Finite Group Databasebeta

← #13256824

Subgroups — #13256824

Order 1792.

500 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731170C211yesmaximal, characteristic, complemented
2#11728211C211yesmaximal, characteristic
3#11727526C211yesmaximal, characteristic
4#11727241C211yesmaximal, characteristic
5C7×#1279C211yesmaximal, characteristic, complemented
6#11725508C211yesmaximal, characteristic, complemented
7#11725461C211yesmaximal, characteristic, complemented
8C2×#6437C2211yescharacteristic
9C2×C7×C4⋊C8C2211yescharacteristic
10C2×C4×C7×C8C2211yescharacteristic
11C4×C7×C4⋊C4C2211yescharacteristic
12C2×#6395—21no—
13C2×#6395—21no—
14C2×#6395—22no—
15#89749—21no—
16#89745—21no—
17C2×#6383—21no—
18C2×#6383—21no—
19#89708—21no—
20#89707C2211yescharacteristic
21C4×C4⋊Dic7C2211yescharacteristic
22#89700—21no—
23C2×C4⋊(C7⋊C8)C2211yescharacteristic
24#89432—22no—
25#10351—71nomaximal, Sylow
26C22×D28—21no—
27C22×D28—21no—
28C22×D28—21no—
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C42×C7—21no—
34C2×C42×C7C2311yescharacteristic
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7D411yescharacteristic
38C2×C4⋊Dic7D411yescharacteristic
39C2×C4⋊Dic7D412yes—
40C2×Dic7⋊C4—22no—
41C2×C4×Dic7—21no—
42C22×C7⋊C8—21no—
43#6437—21no—
44#6437—21no—
45#6437—21no—
46#6437Q812yes—
47C7×C4⋊C8—21no—
48C7×C4⋊C8—21no—
49C4×C7×C8—21no—
50C4×C7×C8—21no—
51C7×#95—21no—
52C7⋊#95—22no—
53C7⋊#95—22no—
54C7⋊#95—22no—
55C7⋊#95—21no—
56#6395—42no—
57#6395—42no—
58#6395—44no—
59#6383—42no—
60#6383—42no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63#2571—71no—
64#1982—71no—
65#1982—71no—
66#1957—71no—
67#1296—71no—
68#1279—71no—
69#1279D711yescharacteristic, complemented
70C23×D7—41no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—22no—
79C2×D28—22no—
80C2×D28—22no—
81C2×D28—22no—
82C2×D28—42no—
83C2×D28—42no—
84C2×D28—41no—
85C2×D28—41no—
86C2×D28—41no—
87C2×D28—41no—
88C2×D28—41no—
89C2×D28—41no—
90C2×D28—41no—
91C2×D28—41no—
92C2×D28—41no—
93C2×D28—41no—
94C2×D28—41no—
95C2×C7×C8—22no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C42×C7—41no—
106C42×C7—41no—
107C42×C7C2×Q811yescharacteristic
108C42×C7#4111yescharacteristic
109C42×C7#4111yescharacteristic
110C42×C7#4111yescharacteristic
111C4⋊Dic7—21no—
112C4⋊Dic7—21no—
113C4⋊Dic7—21no—
114C4⋊Dic7—21no—
115C4⋊Dic7—41no—
116C4⋊Dic7—41no—
117C4⋊Dic7—41no—
118C4⋊Dic7—41no—
119C4⋊Dic7—42no—
120Dic7⋊C4—82no—
121C4×Dic7—41no—
122C4×Dic7—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126C2×#127—71no—
127C2×C4⋊C8—71no—
128C2×C4⋊C8C2×D711yescharacteristic
129C2×#102—142no—
130C2×#102—141no—
131C2×#102—141no—
132C2×#102—141no—
133C2×#102—141no—
134C2×C4×C8C2×D711yescharacteristic
135#400—142no—
136#389—141no—
137#389—141no—
138#384—71no—
139#382—141no—
140#382—141no—
141C4×C4⋊C4—71no—
142C4×C4⋊C4C2×D711yescharacteristic
143C23×C7#12011yescharacteristic
144C22×D7—42no—
145C22×D7—81no—
146C22×D7—81no—
147C22×D7—81no—
148C22×D7—81no—
149C22×D7—81no—
150C22×D7—81no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7C22⋊Q811yescharacteristic
160C2×C4×C7C22⋊Q811yescharacteristic
161C2×C4×C7C22⋊Q811yescharacteristic
162C2×C4×C7#12411yescharacteristic
163C2×C4×C7#12411yescharacteristic
164C2×C4×C7#12411yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×C4×C7#13511yescharacteristic
167C2×C4×C7#13511yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—42no—
178C2×Dic7—42no—
179C2×Dic7—41no—
180C2×Dic7—42no—
181C2×Dic7—41no—
182D28—44no—
183D28—42no—
184D28—42no—
185D28—42no—
186D28—42no—
187D28—82no—
188D28—81no—
189D28—81no—
190D28—81no—
191D28—81no—
192C7×C8—42no—
193C7×C8—81no—
194C7⋊C8—81no—
195C22×D4—141no—
196C22×D4—141no—
197C22×D4—141no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—141no—
201#127—72no—
202#127—141no—
203#127—141no—
204#127—141no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—72no—
208C2×C4⋊C4—141no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—142no—
211C2×C4⋊C4D2811yescharacteristic
212C2×C4⋊C4D2811yescharacteristic
213C2×C42—21no—
214C2×C42—141no—
215C2×C42C22×D711yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—141no—
219C4⋊C8—141no—
220#102—282no—
221#102—282no—
222#102—282no—
223#102—282no—
224#102—284no—
225C4×C8—21no—
226C4×C8—21no—
227#95—21no—
228#95—142no—
229#95—142no—
230#95—142no—
231#95—141no—
