Finite Group Databasebeta

← #13337254

Subgroups — #13337254

Order 1792.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2966C211yesmaximal, characteristic
2#11740600C212yesmaximal
3#11735920C211yesmaximal, characteristic, complemented
4#11728718C211yesmaximal, characteristic, complemented
5#11728716C211yesmaximal, characteristic, complemented
6#11728700C212yesmaximal, complemented
7#11728627C211yesmaximal, characteristic, complemented
8#11728625C211yesmaximal, characteristic, complemented
9#11728608C212yesmaximal, complemented
10#11728574C212yesmaximal, complemented
11C2×#89252C211yesmaximal, characteristic, complemented
12C7×#529C2211yescharacteristic
13#90161C2212yes—
14C7×#446C2211yescharacteristic
15C7×C8×Q8C2211yescharacteristic
16C7×#436C2211yescharacteristic
17C7×#435—21no—
18C7×C8×D4—21no—
19C7×C8×D4C2211yescharacteristic
20C7×#433—21no—
21C7×#432C2211yescharacteristic
22C7×#405—21no—
23C2×C4×C7×C8C2211yescharacteristic
24#89888—21no—
25#89888—21no—
26#89835—22no—
27#89826—22no—
28#89823—22no—
29#89813—22no—
30#89803C2212yes—
31#89792C2212yes—
32#89791C2212yes—
33#89779C2212yes—
34C2×C4⋊(C7⋊C8)C2212yes—
35#89471C2211yescharacteristic
36C4×Dic28—21no—
37C4×Dic28—21no—
38#89454C2211yescharacteristic
39#89453C2212yes—
40#89281C2212yes—
41#89281C2212yes—
42#89277C2212yes—
43#89277C2212yes—
44#89252C2212yes—
45#89252C2211yescharacteristic
46#89252C2211yescharacteristic
47#19228—71nomaximal, Sylow
48C2×C7×M16—21no—
49C22×C7×C8—21no—
50C22×C7×C8D411yescharacteristic
51C22×C7×C8D411yescharacteristic
52C7×#126—21no—
53C7×#125D411yescharacteristic
54C7×#124D411yescharacteristic
55C7×#123—21no—
56C7×C22⋊Q8D411yescharacteristic
57C7×#121D411yescharacteristic
58C4×C7×Q8C2311yescharacteristic
59C4×C7×D4—21no—
60C4×C7×D4C2311yescharacteristic
61C7×#117—21no—
62C2×C42×C7C2311yescharacteristic
63C2×#1144—22no—
64C2×#1142—22no—
65#6489—22no—
66#6487—22no—
67#6487—22no—
68C22×C7⋊C8—22no—
69C2×Dic28—21no—
70C2×Dic28—21no—
71#6433—22no—
72#6432C2312yes—
73#6432C2312yes—
74C4×Dic14—22no—
75C4×Dic14—22no—
76C7×C4⋊C8—21no—
77C7×C4⋊C8—21no—
78C7×C4⋊C8C2311yescharacteristic
79C7×C4⋊C8C2311yescharacteristic
80C7×C22⋊C8—21no—
81C7×C22⋊C8—21no—
82C7×C22⋊C8—21no—
83C7×#97—21no—
84C7×#97—21no—
85C4×C7×C8—21no—
86C4×C7×C8D411yescharacteristic
87C4×C7×C8D411yescharacteristic
88C4×C7×C8C2311yescharacteristic
89C4×C7×C8C2311yescharacteristic
90#6409—22no—
91#6406—22no—
92#6392—21no—
93#6392—21no—
94#6390—22no—
95#6390—22no—
96#6384—22no—
97#6384—22no—
98#6382—22no—
99#6381—22no—
100C4⋊(C7⋊C8)—22no—
101C4⋊(C7⋊C8)C2312yes—
102C4⋊(C7⋊C8)C2312yes—
103#3248—72no—
104#3170—71no—
105#2966D711yescharacteristic
106#1694—71no—
107#1693—71no—
108#1690—71no—
109#1689—71no—
110#1680—72no—
111#1679—72no—
112C2×#334—71no—
113#1552—72no—
114C2×C7×Q8C2×D411yescharacteristic
115C2×C7×D4—21no—
116C2×C7×D4C2×D411yescharacteristic
117C22×C4×C7—21no—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7#4111yescharacteristic
121C2×Dic14—22no—
122C2×Dic14—22no—
123C7×M16—42no—
124C2×C7×C8—22no—
125C2×C7×C8—21no—
126C2×C7×C8—21no—
127C2×C7×C8—41no—
