Finite Group Databasebeta

← #13272952

Subgroups — #13272952

Order 1792.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89887C211yesmaximal, characteristic
2C2×C7×C22⋊C16C211yesmaximal, characteristic, complemented
3#11729725C211yesmaximal, characteristic, complemented
4C2×#89304C211yesmaximal, characteristic, complemented
5C2×#89304C211yesmaximal, characteristic, complemented
6C2×#89299C211yesmaximal, characteristic, complemented
7#11725612C211yesmaximal, characteristic, complemented
8C23×C7×C8C2211yescharacteristic
9C2×#6493—21no—
10C22×D56C2211yescharacteristic
11C22×C7×C16C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13#89984—21no—
14#89887—22no—
15#89887—22no—
16C2×#6395—21no—
17C2×#6392C2211yescharacteristic
18C2×#6392C2211yescharacteristic
19C2×#6392C2211yescharacteristic
20C7×C22⋊C16—22no—
21#89346—21no—
22#89304—22no—
23#89304—22no—
24#89299—22no—
25#13380—71nomaximal, Sylow
26C23×C4×C7D411yescharacteristic
27C22×C22⋊D7—41no—
28C22×D28—21no—
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C22×C7×C8—21no—
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C22×C7×C8—21no—
35C22×C7×C8D411yescharacteristic
36C22×C7×C8C2311yescharacteristic
37C2×C22⋊Dic7—41no—
38#6493—24no—
39#6493—42no—
40C2×(C2×D7)⋊C4—41no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—21no—
43C2×C4⋊Dic7—21no—
44C2×D56—21no—
45C2×D56—21no—
46C2×D56—21no—
47C2×D56—21no—
48C2×D56—21no—
49C2×D56D412yes—
50C2×C7×C16—21no—
51C2×C7×C16—21no—
52C2×C7×C16—21no—
53C2×C7×C16—21no—
54C2×C7×C16—21no—
55C2×C7×C16D412yescomplemented
56C7⋊#95—41no—
57#6395—24no—
58#6392—21no—
59#6392—21no—
60#6392—21no—
61#6392—21no—
62#6392—21no—
63#6392D412yes—
64C2×#466—71no—
65C2×#366—71no—
66C2×#357—71no—
67C2×#357—71no—
68C2×C22⋊C16D711yescharacteristic, complemented
69#1808—71no—
70#1380—71no—
71C24×C7D811yescharacteristic
72C23×D7—41no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7D811yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C2×C22⋊D7—44no—
82C2×C22⋊D7—82no—
83C2×C22⋊D7—82no—
84C22×Dic7—41no—
85C22×Dic7—41no—
86C22×Dic7—41no—
87C2×D28—22no—
88C2×D28—41no—
89C2×D28—41no—
90C2×D28—41no—
91C2×D28—41no—
92C2×D28—41no—
93C2×C7×C8—21no—
94C2×C7×C8—21no—
95C2×C7×C8—21no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8C2×D411yescharacteristic
101C2×C7×C8C2×D411yescharacteristic
102C2×C7×C8C2×D411yescharacteristic
103C2×C7×C8#4111yescharacteristic
104C2×C7×C8#4111yescharacteristic
105C2×C7×C8#4111yescharacteristic
106C22⋊Dic7—82no—
107(C2×D7)⋊C4—44no—
108C4⋊Dic7—22no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111C4⋊Dic7—41no—
112C4⋊Dic7—41no—
113C4⋊Dic7—41no—
114D56—22no—
115D56—22no—
116D56—41no—
117D56—41no—
118C7×C16—22no—
119C7×C16—41no—
120C22×D8—71no—
121C23×C8C2×D711yescharacteristic
122C2×#121—141no—
123C22×C16C2×D711yescharacteristic
124C22×C16C2×D711yescharacteristic
125#466—142no—
126#466—142no—
127C2×#107—71no—
128C2×#107—71no—
129C2×#107—71no—
130C2×#102—141no—
131#380—141no—
132#366—142no—
133#357—142no—
134#357—142no—
135C22⋊C16—22no—
136#340—141no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7D1612yes—
144C23×C7C2×D811yescharacteristic
145C22×D7—42no—
146C22×D7—81no—
147C22×D7—81no—
148C22×D7—81no—
149C22×D7—81no—
150C22×D7—81no—
151C22×D7—81no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7#12011yescharacteristic
160C2×C4×C7#12111yescharacteristic
161C2×C4×C7#12311yescharacteristic
162C2×C4×C7#13511yescharacteristic
163C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C7#13711yescharacteristic
165C22⋊D7—88no—
166C2×Dic7—42no—
167C2×Dic7—81no—
168C2×Dic7—81no—
169C2×Dic7—81no—
170C2×Dic7—81no—
171C2×Dic7—81no—
172C2×Dic7—81no—
173C2×Dic7—81no—
174C2×Dic7—81no—
175D28—42no—
176D28—42no—
177D28—81no—
178D28—81no—
179C7×C8—41no—
180C7×C8#12111yescharacteristic
181C7×C8#12111yescharacteristic
182C7×C8#12311yescharacteristic
183C7×C8#12411yescharacteristic
184C22×D4—141no—
185C22×D4—281no—
186C23×C4D2811yescharacteristic
187C2×D8—72no—
188C2×D8—141no—
189C2×D8—141no—
190C2×D8—141no—
191C2×D8—141no—
192C2×D8—141no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8D2811yescharacteristic
200C22×C8C22×D711yescharacteristic
201#121—144no—
202#121—282no—
