⋊C4C2×C418yes—
49C2×C42×C7C2×C414yes—
50C2×C42×C7D414yes—
51C2×C42×C7D414yes—
52C2×C22⋊Dic7—216no—
53C2×C22⋊Dic7—48no—
54C2×C22⋊Dic7—48no—
55C2×C4⋊Dic7—24no—
56C2×C4⋊Dic7—44no—
57C7⋊#95—24no—
58C7⋊#95—24no—
59C7⋊#95—24no—
60C7⋊#95—24no—
61C7⋊#95—24no—
62C7⋊#95—48no—
63C22×C4×D4D711yescharacteristic
64C2×#383—71no—
65C2×#382—71no—
66#1445—72no—
67#1439—74no—
68#1438—72no—
69C22⋊#95—74no—
70C24×C7—22no—
71C24×C7—22no—
72C24×C7—24no—
73C24×C7—24no—
74C24×C7—28no—
75C24×C7—24no—
76C24×C7C22×C414yes—
77C24×C7#4118yes—
78C24×C7C2411yescharacteristic
79C2×C7×D4—216no—
80C2×C7×D4—216no—
81C2×C7×D4—216no—
82C2×C7×D4—28no—
83C22×C4×C7—24no—
84C22×C4×C7—22no—
85C22×C4×C7—24no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—28no—
90C22×C4×C7—216no—
91C22×C4×C7—416no—
92C22×C4×C7—416no—
93C22×C4×C7—416no—
94C22×C4×C7C22×C414yes—
95C22×C4×C7C22×C412yes—
96C22×C4×C7C22×C412yes—
97C22×C4×C7C22×C412yes—
98C22×C4×C7C2×D412yes—
99C22×C4×C7C2×D412yes—
100C22×C4×C7C2×D412yes—
101C22×C4×C7C2×D412yes—
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D412yes—
104C22×Dic7—24no—
105C22×Dic7—24no—
106C22×Dic7—28no—
107C22×Dic7—28no—
108C22×Dic7—24no—
109C22×Dic7—28no—
110C22×Dic7—28no—
111C22×Dic7—24no—
112C22×Dic7—24no—
113C22×Dic7—24no—
114C7×C4⋊C4—28no—
115C7×C22⋊C4—216no—
116C42×C7C22⋊C4116yescomplemented
117C22⋊Dic7—432no—
118C4⋊Dic7—48no—
119C23×D4C2×D711yescharacteristic
120C24×C4C2×D712yes—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4Dic718yescomplemented
124C22×C4⋊C4—72no—
125C22×C4⋊C4C2×D711yescharacteristic
126C22×C22⋊C4—78no—
127C22×C22⋊C4C2×D712yes—
128C22×C42C2×D711yescharacteristic
129#383—144no—
130#382—144no—
131C2×#95—72no—
132C2×#95—72no—
133C2×#95—72no—
134C2×#95—72no—
135C2×#95—72no—
136C2×#95—78no—
137C23×C7—24no—
138C23×C7—24no—
139C23×C7—28no—
140C23×C7—28no—
141C23×C7—28no—
142C23×C7—28no—
143C23×C7—216no—
144C23×C7—216no—
145C23×C7—44no—
146C23×C7—44no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—44no—
150C23×C7—48no—
151C23×C7—48no—
152C23×C7#11718yes—
153C23×C7C23×C411yescharacteristic
154C23×C7C22×D411yescharacteristic
155C23×C7C22×D411yescharacteristic
156C23×C7C2×#4114yes—
157C23×C7H(5,𝔽2)14yes—
158C23×C7H(5,𝔽2)12yes—
159C23×C72−1+412yes—
160C7×D4—232no—
161C2×C4×C7—24no—
162C2×C4×C7—24no—
163C2×C4×C7—24no—
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—22no—
167C2×C4×C7—24no—
168C2×C4×C7—24no—
169C2×C4×C7—22no—
170C2×C4×C7—24no—
171C2×C4×C7—24no—
172C2×C4×C7—28no—
173C2×C4×C7—432no—
174C2×C4×C7—44no—
175C2×C4×C7—44no—
176C2×C4×C7C2×C22⋊C414yes—
177C2×C4×C7C2×C22⋊C414yes—
178C2×C4×C7C2×C22⋊C414yes—
179C2×Dic7—44no—
180C2×Dic7—44no—
181C2×Dic7—44no—
182C2×Dic7—44no—
183C2×Dic7—44no—
184C2×Dic7—44no—
185C2×Dic7—48no—
186C2×Dic7—48no—
187C2×Dic7—48no—
188C2×Dic7—48no—
189C25C22×D712yes—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4C2×Dic714yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—48no—
197C23×C4—74no—
198C23×C4—74no—
199C23×C4C2×Dic718yes—
200C23×C4C22×D712yes—
201C23×C4C22×D711yescharacteristic
202C23×C4C22×D711yescharacteristic
203C23×C4C22×D711yescharacteristic
204C4×D4—232no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—144no—
208C2×C4⋊C4—284no—
209C2×C4⋊C4C2×Dic714yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—1416no—
213C2×C22⋊C4—288no—
214C2×C22⋊C4—288no—
215C2×C22⋊C4C2×Dic718yes—
216C2×C42C2×Dic714yes—
217C2×C42C22⋊D714yes—
218C2×C42C22⋊D714yes—
219#95—144no—
220#95—144no—
221#95—144no—
222#95—144no—
223#95—144no—
224#95—288no—
225C22×C7—216no—
226C22×C7—28no—
227C22×C7—44no—
228C22×C7—44no—
