Finite Group Databasebeta

← #13372965

Subgroups — #13372965

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1132C211yesmaximal, characteristic
2#11731083C211yesmaximal, characteristic
3C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
4C7×#1770C211yesmaximal, characteristic, complemented
5#11728239C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic
7#11727572C212yesmaximal
8#11727567C212yesmaximal, complemented
9#11727516C211yesmaximal, characteristic, complemented
10#11726440C212yesmaximal
11#11726376C212yesmaximal, complemented
12C22×#1132C2211yescharacteristic
13C2×#6455—21no—
14C2×C4×D28C2211yescharacteristic
15C2×C4×Dic14C2212yes—
16C2×C4×Dic14C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#97C2212yes—
19C2×C4×C7×C8C2211yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C2×#6395C2212yes—
22C2×#6395C2211yescharacteristic
23C2×#6391C2211yescharacteristic
24C2×#6390C2212yes—
25C2×#6390C2211yescharacteristic
26#89759—21no—
27#89748—21no—
28C4⋊C4×Dic7—21no—
29#89711C2212yes—
30#89707C2211yescharacteristic
31#89707C2211yescharacteristic
32C4×C4⋊Dic7C2212yes—
33#89700C2212yes—
34C4×#1132C418yescomplemented
35#89442—22no—
36#89419—22no—
37C7×#336C2212yes—
38#89346C2212yes—
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#22759—71nomaximal, Sylow
43C22×D28C2311yescharacteristic
44C22×C4×D7—21no—
45C22×Dic14C2311yescharacteristic
46C22×C7×C8C2312yes—
47C22×C7×C8C2311yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C42×C7C2312yes—
52C2×C42×C7C2311yescharacteristic
53C2×(C2×D7)⋊C4—21no—
54C2×(C2×D7)⋊C4—22no—
55C2×(C2×D7)⋊C4—22no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7C2312yes—
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×Dic7⋊C4—21no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—22no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7—22no—
69C2×C4×Dic7—22no—
70C2×#1132—22no—
71C2×#1132—22no—
72C2×#1132C2×C414yes—
73#6455—44no—
74C4×D28—22no—
75C4×D28C2×C414yes—
76C4×Dic14—22no—
77C4×Dic14—24no—
78C4×Dic14—42no—
79C4×Dic14C2×C414yes—
80C7×C4⋊C8—22no—
81C7×#97—42no—
82C4×C7×C8C2×C414yes—
83C7×#95D412yes—
84C7⋊#95—22no—
85C7⋊#95—22no—
86#6395—42no—
87#6395C2×C414yes—
88#6391C2×C414yes—
89#6390—24no—
90#6390C2×C414yes—
91C2×C4×SD16—71no—
92#2354—71no—
93Q8×C4⋊C4—71no—
94#1936—72no—
95#1919—72no—
96#1880—72no—
97#1834—71no—
98#1803—71no—
99#1802—71no—
100#1770D711yescharacteristic, complemented
101#1763—72no—
102C23×D7—21no—
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2411yescharacteristic
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7—21no—
111C22×Dic7—22no—
112C22×Dic7—22no—
113C2×D28—21no—
114C2×D28—21no—
115C2×D28—21no—
116C2×D28—21no—
117C2×D28—21no—
118C2×D28C22×C412yes—
119C2×C4×D7—24no—
120C2×C4×D7—42no—
121C2×C4×D7—42no—
122C2×Dic14—21no—
123C2×Dic14—21no—
124C2×Dic14—21no—
125C2×Dic14C22×C412yes—
126C2×Dic14#4112yes—
127C2×Dic14#4112yes—
128C2×C7×C8—21no—
129C2×C7×C8—21no—
130C2×C7×C8—21no—
131C2×C7×C8—21no—
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8—22no—
135C2×C7×C8C22×C412yes—
136C2×C7×C8C22×C412yes—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C42×C7—22no—
140C42×C7—22no—
141C42×C7C22×C412yes—
142C42×C7#4112yes—
143(C2×D7)⋊C4—24no—
144(C2×D7)⋊C4—44no—
145(C2×D7)⋊C4—82no—
146C4⋊Dic7—21no—
147C4⋊Dic7—21no—
148C4⋊Dic7—22no—
149C4⋊Dic7—22no—
150C4⋊Dic7—22no—
151C4⋊Dic7—22no—
152C4⋊Dic7—42no—
153C4⋊Dic7—42no—