128C2×C7×C8—42no—
129C2×C7×C8C2×D411yescharacteristic
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8C2×D411yescharacteristic
132C2×C7×C8C2×D411yescharacteristic
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4C2×D411yescharacteristic
138C7×C4⋊C4C2×D411yescharacteristic
139C7×C22⋊C4—21no—
140C7×C22⋊C4—21no—
141C7×C22⋊C4—21no—
142C7×C22⋊C4—21no—
143C7×C22⋊C4—21no—
144C42×C7—21no—
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7#4111yescharacteristic
148C42×C7C2411yescharacteristic
149C22⋊Dic7—42no—
150#1144—82no—
151#1142—82no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156Dic7⋊C4—42no—
157Dic7⋊C4—42no—
158C4×Dic7—42no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—82no—
162Dic28—42no—
163Dic28—42no—
164#533—72no—
165#529C2×D711yescharacteristic
166#500—71no—
167#494—71no—
168#487—72no—
169#479—142no—
170#475—72no—
171#474—72no—
172#467—141no—
173#467—141no—
174#467—142no—
175#465—142no—
176#464—142no—
177#462—72no—
178#461—72no—
179#446C2×D711yescharacteristic
180C8×Q8C2×D711yescharacteristic
181C4×Q16—141no—
182C4×Q16—141no—
183#436C2×D711yescharacteristic
184#435—21no—
185C8×D4—21no—
186C8×D4C2×D711yescharacteristic
187#433—21no—
188#432C2×D711yescharacteristic
189C2×C4⋊C8—72no—
190#405—21no—
191C2×C4×C8C2×D711yescharacteristic
192#334—72no—
193#334—71no—
194#334—71no—
195#332—72no—
196#332—72no—
197#332—72no—
198#332—72no—
199C23×C7—21no—
200C23×C7#12111yescharacteristic
201C7×Q8—22no—
202C7×D4—22no—
203C7×D4—42no—
204C2×C4×C7—22no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—22no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7#12111yescharacteristic
214C2×C4×C7#12111yescharacteristic
215C2×C4×C7#12111yescharacteristic
216C2×C4×C7C22×D411yescharacteristic
217C2×C4×C7C22×D411yescharacteristic
218C2×C4×C7C2×#4111yescharacteristic
219C2×Dic7—42no—
220C2×Dic7—42no—
221Dic14—42no—
222Dic14—42no—
223Dic14—42no—
224Dic14—42no—
225C7×C8—21no—
226C7×C8—21no—
227C7×C8—21no—
228C7×C8—41no—
229C7×C8—41no—
230C7×C8#13512yes—
231C7⋊C8—42no—
232C7⋊C8—42no—
233C2×Q16—142no—
234C2×Q16—141no—
235C2×Q16—141no—
236C2×SD16—142no—
237C2×M16—21no—
238C22×C8—21no—
239C22×C8—142no—
240C22×C8C22⋊D711yescharacteristic
241C22×C8C22⋊D711yescharacteristic
242C4⋊Q8—72no—
243C4⋊Q8—72no—
244#126—21no—
245#125—142no—
246#125D2811yescharacteristic
247#124D2811yescharacteristic
248#123—21no—
249C22⋊Q8—142no—
250C22⋊Q8—142no—
251C22⋊Q8D2811yescharacteristic
252#121D2811yescharacteristic
253C4×Q8—142no—
254C4×Q8—142no—
255C4×Q8C22×D711yescharacteristic
256C4×D4—21no—
257C4×D4C22×D711yescharacteristic
258#117—21no—
259#117—142no—
260C2×C42C22×D711yescharacteristic
261#107—141no—
262#107—141no—
263#107—142no—
264#106—142no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—72no—
268C4⋊C8—72no—
269C4⋊C8—142no—
270C4⋊C8C22×D711yescharacteristic
271C4⋊C8C22×D711yescharacteristic
272#103—142no—
273#103—142no—
274#103—142no—
275#103—142no—
276#103—142no—
277#102—142no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281#97—21no—
282#97—21no—
283C4×C8—21no—
284C4×C8C22⋊D711yescharacteristic
285C4×C8C22⋊D711yescharacteristic
