Finite Group Databasebeta

← #13256841

Subgroups — #13256841

Order 1792.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731162C211yesmaximal, characteristic, complemented
2#11728211C211yesmaximal, characteristic
3#11727526C211yesmaximal, characteristic
4#11727241C211yesmaximal, characteristic
5C7×#1279C211yesmaximal, characteristic, complemented
6#11725508C211yesmaximal, characteristic, complemented
7#11725461C211yesmaximal, characteristic, complemented
8C2×#6453—21no—
9C2×#6437C2211yescharacteristic
10C2×C7×C4⋊C8C2211yescharacteristic
11C2×C4×C7×C8C2211yescharacteristic
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6395—21no—
14C2×#6395—21no—
15C2×#6395—22no—
16#89757—21no—
17C2×#6383—21no—
18C2×#6383—21no—
19#89707—21no—
20#89707C2211yescharacteristic
21C4×C4⋊Dic7C2211yescharacteristic
22#89700—21no—
23C2×C4⋊(C7⋊C8)C2211yescharacteristic
24#89433—22no—
25#10354—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—22no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—22no—
39C2×C4⋊Dic7D411yescharacteristic
40C2×C4⋊Dic7D411yescharacteristic
41C2×C4⋊Dic7D412yes—
42C2×Dic7⋊C4—22no—
43C2×C4×Dic7—21no—
44C22×C7⋊C8—21no—
45#6453—42no—
46#6453—42no—
47#6437—21no—
48#6437—21no—
49#6437—21no—
50#6437Q812yes—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C4×C7×C8—21no—
54C4×C7×C8—21no—
55C7×#95—21no—
56C7⋊#95—22no—
57C7⋊#95—21no—
58#6395—42no—
59#6395—42no—
60#6395—44no—
61#6383—42no—
62#6383—42no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65#2552—71no—
66#1982—71no—
67#1982—71no—
68#1957—71no—
69#1296—71no—
70#1279—71no—
71#1279D711yescharacteristic, complemented
72C23×D7—41no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×Dic7—21no—
79C22×Dic7—21no—
80C22×Dic7—22no—
81C2×D28—22no—
82C2×D28—22no—
83C2×D28—22no—
84C2×D28—42no—
85C2×D28—42no—
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×D28—41no—
96C2×D28—41no—
97C2×C7×C8—22no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4D812yes—
106C7×C4⋊C4SD1612yes—
107C42×C7—41no—
108C42×C7—41no—
109C42×C7C2×Q811yescharacteristic
110C42×C7#4111yescharacteristic
111C42×C7#4111yescharacteristic
112C42×C7#4111yescharacteristic
113C4⋊Dic7—21no—
114C4⋊Dic7—21no—
115C4⋊Dic7—21no—
116C4⋊Dic7—21no—
117C4⋊Dic7—44no—
118C4⋊Dic7—41no—
119C4⋊Dic7—41no—
120C4⋊Dic7—41no—
121C4⋊Dic7—41no—
122C4⋊Dic7—42no—
123C4⋊Dic7—82no—
124Dic7⋊C4—44no—
125C4×Dic7—41no—
126C4×Dic7—41no—
127C2×C7⋊C8—41no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8—41no—
130C2×#127—71no—
131C2×#125—141no—
132C2×C4⋊C8—71no—
133C2×C4⋊C8C2×D711yescharacteristic
134C2×#102—142no—
135C2×#102—141no—
136C2×#102—141no—
137C2×#102—141no—
138C2×#102—141no—
139C2×C4×C8C2×D711yescharacteristic
140#398—142no—
141#389—141no—
142#389—141no—
143#384—71no—
144#384—141no—
145C4×C4⋊C4—71no—
146C4×C4⋊C4C2×D711yescharacteristic
147C23×C7#12011yescharacteristic
148C22×D7—42no—
149C22×D7—81no—
150C22×D7—81no—
151C22×D7—81no—
152C22×D7—81no—
153C22×D7—81no—
154C22×D7—81no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7C22⋊Q811yescharacteristic
164C2×C4×C7C22⋊Q811yescharacteristic
165C2×C4×C7C22⋊Q811yescharacteristic
166C2×C4×C7#12411yescharacteristic
167C2×C4×C7#12411yescharacteristic
168C2×C4×C7#12411yescharacteristic
169C2×C4×C7C2×D811yescharacteristic
170C2×C4×C7C2×SD1611yescharacteristic
171C2×C4×C7#13511yescharacteristic
172C2×C4×C7#13511yescharacteristic
173C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C7#13711yescharacteristic
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—42no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—42no—
183C2×Dic7—42no—
184C2×Dic7—41no—
185C2×Dic7—41no—
186D28—44no—
187D28—42no—
188D28—42no—
189D28—42no—
190D28—42no—
191D28—82no—
192D28—81no—
193D28—81no—
194D28—81no—
195D28—81no—
196C7×C8—42no—
197C7×C8—81no—
198C7⋊C8—81no—
199C22×D4—141no—
200C22×D4—141no—
201C22×D4—141no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—141no—
205#127—72no—
206#127—141no—
207#127—141no—
208#127—141no—
209#125—282no—
210#125—282no—
211C2×C4⋊C4—71no—
212C2×C4⋊C4—71no—
213C2×C4⋊C4—72no—
214C2×C4⋊C4—142no—
215C2×C4⋊C4—142no—
216C2×C4⋊C4—141no—
217C2×C4⋊C4—141no—
218C2×C4⋊C4—142no—
219C2×C4⋊C4D2811yescharacteristic
220C2×C4⋊C4D2811yescharacteristic
221C2×C42—21no—
222C2×C42—141no—
223C2×C42C22×D711yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—141no—
227C4⋊C8—141no—
228#102—282no—
229#102—282no—
230#102—284no—
231#102—282no—
232#102—282no—