203C2×C4⋊C4—141no—
204C2×C4⋊C4—141no—
205C2×C4⋊C4—141no—
206C2×C22⋊C4—281no—
207C2×C22⋊C4—281no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C22⋊D712yescomplemented
214#107—72no—
215#107—141no—
216#107—141no—
217#107—141no—
218#107—141no—
219#107—141no—
220#102—144no—
221#95—281no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7#44711yescharacteristic
230C22×C7#46611yescharacteristic
231C22×C7#48011yescharacteristic
232C22×C7C2×D1611yescharacteristic
233C22×C7#50811yescharacteristic
234C22×C7#50911yescharacteristic
235C22×C7#51011yescharacteristic
236C2×D7—82no—
237C2×D7—82no—
238C2×D7—82no—
239C2×D7—161no—
240C2×D7—161no—
241C2×D7—161no—
242C2×D7—161no—
243C4×C7—41no—
244C4×C7#39411yescharacteristic
245C4×C7#45211yescharacteristic
246C4×C7#46811yescharacteristic
247C4×C7#48211yescharacteristic
248Dic7—82no—
249Dic7—161no—
250Dic7—161no—
251C24—281no—
252C24D5611yescharacteristic
253C2×D4—142no—
254C2×D4—284no—
255C2×D4—281no—
256C2×D4—281no—
257C2×D4—281no—
258C2×D4—281no—
259C2×D4—281no—
260C2×D4—562no—
261C2×D4—562no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—281no—
269C22×C4—281no—
270C22×C4—281no—
271C22×C4D5611yescharacteristic
272C22×C4C2×D2811yescharacteristic
273D8—142no—
274D8—142no—
275D8—281no—
276D8—281no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8#115711yescharacteristic
285C2×C8D4×D711yescharacteristic
286C2×C8D4×D711yescharacteristic
287C2×C8#115911yescharacteristic
288C2×C8#115911yescharacteristic
289C2×C8C2×C22⋊D711yescharacteristic
290C4⋊C4—142no—
291C4⋊C4—281no—
292C4⋊C4—281no—
293C4⋊C4—281no—
294C4⋊C4—281no—
295C4⋊C4—281no—
296C22⋊C4—284no—
297C22⋊C4—562no—
298C16—22no—
299C16—41no—
300C2×C7—42no—
301C2×C7#203311yescharacteristic
302C2×C7#218411yescharacteristic
303C2×C7#218911yescharacteristic
304C2×C7#221511yescharacteristic
305C2×C7#222011yescharacteristic
306C2×C7#223211yescharacteristic
307C2×C7#223411yescharacteristic
308D7—162no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—282no—
316C23—561no—
317C23—561no—
318C23—561no—
319C23—561no—
320C23—561no—
321C23—561no—
322C23D11212yes—
323C23C2×D5611yescharacteristic
324D4—282no—
325D4—282no—
326D4—568no—
327D4—561no—
328D4—561no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—282no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4—561no—
341C2×C4—561no—
342C2×C4—561no—
343C2×C4—561no—
344C2×C4—561no—
345C2×C4#644511yescharacteristic
346C2×C4#644911yescharacteristic
347C2×C4#646711yescharacteristic
348C2×C4#647111yescharacteristic
349C2×C4#647211yescharacteristic
350C2×C4#649311yescharacteristic
351C8—41no—
352C8#644611yescharacteristic
353C8#644811yescharacteristic
354C8#650111yescharacteristic
355C8#650211yescharacteristic
356C7#1338011yescharacteristic, complemented, Sylow
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—562no—
365C22—562no—
366C22—562no—
367C22—1121no—
368C22—1121no—
369C22—1121no—
370C22—1121no—
371C22#8950411yescharacteristic
372C22#8950811yescharacteristic
373C22C2×D11211yescharacteristic
374C22#8967611yescharacteristic
375C22#8968011yescharacteristic
376C22#8968111yescharacteristic
377C22#8988711yescharacteristic
378C4—41no—
379C4—562no—
380C4—1121no—
381C4—1121no—
382C4#8950511yescharacteristic
383C4#8950911yescharacteristic
384C4#8973311yescharacteristic
385C4#8990611yescharacteristic
386C2—42no—
387C2—1122no—
388C2#1172684911yescharacteristic
389C2#1172685111yescharacteristic
390C2#1172686211yescharacteristic
391C2#1172686511yescharacteristic
392C2#1172768811yescharacteristic
393C2#1172916711yescharacteristic
394C2#1172919811yescharacteristic
subgroupG/H
centerC23C2×D56
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×C22⋊C16C2
socleC23×C7C2×D8
prime radicalC2×#89887C2
hypercenterC2×C22⋊C16D7
hypocenterC7#13380
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×C7×C22⋊C16C2semidirect
3. #11729725C2semidirect
4. C2×#89304C2semidirect
5. C2×#89304C2semidirect
6. C2×#89299C2semidirect
7. #11725612C2semidirect
55. C2×C7×C16D4semidirect
68. C2×C22⋊C16D7semidirect
213. C2×C16C22⋊D7semidirect
356. C7#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#898872
C2×C7×C22⋊C162
#117297252
C2×#893042
C2×#892992
#117256122
#133807