Finite Group Databasebeta

← #13382774

Subgroups — #13382774

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89652C212yesmaximal
2#11733408C211yesmaximal, characteristic
3#11731763C211yesmaximal, characteristic
4C7×#2083C211yesmaximal, characteristic, complemented
5#11728374C212yesmaximal, complemented
6#11727362C211yesmaximal, characteristic, complemented
7#11727324C211yesmaximal, characteristic, complemented
8#11726620C212yesmaximal
9#11726591C212yesmaximal, complemented
10D7×#336C211yesmaximal, characteristic
11#11726393C211yesmaximal, characteristic, complemented
12C22×C8×D7C2212yes—
13C2×#6455C2212yes—
14C2×#6455C2211yescharacteristic
15C2×#6455C2211yescharacteristic
16C2×C42×D7C2211yescharacteristic
17C2×C7×#107C2212yes—
18C7×#389C2211yescharacteristic
19C7×#384C2211yescharacteristic
20C2×#6392C2212yes—
21#89761—21no—
22#89759—21no—
23#89759—21no—
24#89757C2211yescharacteristic
25#89749C2212yes—
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7—21no—
29#89745C2211yescharacteristic
30C2×#6382C2212yes—
31C2×#6382C2212yes—
32#89707C2211yescharacteristic
33#89700C2211yescharacteristic
34#89652—28no—
35#89440—21no—
36#89421—21no—
37C7×#340C2212yes—
38C7×#336C2211yescharacteristic
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2211yescharacteristic
44#23656—71nomaximal, Sylow
45C22×C4×D7D411yescharacteristic
46C22×C4×D7D411yescharacteristic
47C22×C4×D7C2311yescharacteristic
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C42×C7C2311yescharacteristic
56C2×(C2×D7)⋊C4—21no—
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—22no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2312yes—
64C2×C4⋊Dic7C2311yescharacteristic
65C2×Dic7⋊C4—21no—
66C2×Dic7⋊C4—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7—21no—
69C2×C4×Dic7—22no—
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74C2×C8×D7—24no—
75C2×C8×D7—24no—
76C2×C8×D7Q818yes—
77#6455—22no—
78#6455—24no—
79#6455—24no—
80#6455—24no—
81#6455D414yes—
82C42×D7—24no—
83C7×#107—24no—
84C7×#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90#6392—24no—
91#6382—24no—
92#6382—24no—
93C2×#427—72no—
94#2296—71no—
95#2285—71no—
96#2085—72no—
97#2085—72no—
98#2084—71no—
99#2084—71no—
100#2083—71no—
101#2083D711yescharacteristic, complemented
102#1735—72no—
103C2×#336—71no—
104C23×D7C2×D411yescharacteristic
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7—22no—
114C22×Dic7C2×D411yescharacteristic
115C22×Dic7C2×D411yescharacteristic
116C22×Dic7C2×D411yescharacteristic
117C2×C4×D7—22no—
118C2×C4×D7—22no—
119C2×C4×D7—22no—
120C2×C4×D7—22no—
121C2×C4×D7—41no—
122C2×C4×D7—41no—
123C2×C4×D7C2×D412yes—
124C2×C4×D7C2×Q814yes—
125C2×C4×D7#4114yes—
126C2×C4×D7#4112yes—
127C2×C7×C8—22no—
128C2×C7×C8—22no—
129C2×C7×C8C2×Q814yes—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—42no—
135C7×C4⋊C4—42no—
136C7×C4⋊C4—42no—
137C7×C4⋊C4C2×D412yes—
138C42×C7—22no—
139(C2×D7)⋊C4—24no—
140(C2×D7)⋊C4—44no—
141(C2×D7)⋊C4—42no—
142C4⋊Dic7—21no—
143C4⋊Dic7—22no—
144C4⋊Dic7—22no—
145C4⋊Dic7—22no—
146C4⋊Dic7—42no—
147C4⋊Dic7—42no—
148C4⋊Dic7—42no—
149C4⋊Dic7C2×D412yes—
150Dic7⋊C4—42no—
151Dic7⋊C4—42no—
152C4×Dic7—22no—
153C4×Dic7—24no—