232C22×C7#39311yescharacteristic
233C22×C7#44911yescharacteristic
234C22×C7#44911yescharacteristic
235C22×C7#44911yescharacteristic
236C22×C7#44911yescharacteristic
237C22×C7#45311yescharacteristic
238C22×C7#45411yescharacteristic
239C2×D7—82no—
240C2×D7—82no—
241C2×D7—82no—
242C2×D7—161no—
243C2×D7—161no—
244C2×D7—161no—
245C2×D7—161no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—81no—
249C4×C7—81no—
250C4×C7#47811yescharacteristic
251C4×C7#47811yescharacteristic
252C4×C7#47811yescharacteristic
253C4×C7#47811yescharacteristic
254C4×C7#48811yescharacteristic
255C4×C7#48811yescharacteristic
256C4×C7#49011yescharacteristic
257C4×C7#49011yescharacteristic
258Dic7—82no—
259Dic7—81no—
260Dic7—81no—
261C24—281no—
262C2×D4—142no—
263C2×D4—142no—
264C2×D4—142no—
265C2×D4—282no—
266C2×D4—282no—
267C2×D4—281no—
268C2×D4—281no—
269C2×D4—281no—
270C2×D4—281no—
271C2×D4—281no—
272C2×D4—281no—
273C2×D4—281no—
274C2×D4—281no—
275C2×D4—281no—
276C2×D4—281no—
277C2×D4—281no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—141no—
281C22×C4—141no—
282C22×C4—142no—
283C22×C4C2×D2811yescharacteristic
284C22×C4D4×D711yescharacteristic
285C22×C4D4×D711yescharacteristic
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—281no—
293C2×C8—281no—
294C2×C8—281no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—141no—
300C4⋊C4—141no—
301C4⋊C4—141no—
302C4⋊C4—141no—
303C4⋊C4—282no—
304C4⋊C4—281no—
305C4⋊C4—281no—
306C4⋊C4—281no—
307C4⋊C4—281no—
308C4⋊C4—562no—
309C42—41no—
310C42—41no—
311C42—281no—
312C42—281no—
313C42#115711yescharacteristic
314C42#115711yescharacteristic
315C42#115911yescharacteristic
316C42Q8×D711yescharacteristic
317C2×C7#162011yescharacteristic
318C2×C7#162411yescharacteristic
319C2×C7#162811yescharacteristic
320C2×C7#162811yescharacteristic
321C2×C7#202511yescharacteristic
322C2×C7#202511yescharacteristic
323C2×C7#202711yescharacteristic
324D7—162no—
325C23—282no—
326C23—561no—
327C23—561no—
328C23—561no—
329C23—561no—
330C23—561no—
331C23—561no—
332C23#644511yescharacteristic
333D4—284no—
334D4—282no—
335D4—282no—
336D4—282no—
337D4—282no—
338D4—562no—
339D4—561no—
340D4—561no—
341D4—561no—
342D4—561no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—281no—
352C2×C4—281no—
353C2×C4—282no—
354C2×C4—281no—
355C2×C4—281no—
356C2×C4—282no—
357C2×C4—282no—
358C2×C4—281no—
359C2×C4—281no—
360C2×C4#643811yescharacteristic
361C2×C4#644811yescharacteristic
362C2×C4#644811yescharacteristic
363C2×C4#645911yescharacteristic
364C2×C4#645911yescharacteristic
365C2×C4#646011yescharacteristic
366C2×C4#646711yescharacteristic
367C2×C4#646711yescharacteristic
368C2×C4#647111yescharacteristic
369C2×C4#647111yescharacteristic
370C2×C4#647411yescharacteristic
371C2×C4#647711yescharacteristic
372C2×C4#647911yescharacteristic
373C2×C4#648211yescharacteristic
374C8—42no—
375C8—81no—
376C8—561no—
377C7#1035111yescharacteristic, complemented, Sylow
378C22—562no—
379C22—562no—
380C22—562no—
381C22—1121no—
382C22—1121no—
383C22—1121no—
384C22—1121no—
385C22#8944511yescharacteristic
386C22#8950611yescharacteristic
387C22#8950611yescharacteristic
388C22#8955311yescharacteristic
389C22#8958311yescharacteristic
390C22#8959411yescharacteristic
391C22#8959911yescharacteristic
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—561no—
397C4—561no—
398C4—562no—
399C4#8946911yescharacteristic
400C4#8948511yescharacteristic
401C4#8953611yescharacteristic
402C4#8957211yescharacteristic
403C4#8961911yescharacteristic
404C4#8961911yescharacteristic
405C4#8964311yescharacteristic
406C4#8966311yescharacteristic
407C2—1122no—
408C2#1172595311yescharacteristic
409C2#1172595511yescharacteristic
410C2#1172599911yescharacteristic
411C2#1172601211yescharacteristic
412C2#1172643311yescharacteristic
413C2#1172651711yescharacteristic
414C2#1172665311yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1279C2
socleC23×C7#120
prime radicalC2×#6437C22
hypercenter#1279D7
hypocenterC7#10351
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #11731170C2semidirect
5. C7×#1279C2semidirect
6. #11725508C2semidirect
7. #11725461C2semidirect
69. #1279D7semidirect
377. C7#10351semidirect

Maximal subgroups

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

subgroupindex
#117311702
#117282112
#117275262
#117272412
C7×#12792
#117255082
#117254612
#103517