Finite Group Databasebeta

← #13453629

Subgroups — #13453629

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6510C211yesmaximal, characteristic
2#11731800C211yesmaximal, characteristic, complemented
3C7×#1803C211yesmaximal, characteristic, complemented
4#11729380C211yesmaximal, characteristic, complemented
5#11728233C211yesmaximal, characteristic
6Dic7×C4⋊C8C211yesmaximal, characteristic, complemented
7#11727986C212yesmaximal, complemented
8#11727980C212yesmaximal
9#11727338C211yesmaximal, characteristic
10#11726653C212yesmaximal
11#11726602C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6510C2211yescharacteristic
14C2×#6453C2212yes—
15C2×C4×D28C2211yescharacteristic
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×#103C2212yes—
18C7×#384C2211yescharacteristic
19#90002C2211yescharacteristic
20C2×#6409C2212yes—
21C2×#6395C2212yes—
22C2×C8×Dic7C2212yes—
23#89762—21no—
24#89762—21no—
25#89760C2211yescharacteristic
26#89757—21no—
27#89757—21no—
28#89749C2211yescharacteristic
29#89749C2212yes—
30#89745—21no—
31#89745—21no—
32C2×#6383C2212yes—
33C4×(C2×D7)⋊C4—21no—
34C4×(C2×D7)⋊C4—21no—
35#89707C2211yescharacteristic
36C42×Dic7C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C7×#340C2212yes—
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#29140—71nomaximal, Sylow
44C22×C7×Q8C2311yescharacteristic
45C22×D28C2311yescharacteristic
46C22×C4×D7—21no—
47C22×C7×C8C2312yes—
48C4×C7×Q8—22no—
49C4×C7×Q8—22no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2312yes—
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57#6510—22no—
58#6510Q814yes—
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—21no—
61C2×(C2×D7)⋊C4—21no—
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7C2311yescharacteristic
68C2×C4⋊Dic7C2312yes—
69C2×Dic7⋊C4—22no—
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7D411yescharacteristic
73C2×C4×Dic7D411yescharacteristic
74C2×C4×Dic7C2311yescharacteristic
75C2×C4×Dic7C2311yescharacteristic
76C22×C7⋊C8C2312yes—
77#6453—42no—
78#6453—42no—
79C4×D28—22no—
80C4×D28—22no—
81C7×C4⋊C8—22no—
82C7×#103—42no—
83#6409—42no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90#6395—42no—
91C8×Dic7—24no—
92#6383—42no—
93C4⋊(C7⋊C8)—22no—
94#2859—71no—
95C4×#124—71no—
96#2026—72no—
97#2025—72no—
98#1958—72no—
99#1835—72no—
100#1803—71no—
101#1803D711yescharacteristic, complemented
102#1802—71no—
103#1802—71no—
104C4×C4⋊C8—71no—
105C23×D7—21no—
106C2×C7×Q8—21no—
107C2×C7×Q8—21no—
108C2×C7×Q8—21no—
109C2×C7×Q8—21no—
110C2×C7×Q8—21no—
111C2×C7×Q8C2×Q812yes—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2411yescharacteristic
118C22×Dic7—21no—
119C22×Dic7—22no—
120C22×Dic7C2×D411yescharacteristic
121C22×Dic7C2×D411yescharacteristic
122C22×Dic7C2×D411yescharacteristic
123C22×Dic7C2×D411yescharacteristic
124C2×D28—21no—
125C2×D28—21no—
126C2×D28—21no—
127C2×D28—21no—
128C2×D28—21no—
129C2×D28C2×Q812yes—
130C2×C4×D7—42no—
131C2×C4×D7—42no—
132C2×C4×D7—42no—
133C2×C7×C8—22no—
134C2×C7×C8—22no—
135C2×C7×C8—22no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—42no—
141C7×C4⋊C4—42no—
142C7×C4⋊C4—42no—
143C42×C7—42no—
144C42×C7#4112yes—
145C42×C7#4112yes—
146(C2×D7)⋊C4—24no—
147(C2×D7)⋊C4—24no—
148(C2×D7)⋊C4—42no—
149(C2×D7)⋊C4—42no—
150(C2×D7)⋊C4—42no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—44no—
156C4⋊Dic7—42no—
157C4⋊Dic7—42no—
158Dic7⋊C4—44no—
159C4×Dic7—22no—