154C4×Dic7—22no—
155C4×Dic7—22no—
156C4×Dic7—44no—
157C4×Dic7—42no—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8C2×Q814yes—
161C8×D7—216no—
162C23×C8—72no—
163C2×#117—71no—
164C2×#117—72no—
165C2×#117—71no—
166C22×C42—71no—
167#427—148no—
168C2×#107—72no—
169C2×#107C2×D712yes—
170C2×#106—72no—
171C2×#106—72no—
172#389—71no—
173#389—71no—
174#389—71no—
175#389C2×D711yescharacteristic
176#385—141no—
177#385—141no—
178#384—71no—
179#384C2×D711yescharacteristic
180#382—72no—
181#380—141no—
182#380—141no—
183C4×C4⋊C4—141no—
184C4×C4⋊C4—141no—
185C4×C4⋊C4—141no—
186C4×C4⋊C4—141no—
187#340—72no—
188#340—72no—
189#340—72no—
190#340C2×D712yes—
191#336—72no—
192#336—71no—
193#336C2×D711yescharacteristic
194C23×C7C22×D411yescharacteristic
195C22×D7—21no—
196C22×D7—21no—
197C22×D7—21no—
198C22×D7—21no—
199C22×D7#12112yes—
200C22×D7#12312yes—
201C22×D7C4⋊Q812yes—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—22no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—42no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7#13512yes—
216C2×C4×C7#13512yes—
217C2×C4×C7C22×D411yescharacteristic
218C2×C4×C7C22×Q812yes—
219C2×C4×C7C2×#4112yes—
220C2×C4×C7C2×#4111yescharacteristic
221C2×Dic7—22no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—21no—
225C2×Dic7—21no—
226C2×Dic7—21no—
227C2×Dic7—21no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—42no—
231C2×Dic7—41no—
232C2×Dic7—41no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—42no—
236C2×Dic7#12112yes—
237C2×Dic7#12312yes—
238C2×Dic7C4⋊Q812yes—
239C4×D7—44no—
240C4×D7—44no—
241C4×D7C22⋊Q818yes—
242C4×D7#12518yes—
243C7×C8—24no—
244C7⋊C8—24no—
245C23×C4—71no—
246C23×C4—71no—
247C23×C4—71no—
248C22×C8—72no—
249C22×C8—78no—
250C22×C8—144no—
251C22×C8—144no—
252C22×C8C22×D712yes—
253#117—74no—
254#117—142no—
255#117—144no—
256#117—144no—
257#117—144no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—71no—
262C2×C4⋊C4—71no—
263C2×C4⋊C4—72no—
264C2×C4⋊C4—141no—
265C2×C4⋊C4—141no—
266C2×C4⋊C4—141no—
267C2×C4⋊C4—141no—
268C2×C4⋊C4—141no—
269C2×C4⋊C4C22×D711yescharacteristic
270C2×C4⋊C4C22×D712yes—
271C2×C4⋊C4C22×D711yescharacteristic
272C2×C22⋊C4—141no—
273C2×C22⋊C4—142no—
274C2×C22⋊C4—141no—
275C2×C42—71no—
276C2×C42—71no—
277C2×C42—71no—
278C2×C42—144no—
279C2×C42—141no—
280C2×C42—141no—
281C2×C42—142no—
282C2×C42C22×D711yescharacteristic
283#107—24no—
284#107—144no—
285#106—144no—
286#106—144no—
287#95—21no—
288#95—142no—
289#95—142no—
290#95—141no—
291#95—141no—
292#95—141no—
293C22×C7C2×#12111yescharacteristic
294C22×C7C2×#12311yescharacteristic
295C22×C7C2×C4⋊Q811yescharacteristic
296C22×C7C2×#13511yescharacteristic
297C22×C7C2×#13511yescharacteristic
298C22×C7#57511yescharacteristic
299C22×C7#57511yescharacteristic
300C2×D7—22no—
301C2×D7—22no—
302C2×D7—22no—
303C2×D7—22no—
304C2×D7—22no—
305C2×D7—22no—
306C2×D7#39814yes—
307C4×C7—22no—
308C4×C7—22no—
309C4×C7—42no—
310C4×C7—81no—
311C4×C7—82no—
312C4×C7C2×C22⋊Q812yes—
313C4×C7C2×#12512yes—
314Dic7—41no—
315Dic7—41no—
316Dic7—42no—
317Dic7—81no—
318Dic7—82no—
319Dic7#39814yes—
320C24—71no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—71no—
325C22×C4—72no—