229C22×C7—48no—
230C22×C7—48no—
231C22×C7—48no—
232C22×C7—416no—
233C22×C7C22×C22⋊C411yescharacteristic
234C22×C7C2×#11712yes—
235C22×C7#51812yes—
236C22×C7#52012yes—
237C22×C7#52512yes—
238C22×C7#52512yes—
239C22×C7#53512yes—
240C22×C7#53512yes—
241C22×C7#53511yescharacteristic
242C22×C7#53611yescharacteristic
243C22×C7#53712yes—
244C22×C7#53818yes—
245C22×C7#53918yes—
246C4×C7—24no—
247C4×C7—24no—
248C4×C7—24no—
249C4×C7—44no—
250Dic7—84no—
251Dic7—84no—
252C24—24no—
253C24—28no—
254C24—24no—
255C24—24no—
256C24—22no—
257C24—22no—
258C24#115718yes—
259C24C22×Dic714yes—
260C24C23×D711yescharacteristic
261C2×D4—216no—
262C2×D4—216no—
263C2×D4—216no—
264C2×D4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—216no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—416no—
276C22×C4—148no—
277C22×C4—148no—
278C22×C4—144no—
279C22×C4—148no—
280C22×C4—144no—
281C22×C4—144no—
282C22×C4—148no—
283C22×C4—144no—
284C22×C4—144no—
285C22×C4—144no—
286C22×C4C22×Dic714yes—
287C22×C4C22×Dic712yes—
288C22×C4C22×Dic712yes—
289C22×C4C22×Dic712yes—
290C22×C4C2×C22⋊D712yes—
291C22×C4C2×C22⋊D712yes—
292C22×C4C2×C22⋊D712yes—
293C22×C4C2×C22⋊D712yes—
294C22×C4C2×C22⋊D712yes—
295C22×C4C2×C22⋊D712yes—
296C4⋊C4—28no—
297C4⋊C4—288no—
298C22⋊C4—216no—
299C22⋊C4—2832no—
300C42C22⋊Dic7116yescomplemented
301C2×C7—48no—
302C2×C7#228011yescharacteristic
303C2×C7#230911yescharacteristic
304C2×C7#231111yescharacteristic
305C2×C7#231814yes—
306C2×C7#263112yes—
307C2×C7#263312yes—
308C2×C7#263614yes—
309C23—24no—
310C23—28no—
311C23—28no—
312C23—28no—
313C23—216no—
314C23—216no—
315C23—28no—
316C23—24no—
317C23—44no—
318C23—44no—
319C23—48no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—44no—
324C23#648918yes—
325C23C2×#115714yes—
326C23#654814yes—
327C23#655312yes—
328C23#655412yes—
329C23C23×Dic711yescharacteristic
330C23C22×C22⋊D711yescharacteristic
331C23C22×C22⋊D711yescharacteristic
332D4—232no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—432no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—284no—
349C2×C4—284no—
350C2×C4—284no—
351C2×C4—284no—
352C2×C4—288no—
353C2×C4—288no—
354C2×C4—288no—
355C2×C4—288no—
356C2×C4—284no—
357C2×C4—284no—
358C2×C4C2×C22⋊Dic714yes—
359C2×C4C2×C22⋊Dic714yes—
360C2×C4C2×C22⋊Dic714yes—
361C7#1434511yescharacteristic, complemented, Sylow
362C22—28no—
363C22—216no—
364C22—44no—
365C22—44no—
366C22—48no—
367C22—48no—
368C22—48no—
369C22—416no—
370C22#9025118yes—
371C22#9025518yes—
372C22C2×#648912yes—
373C22#9048212yes—
374C22#9048212yes—
375C22#9048912yes—
376C22#9049612yes—
377C22#9049612yes—
378C22#9051611yescharacteristic
379C22#9051812yes—
380C22#9052111yescharacteristic
381C22#9052412yes—
382C22C22×C22⋊Dic711yescharacteristic
383C4—24no—
384C4—24no—
385C4—24no—
386C4—44no—
387C4—564no—
388C4—564no—
389C2—48no—
390C2#1173976914yes—
391C2#1173983612yes—
392C2#1173986612yes—
393C2#1173986814yes—
394C2#1174147211yescharacteristic
395C2#1174148911yescharacteristic
396C2#1174153211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC22×C4×C7×D4C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC22×C4×D4D7
hypocenterC7#14345
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#89709C2semidirect
3. C2×#89708C2semidirect
5. #11729881C2semidirect
6. #11729323C2semidirect
7. #11729300C2semidirect
12. C2×C4×C7×D4C4semidirect
116. C42×C7C22⋊C4semidirect
123. C2×C4×D4Dic7semidirect
300. C42C22⋊Dic7semidirect
361. C7#14345semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×#897092
C2×#897082
#117298852
#117298812
#117293232
#117293002
#143457