160C4×Dic7—22no—
161C4×Dic7—21no—
162C4×Dic7—21no—
163C4×Dic7—41no—
164C4×Dic7—41no—
165C4×Dic7C2×Q812yes—
166C4×Dic7#4112yes—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×#125—72no—
171C2×#124—71no—
172C2×C4×Q8C2×D711yescharacteristic
173C2×C4×D4—71no—
174C2×C4⋊C8—71no—
175C2×C4⋊C8C2×D711yescharacteristic
176C2×#103—72no—
177C2×#103C2×D712yes—
178C2×#102—72no—
179C2×#102—72no—
180C2×C4×C8—72no—
181#391—71no—
182#390—71no—
183#389—141no—
184#389—141no—
185#389—141no—
186#389—141no—
187#388—141no—
188#388—141no—
189#384—71no—
190#384C2×D711yescharacteristic
191#382—71no—
192#382—72no—
193C4×C22⋊C4—141no—
194C4×C22⋊C4—141no—
195C43—71no—
196#340—72no—
197#340—72no—
198#340—72no—
199#340C2×D712yes—
200#336—72no—
201C23×C7C22×D411yescharacteristic
202C22×D7—22no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C22×D7—41no—
207C22×D7—41no—
208C22×D7—41no—
209C7×Q8—22no—
210C7×Q8—22no—
211C7×Q8—41no—
212C7×Q8—41no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—22no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—42no—
224C2×C4×C7#12412yes—
225C2×C4×C7#12412yes—
226C2×C4×C7#13512yes—
227C2×C4×C7#13512yes—
228C2×C4×C7C22×Q811yescharacteristic
229C2×C4×C7C2×#4111yescharacteristic
230C2×C4×C7C2×#4111yescharacteristic
231C2×C4×C7C2×#4111yescharacteristic
232C2×C4×C7H(5,𝔽2)11yescharacteristic
233C2×C4×C7H(5,𝔽2)11yescharacteristic
234C2×Dic7—21no—
235C2×Dic7—21no—
236C2×Dic7—21no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—21no—
241C2×Dic7—21no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—42no—
250C2×Dic7C22⋊Q812yes—
251C2×Dic7C22⋊Q812yes—
252D28—22no—
253D28—22no—
254D28—41no—
255D28—41no—
256C4×D7—84no—
257C7×C8—42no—
258C7⋊C8—42no—
259C22×Q8C22×D711yescharacteristic
260C22×D4—71no—
261C23×C4—141no—
262C22×C8—72no—
263C22×C8C22×D712yes—
264#125—282no—
265#125—282no—
266#124—74no—
267#124—142no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×D4—142no—
271C4×D4—142no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—71no—
276C2×C4⋊C4—72no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—142no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4C22×D711yescharacteristic
282C2×C4⋊C4C22×D712yes—
283C2×C22⋊C4—141no—
284C2×C22⋊C4—141no—
285C2×C22⋊C4—141no—
286C2×C22⋊C4—141no—
287C2×C22⋊C4—141no—
288C2×C42—21no—
289C2×C42—71no—
290C2×C42—71no—
291C2×C42—71no—
292C2×C42—71no—
293C2×C42—71no—
294C2×C42—71no—
295C2×C42C22×D711yescharacteristic
296C4⋊C8—22no—
297C4⋊C8—142no—
298#103—42no—
299#103—282no—
300#102—282no—
301#102—282no—
302C4×C8—144no—
303#95—141no—
304#95—141no—
305#95—141no—
306#95—141no—
307#95—142no—
308#95—142no—
309C22×C7C2×C22⋊Q811yescharacteristic
310C22×C7C2×#12411yescharacteristic
311C22×C7#53511yescharacteristic
312C22×C7C2×#13511yescharacteristic
313C22×C7C2×#13511yescharacteristic
314C22×C7#57511yescharacteristic
315C22×C7#57511yescharacteristic
316C2×D7—42no—
317C2×D7—42no—
318C2×D7—42no—
319C2×D7—81no—
320C2×D7—81no—
321C2×D7—81no—
322C2×D7—81no—
323C4×C7—22no—
324C4×C7—42no—
325C4×C7—81no—
326C4×C7—82no—
327C4×C7#48814yes—
328C4×C7#55411yescharacteristic
329C4×C7#55411yescharacteristic
330C4×C7#55511yescharacteristic
331C4×C7#55511yescharacteristic
332Dic7—22no—
333Dic7—22no—
334Dic7—41no—
335Dic7—41no—
336Dic7—81no—
337Dic7—82no—
338C24—141no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8Q8×D712yes—
345C2×D4—72no—
346C2×D4—141no—
347C2×D4—141no—