154C4⋊Dic7—81no—
155C4⋊Dic7—81no—
156C4⋊Dic7C22×C412yes—
157C4⋊Dic7C22×C412yes—
158Dic7⋊C4—24no—
159Dic7⋊C4—44no—
160Dic7⋊C4—82no—
161C4×Dic7—24no—
162C4×Dic7—44no—
163C4×Dic7—44no—
164#1132—28no—
165C22×SD16—71no—
166C2×C4×Q8—71no—
167C2×C4×Q8—72no—
168C2×C4×D4—71no—
169C2×#117—141no—
170C4×SD16—78no—
171C2×#106—71no—
172C2×C4⋊C8C2×D711yescharacteristic
173C2×#103—71no—
174C2×#103—72no—
175C2×#102—71no—
176C2×#102—72no—
177C2×#97C2×D712yes—
178C2×C4×C8C2×D711yescharacteristic
179#390—72no—
180#389—72no—
181#389—142no—
182#388—142no—
183#384—71no—
184#384—71no—
185#384—141no—
186#380—141no—
187C4×C4⋊C4—72no—
188C4×C4⋊C4—141no—
189C4×C4⋊C4C2×D711yescharacteristic
190#340—72no—
191#340—72no—
192#340—72no—
193#340—72no—
194#336C2×D712yes—
195C23×C7C22×D411yescharacteristic
196C22×D7—22no—
197C22×D7—41no—
198C22×D7—41no—
199C22×D7—41no—
200C22×D7—41no—
201C22×D7—41no—
202C22×D7—41no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7C4×D412yes—
212C2×C4×C7C4×D412yes—
213C2×C4×C7AGL(1,ℤ/8)12yes—
214C2×C4×C7#13712yes—
215C2×C4×C7C23×C411yescharacteristic
216C2×C4×C7C2×#4111yescharacteristic
217C2×C4×C7C2×#4111yescharacteristic
218C2×C4×C7C2×#4111yescharacteristic
219C2×C4×C7H(5,𝔽2)11yescharacteristic
220C2×C4×C72−1+411yescharacteristic
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—22no—
226C2×Dic7—41no—
227C2×Dic7—41no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—42no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236D28—22no—
237D28—22no—
238D28—41no—
239D28—41no—
240C4×D7—48no—
241Dic14—22no—
242Dic14—22no—
243Dic14—22no—
244Dic14#11714yescomplemented
245C7×C8—22no—
246C7×C8—22no—
247C7×C8—42no—
248C22×Q8—71no—
249C22×D4—71no—
250C23×C4—141no—
251C2×SD16—74no—
252C2×SD16—142no—
253C2×SD16—142no—
254C22×C8C22×D712yes—
255C22×C8C22×D711yescharacteristic
256C22×C8C22×D711yescharacteristic
257C4×Q8—74no—
258C4×Q8—144no—
259C4×Q8—142no—
260C4×Q8—282no—
261C4×D4—74no—
262C4×D4—142no—
263#117—284no—
264C2×C4⋊C4—71no—
265C2×C4⋊C4—71no—
266C2×C4⋊C4—72no—
267C2×C4⋊C4—72no—
268C2×C4⋊C4—141no—
269C2×C4⋊C4—141no—
270C2×C4⋊C4—142no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—142no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4D2811yescharacteristic
276C2×C4⋊C4D2811yescharacteristic
277C2×C22⋊C4—141no—
278C2×C22⋊C4—142no—
279C2×C22⋊C4—142no—
280C2×C42—141no—
281C2×C42—142no—
282C2×C42—142no—
283C2×C42C22×D712yes—
284C2×C42C22×D711yescharacteristic
285#106—74no—
286C4⋊C8—22no—
287#103—74no—
288#103—144no—
289#102—74no—
290#102—282no—
291#97—42no—
292C4×C8C4×D714yes—
293#95—142no—
294#95—142no—
295#95D2812yes—
296C22×C7C2×C4×D411yescharacteristic
297C22×C7#54611yescharacteristic
298C22×C7#54811yescharacteristic
299C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C7C2×#13711yescharacteristic
301C22×C7#57711yescharacteristic
302C22×C7#57711yescharacteristic
303C2×D7—42no—
304C2×D7—42no—
305C2×D7—42no—
306C2×D7—81no—
307C2×D7—81no—
308C2×D7—81no—
309C2×D7—81no—
310C4×C7—22no—
311C4×C7—42no—
312C4×C7—42no—
313C4×C7#44014yes—
314C4×C7C2×#11711yescharacteristic
315C4×C7#52011yescharacteristic
316C4×C7#55211yescharacteristic
317C4×C7#55311yescharacteristic
318Dic7—24no—
319Dic7—42no—
320Dic7—42no—
321Dic7—82no—
322Dic7—82no—
323C24—141no—
324C2×Q8—72no—
325C2×Q8—72no—
326C2×Q8—72no—
327C2×Q8—141no—
328C2×Q8—141no—
329C2×Q8—141no—
330C2×D4—72no—
331C2×D4—141no—
332C2×D4—141no—
333C2×D4—141no—