233C4×C8—21no—
234C4×C8—21no—
235#95—21no—
236#95—142no—
237#95—141no—
238C22×C7#39311yescharacteristic
239C22×C7#44711yescharacteristic
240C22×C7#44811yescharacteristic
241C22×C7#44911yescharacteristic
242C22×C7#44911yescharacteristic
243C22×C7#45311yescharacteristic
244C22×C7#45411yescharacteristic
245C2×D7—82no—
246C2×D7—82no—
247C2×D7—82no—
248C2×D7—161no—
249C2×D7—161no—
250C2×D7—161no—
251C2×D7—161no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—42no—
255C4×C7—42no—
256C4×C7#47411yescharacteristic
257C4×C7#47611yescharacteristic
258C4×C7#47811yescharacteristic
259C4×C7#47811yescharacteristic
260C4×C7#48611yescharacteristic
261C4×C7#48811yescharacteristic
262C4×C7#48911yescharacteristic
263C4×C7#49011yescharacteristic
264Dic7—81no—
265Dic7—82no—
266Dic7—81no—
267C24—281no—
268C2×D4—142no—
269C2×D4—142no—
270C2×D4—142no—
271C2×D4—282no—
272C2×D4—282no—
273C2×D4—281no—
274C2×D4—281no—
275C2×D4—281no—
276C2×D4—281no—
277C2×D4—281no—
278C2×D4—281no—
279C2×D4—281no—
280C2×D4—281no—
281C2×D4—281no—
282C2×D4—281no—
283C2×D4—281no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4—142no—
289C22×C4C2×D2811yescharacteristic
290C22×C4D4×D711yescharacteristic
291C22×C4D4×D711yescharacteristic
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—281no—
299C2×C8—281no—
300C2×C8—281no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—141no—
304C4⋊C4—141no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C4⋊C4—282no—
308C4⋊C4—284no—
309C4⋊C4—284no—
310C4⋊C4—281no—
311C4⋊C4—281no—
312C4⋊C4—281no—
313C4⋊C4—281no—
314C4⋊C4—562no—
315C4⋊C4#113212yes—
316C4⋊C4D5612yes—
317C42—41no—
318C42—41no—
319C42—281no—
320C42—281no—
321C42#115711yescharacteristic
322C42#115711yescharacteristic
323C42#115911yescharacteristic
324C42Q8×D711yescharacteristic
325C2×C7#162111yescharacteristic
326C2×C7#162211yescharacteristic
327C2×C7#162511yescharacteristic
328C2×C7#162611yescharacteristic
329C2×C7#202311yescharacteristic
330C2×C7#202511yescharacteristic
331C2×C7#202711yescharacteristic
332D7—162no—
333C23—282no—
334C23—561no—
335C23—561no—
336C23—561no—
337C23—561no—
338C23—561no—
339C23—561no—
340C23#644511yescharacteristic
341D4—284no—
342D4—282no—
343D4—282no—
344D4—282no—
345D4—282no—
346D4—562no—
347D4—561no—
348D4—561no—
349D4—561no—
350D4—561no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—281no—
360C2×C4—281no—
361C2×C4—282no—
362C2×C4—281no—
363C2×C4—281no—
364C2×C4—282no—
365C2×C4—282no—
366C2×C4—281no—
367C2×C4—281no—
368C2×C4#643811yescharacteristic
369C2×C4#644811yescharacteristic
370C2×C4#644811yescharacteristic
371C2×C4#645911yescharacteristic
372C2×C4#645911yescharacteristic
373C2×C4#646011yescharacteristic
374C2×C4C2×#113211yescharacteristic
375C2×C4C2×D5611yescharacteristic
376C2×C4#647111yescharacteristic
377C2×C4#647211yescharacteristic
378C2×C4#647411yescharacteristic
379C2×C4#647711yescharacteristic
380C2×C4#647911yescharacteristic
381C2×C4#648211yescharacteristic
382C8—42no—
383C8—81no—
384C8—561no—
385C7#1035411yescharacteristic, complemented, Sylow
386C22—562no—
387C22—562no—
388C22—562no—
389C22—1121no—
390C22—1121no—
391C22—1121no—
392C22—1121no—
393C22#8944511yescharacteristic
394C22#8950411yescharacteristic
395C22#8951011yescharacteristic
396C22#8955311yescharacteristic
397C22#8958311yescharacteristic
398C22#8959411yescharacteristic
399C22#8959911yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—561no—
405C4—561no—
406C4—562no—
407C4#8946611yescharacteristic
408C4#8948611yescharacteristic
409C4#8953611yescharacteristic
410C4#8957211yescharacteristic
411C4#8961411yescharacteristic
412C4#8961811yescharacteristic
413C4#8964311yescharacteristic
414C4#8966311yescharacteristic
415C2—1122no—
416C2#1172595211yescharacteristic
417C2#1172595411yescharacteristic
418C2#1172600611yescharacteristic
419C2#1172601311yescharacteristic
420C2#1172643211yescharacteristic
421C2#1172651711yescharacteristic
422C2#1172665311yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1279C2
socleC23×C7#120
prime radicalC2×#6437C22
hypercenter#1279D7
hypocenterC7#10354
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. #11731162C2semidirect
5. C7×#1279C2semidirect
6. #11725508C2semidirect
7. #11725461C2semidirect
71. #1279D7semidirect
385. C7#10354semidirect

Maximal subgroups

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

subgroupindex
#117311622
#117282112
#117275262
#117272412
C7×#12792
#117255082
#117254612
#103547