348C2×D4—141no—
349C2×D4—141no—
350C2×D4—141no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—71no—
355C22×C4—71no—
356C22×C4—71no—
357C22×C4—71no—
358C22×C4—141no—
359C22×C4—142no—
360C22×C4—282no—
361C22×C4—282no—
362C22×C4—282no—
363C22×C4D4×D711yescharacteristic
364C22×C4D4×D711yescharacteristic
365C22×C4C23×D711yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—142no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4—142no—
382C4⋊C4—142no—
383C4⋊C4—282no—
384C4⋊C4—284no—
385C4⋊C4—284no—
386C4⋊C4—282no—
387C22⋊C4—144no—
388C22⋊C4—144no—
389C22⋊C4—282no—
390C22⋊C4—282no—
391C22⋊C4—282no—
392C42—42no—
393C42—72no—
394C42—72no—
395C42—142no—
396C42—142no—
397C42—141no—
398C42—141no—
399C42—281no—
400C42—281no—
401C42#115712yes—
402C42#115912yes—
403C2×C7#256311yescharacteristic
404C2×C7#307611yescharacteristic
405C2×C7#307611yescharacteristic
406C2×C7C2×#48811yescharacteristic
407C2×C7#314211yescharacteristic
408C2×C7#321211yescharacteristic
409C2×C7#321211yescharacteristic
410D7—82no—
411C23—142no—
412C23—281no—
413C23—281no—
414C23—281no—
415C23—281no—
416C23—281no—
417C23—281no—
418C23C2×D4×D711yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423D4—142no—
424D4—142no—
425D4—281no—
426D4—281no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—72no—
439C2×C4—72no—
440C2×C4—141no—
441C2×C4—141no—
442C2×C4—141no—
443C2×C4—141no—
444C2×C4—141no—
445C2×C4—141no—
446C2×C4—141no—
447C2×C4—141no—
448C2×C4—141no—
449C2×C4—141no—
450C2×C4—281no—
451C2×C4—281no—
452C2×C4—282no—
453C2×C4—281no—
454C2×C4—282no—
455C2×C4—282no—
456C2×C4—564no—
457C2×C4#644812yes—
458C2×C4#644812yes—
459C2×C4#647912yes—
460C2×C4#648212yes—
461C2×C4C2×#115711yescharacteristic
462C2×C4C2×#115911yescharacteristic
463C2×C4#654811yescharacteristic
464C2×C4C2×Q8×D711yescharacteristic
465C2×C4D7×#4111yescharacteristic
466C2×C4#655311yescharacteristic
467C8—42no—
468C8—282no—
469C7#2914011yescharacteristic, complemented, Sylow
470C22—282no—
471C22—282no—
472C22—282no—
473C22—561no—
474C22—561no—
475C22—561no—
476C22—561no—
477C22C2×#644811yescharacteristic
478C22#9029811yescharacteristic
479C22D7×C22⋊Q811yescharacteristic
480C22C2×#647911yescharacteristic
481C22C2×#648211yescharacteristic
482C22#9046411yescharacteristic
483C22#9046811yescharacteristic
484C4—22no—
485C4—42no—
486C4—81no—
487C4—82no—
488C4—142no—
489C4—142no—
490C4—281no—
491C4—281no—
492C4—562no—
493C4—561no—
494C4#8957214yes—
495C4#9020211yescharacteristic
496C4#9022311yescharacteristic
497C4#9030711yescharacteristic
498C4#9032611yescharacteristic
499C2—562no—
500C2#1173138711yescharacteristic
501C2C2×#8957211yescharacteristic
502C2#1173632411yescharacteristic
503C2#1173783311yescharacteristic
504C2#1173792611yescharacteristic
505C2#1173845511yescharacteristic
506C2#1173855211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1803C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1803D7
hypocenterC7#29140
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #11731800C2semidirect
3. C7×#1803C2semidirect
4. #11729380C2semidirect
6. Dic7×C4⋊C8C2semidirect
7. #11727986C2semidirect
11. #11726602C2semidirect
101. #1803D7semidirect
469. C7#29140semidirect

Maximal subgroups

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

subgroupindex
C4×#65102
#117318002
C7×#18032
#117293802
#117282332
Dic7×C4⋊C82
#117279862
#117279802
#117273382
#117266532
#117266022
#291407