326C22×C4—74no—
327C22×C4—74no—
328C22×C4—72no—
329C22×C4—71no—
330C22×C4—71no—
331C22×C4—142no—
332C22×C4—142no—
333C22×C4—141no—
334C22×C4—142no—
335C22×C4—142no—
336C22×C4—141no—
337C22×C4—142no—
338C22×C4—281no—
339C22×C4—281no—
340C22×C4D4×D711yescharacteristic
341C22×C4D4×D711yescharacteristic
342C22×C4C23×D711yescharacteristic
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—74no—
346C2×C8—1416no—
347C2×C8—142no—
348C2×C8—142no—
349C2×C8Q8×D714yes—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—72no—
358C4⋊C4—141no—
359C4⋊C4—142no—
360C4⋊C4—142no—
361C4⋊C4—142no—
362C4⋊C4—282no—
363C4⋊C4—282no—
364C4⋊C4—282no—
365C4⋊C4—282no—
366C4⋊C4—282no—
367C4⋊C4D4×D712yes—
368C22⋊C4—144no—
369C22⋊C4—284no—
370C22⋊C4—282no—
371C42—22no—
372C42—142no—
373C42—142no—
374C42—144no—
375C42—142no—
376C42—284no—
377C42—282no—
378C2×C7C2×#39811yescharacteristic
379C2×C7#303511yescharacteristic
380C2×C7#303511yescharacteristic
381C2×C7#309111yescharacteristic
382C2×C7#309111yescharacteristic
383C2×C7#316011yescharacteristic
384C2×C7#316211yescharacteristic
385D7—24no—
386C23—72no—
387C23—72no—
388C23—72no—
389C23—141no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23C2×D4×D711yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—72no—
408C2×C4—72no—
409C2×C4—72no—
410C2×C4—78no—
411C2×C4—78no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—142no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—284no—
420C2×C4—284no—
421C2×C4—281no—
422C2×C4—281no—
423C2×C4—281no—
424C2×C4—282no—
425C2×C4—281no—
426C2×C4—282no—
427C2×C4—282no—
428C2×C4—281no—
429C2×C4#647512yes—
430C2×C4#648212yes—
431C2×C4C2×D4×D711yescharacteristic
432C2×C4C2×Q8×D712yes—
433C2×C4D7×#4111yescharacteristic
434C2×C4D7×#4112yes—
435C8—24no—
436C8—144no—
437C7#2365611yescharacteristic, complemented, Sylow
438C22—74no—
439C22—142no—
440C22—142no—
441C22—142no—
442C22—142no—
443C22—142no—
444C22—142no—
445C22D7×#12111yescharacteristic
446C22D7×#12311yescharacteristic
447C22D7×C4⋊Q811yescharacteristic
448C22C2×#647511yescharacteristic
449C22C2×#648211yescharacteristic
450C22#9046411yescharacteristic
451C22#9046811yescharacteristic
452C4—22no—
453C4—22no—
454C4—42no—
455C4—74no—
456C4—81no—
457C4—82no—
458C4—281no—
459C4—282no—
460C4—281no—
461C4—561no—
462C4—562no—
463C4D7×C22⋊Q812yes—
464C4D7×#12512yes—
465C2—144no—
466C2D7×#39811yescharacteristic
467C2#1173777611yescharacteristic
468C2#1173792011yescharacteristic
469C2#1173872611yescharacteristic
470C2#1173891211yescharacteristic
471C2#1173938311yescharacteristic
472C2#1173941811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2083C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#2083D7
hypocenterC7#23656
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
4. C7×#2083C2semidirect
5. #11728374C2semidirect
6. #11727362C2semidirect
7. #11727324C2semidirect
9. #11726591C2semidirect
11. #11726393C2semidirect
101. #2083D7semidirect
437. C7#23656semidirect

Maximal subgroups

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

subgroupindex
C2×#896522
#117334082
#117317632
C7×#20832
#117283742
#117273622
#117273242
#117266202
#117265912
D7×#3362
#117263932
#236567