334C2×D4—141no—
335C2×D4—141no—
336C22×C4—141no—
337C22×C4—141no—
338C22×C4—141no—
339C22×C4—144no—
340C22×C4—142no—
341C22×C4—142no—
342C22×C4—282no—
343C22×C4—282no—
344C22×C4C2×D2812yes—
345C22×C4C2×D2812yes—
346C22×C4C2×D2811yescharacteristic
347C22×C4C2×D2811yescharacteristic
348C22×C4C23×D711yescharacteristic
349SD16—148no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8C2×C4×D712yes—
358C2×C8C2×C4×D712yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—72no—
362C4⋊C4—72no—
363C4⋊C4—144no—
364C4⋊C4—141no—
365C4⋊C4—141no—
366C4⋊C4—142no—
367C4⋊C4—142no—
368C4⋊C4—142no—
369C4⋊C4—142no—
370C4⋊C4—284no—
371C4⋊C4—282no—
372C4⋊C4—282no—
373C4⋊C4—562no—
374C4⋊C4—561no—
375C4⋊C4—561no—
376C22⋊C4—144no—
377C22⋊C4—284no—
378C22⋊C4—562no—
379C42—22no—
380C42—22no—
381C42—144no—
382C42—284no—
383C42—284no—
384C42C2×C4×D712yes—
385C42#115712yes—
386C2×C7#235211yescharacteristic
387C2×C7C2×#44011yescharacteristic
388C2×C7#295211yescharacteristic
389C2×C7#332911yescharacteristic
390C2×C7#333211yescharacteristic
391C2×C7#336811yescharacteristic
392C2×C7#336911yescharacteristic
393D7—82no—
394C23—142no—
395C23—281no—
396C23—281no—
397C23—281no—
398C23—281no—
399C23—281no—
400C23—281no—
401C23C22×D2811yescharacteristic
402Q8—74no—
403Q8—142no—
404Q8—142no—
405Q8—142no—
406D4—142no—
407D4—142no—
408D4—281no—
409D4—281no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—142no—
419C2×C4—142no—
420C2×C4—142no—
421C2×C4—142no—
422C2×C4—142no—
423C2×C4—281no—
424C2×C4—281no—
425C2×C4—281no—
426C2×C4—281no—
427C2×C4—288no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4C4×D2812yes—
435C2×C4C4×D2812yes—
436C2×C4#647112yes—
437C2×C4#647212yes—
438C2×C4C22×C4×D711yescharacteristic
439C2×C4C2×#115711yescharacteristic
440C2×C4#654811yescharacteristic
441C2×C4#655111yescharacteristic
442C2×C4D7×#4111yescharacteristic
443C2×C4D7×#4111yescharacteristic
444C8—22no—
445C8—22no—
446C8—42no—
447C7#2275911yescharacteristic, complemented, Sylow
448C22—282no—
449C22—282no—
450C22—282no—
451C22—561no—
452C22—561no—
453C22—561no—
454C22—561no—
455C22C2×C4×D2811yescharacteristic
456C22#9024511yescharacteristic
457C22#9026711yescharacteristic
458C22C2×#647111yescharacteristic
459C22C2×#647211yescharacteristic
460C22#9044211yescharacteristic
461C22#9044211yescharacteristic
462C4—22no—
463C4—42no—
464C4—42no—
465C4—144no—
466C4—282no—
467C4—282no—
468C4—562no—
469C4—562no—
470C4#8948214yes—
471C4#9019511yescharacteristic
472C4D7×#11711yescharacteristic
473C4#9024911yescharacteristic
474C4#9027211yescharacteristic
475C2—562no—
476C2#1173121611yescharacteristic
477C2C2×#8948211yescharacteristic
478C2#1173601911yescharacteristic
479C2#1173700911yescharacteristic
480C2#1173701011yescharacteristic
481C2#1173742511yescharacteristic
482C2#1173742611yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1770C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1770D7
hypocenterC7#22759
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×Dic14C2semidirect
4. C7×#1770C2semidirect
5. #11728239C2semidirect
8. #11727567C2semidirect
9. #11727516C2semidirect
11. #11726376C2semidirect
34. C4×#1132C4semidirect
100. #1770D7semidirect
244. Dic14#117semidirect
447. C7#22759semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11322
#117310832
C4⋊C4×Dic142
C7×#17702
#117282392
#117282332
#117275722
#117275672
#117275162
#117264402
#117263762
#227597