286C4×C8C22×D711yescharacteristic
287C4×C8C22×D711yescharacteristic
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—41no—
291C22×C7—41no—
292C22×C7—41no—
293C22×C7#47312yes—
294C22×C7C2×#12111yescharacteristic
295C4×C7—21no—
296C4×C7—21no—
297C4×C7—21no—
298C4×C7—21no—
299C4×C7—41no—
300C4×C7—41no—
301C4×C7—41no—
302C4×C7C2×#13511yescharacteristic
303C4×C7#57511yescharacteristic
304C4×C7#57712yes—
305Dic7—82no—
306Dic7—82no—
307C2×Q8—142no—
308C2×Q8—142no—
309C2×Q8C2×D2811yescharacteristic
310C2×D4—21no—
311C2×D4C2×D2811yescharacteristic
312C22×C4—21no—
313C22×C4C2×D2811yescharacteristic
314C22×C4#115711yescharacteristic
315C22×C4C2×C22⋊D711yescharacteristic
316Q16—282no—
317Q16—282no—
318Q16—562no—
319SD16—562no—
320M16—42no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—142no—
327C2×C8—142no—
328C2×C8—562no—
329C2×C8C2×C22⋊D711yescharacteristic
330C2×C8C2×C22⋊D711yescharacteristic
331C2×C8C2×C22⋊D711yescharacteristic
332C2×C8C2×C22⋊D711yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—142no—
338C4⋊C4—142no—
339C4⋊C4—142no—
340C4⋊C4—142no—
341C4⋊C4—282no—
342C4⋊C4—282no—
343C4⋊C4C2×D2811yescharacteristic
344C4⋊C4C2×D2811yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—282no—
351C42—21no—
352C42—282no—
353C42C2×D2811yescharacteristic
354C42#115711yescharacteristic
355C42C2×C22⋊D711yescharacteristic
356C42C23×D711yescharacteristic
357C2×C7—22no—
358C2×C7—81no—
359C2×C7#303511yescharacteristic
360C2×C7#306011yescharacteristic
361C2×C7C2×#47311yescharacteristic
362C23—21no—
363C23#649311yescharacteristic
364Q8—22no—
365Q8—282no—
366Q8—282no—
367Q8—282no—
368Q8—282no—
369D4—22no—
370D4—42no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4#649311yescharacteristic
383C2×C4#649311yescharacteristic
384C2×C4#649311yescharacteristic
385C2×C4C22×D2811yescharacteristic
386C2×C4C2×#115711yescharacteristic
387C2×C4C22×C22⋊D711yescharacteristic
388C8—21no—
389C8—21no—
390C8—21no—
391C8—41no—
392C8—41no—
393C8—282no—
394C8—282no—
395C8#651312yes—
396C7#1922811yescharacteristic, complemented, Sylow
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#8991512yes—
403C22C2×#649311yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—562no—
412C4—562no—
413C4#9044212yes—
414C4C2×#651311yescharacteristic
415C4#9051311yescharacteristic
416C2—22no—
417C2—81no—
418C2#1173997011yescharacteristic
419C2#1174124511yescharacteristic
420C2C2×#8991511yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2966C2
socleC22×C7C2×#121
prime radicalC7×#121D4
hypercenter#2966D7
hypocenterC7#19228
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
3. #11735920C2semidirect
4. #11728718C2semidirect
5. #11728716C2semidirect
6. #11728700C2semidirect
7. #11728627C2semidirect
8. #11728625C2semidirect
9. #11728608C2semidirect
10. #11728574C2semidirect
11. C2×#89252C2semidirect
396. C7#19228semidirect

Maximal subgroups

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

subgroupindex
C7×#29662
#117406002
#117359202
#117287182
#117287162
#117287002
#117286272
#117286252
#117286082
#117285742
C2×#892522
#192287