Finite Group Databasebeta

← #13365849

Subgroups — #13365849

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic7×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3#11731062C211yesmaximal, characteristic
4C7×#1987C211yesmaximal, characteristic, complemented
5#11728233C211yesmaximal, characteristic
6Dic7×C4⋊C8C211yesmaximal, characteristic, complemented
7#11727221C211yesmaximal, characteristic, complemented
8#11726612C212yesmaximal, complemented
9#11726607C212yesmaximal
10#11726546C212yesmaximal
11#11726542C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×Q8×Dic7—21no—
14C2×C4×D28C2211yescharacteristic
15C2×#6435—21no—
16C2×C7×C4⋊C8C2211yescharacteristic
17C4×C7×C4⋊C4C2211yescharacteristic
18#90002—21no—
19C2×#6395C2212yes—
20C2×C8×Dic7C2212yes—
21#89856C418yescomplemented
22#89760C2211yescharacteristic
23#89760C2211yescharacteristic
24#89757—21no—
25#89757C2212yes—
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7C2212yes—
28#89745C2211yescharacteristic
29#89745C2211yescharacteristic
30C2×#6383C2212yes—
31C2×#6382C2212yes—
32C2×#6382C2212yes—
33C4×(C2×D7)⋊C4—21no—
34#89707C2211yescharacteristic
35C42×Dic7C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89441—21no—
38#89441—21no—
39C7×#340C2212yes—
40C7×#340C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22111—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C22×C7×C8C2312yes—
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8C2×C414yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2312yes—
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57Q8×Dic7—44no—
58C2×(C2×D7)⋊C4—21no—
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—21no—
61C2×(C2×D7)⋊C4—21no—
62C2×(C2×D7)⋊C4—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—22no—
66C2×C4⋊Dic7C2312yes—
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—21no—
69C2×Dic7⋊C4—21no—
70C2×C4×Dic7—22no—
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7D411yescharacteristic
73C2×C4×Dic7D411yescharacteristic
74C2×C4×Dic7D411yescharacteristic
75C2×C4×Dic7C2311yescharacteristic
76C2×C4×Dic7C2311yescharacteristic
77C22×C7⋊C8C2312yes—
78C4×D28—22no—
79C4×D28C2×C414yes—
80#6435—44no—
81C7×C4⋊C8—22no—
82C7×#95—21no—
83C7⋊#95—21no—
84C7⋊#95—21no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87#6395—42no—
88C8×Dic7—42no—
89#6383C2×C418yes—
90#6382—42no—
91#6382C2×C418yes—
92C4⋊(C7⋊C8)C2×C414yes—
93C2×#476—71no—
94C4×C4⋊Q8—71no—
95#2293—71no—
96#1987D711yescharacteristic, complemented
97#1948—71no—
98#1941—72no—
99#1934—72no—
100#1926—72no—
101#1919—72no—
102#1802—71no—
103C4×C4⋊C8—71no—
104C23×D7—21no—
105C2×C7×Q8—24no—
106C2×C7×Q8—42no—
107C2×C7×Q8—42no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2411yescharacteristic
114C22×Dic7—21no—
115C22×Dic7—22no—
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C22×Dic7C2×D411yescharacteristic
120C2×D28—21no—
121C2×D28—21no—
122C2×D28—21no—
123C2×D28—21no—
124C2×D28—21no—
125C2×D28C22×C412yes—
126C2×C4×D7—24no—
127C2×C4×D7—42no—
128C2×C4×D7—42no—
129C2×C7×C8—22no—
130C2×C7×C8—22no—
131C2×C7×C8—22no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4C22×C414yes—
139C42×C7—42no—
140C42×C7C22×C412yes—
141C42×C7#4112yes—
142(C2×D7)⋊C4—24no—
143(C2×D7)⋊C4—42no—
144(C2×D7)⋊C4—42no—
145(C2×D7)⋊C4—42no—
146(C2×D7)⋊C4—42no—
147C4⋊Dic7—21no—
148C4⋊Dic7—22no—
149C4⋊Dic7—22no—
150C4⋊Dic7—42no—
151C4⋊Dic7—42no—
152C4⋊Dic7—44no—
153C4⋊Dic7C22×C412yes—
154Dic7⋊C4—42no—
155Dic7⋊C4—42no—
156C4×Dic7—22no—
157C4×Dic7—21no—
158C4×Dic7—21no—
159C4×Dic7—21no—
160C4×Dic7—21no—
161C4×Dic7—22no—
162C4×Dic7—41no—
163C4×Dic7—44no—
164C4×Dic7SD1614yes—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8C22×C414yes—
168C2×C4⋊Q8C2×D711yescharacteristic
169C2×C4×Q8—141no—
170C2×C4×D4—71no—
171C2×#117—141no—
172#476—78no—
173C2×#106—72no—
174C2×#106—72no—
175C2×C4⋊C8—71no—
176C2×C4⋊C8C2×D711yescharacteristic
177C2×#102—72no—
178C2×#102—72no—
179C2×C4×C8—72no—
180#391—141no—
181#390—71no—
182#390—71no—
183#389—72no—
184#389—71no—
185#389—71no—
186#389—141no—
187#388—141no—
188#388—141no—
189#384—71no—
190C4×C4⋊C4—72no—
191C4×C4⋊C4—141no—
192C4×C4⋊C4C2×D711yescharacteristic
193C4×C22⋊C4—141no—
194C43—71no—
195#340—72no—
196#340—72no—
197#340C2×D712yes—
198#340C2×D712yes—
199#336—72no—
200C23×C7C22×D411yescharacteristic
201C22×D7—22no—
202C22×D7—41no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C22×D7—41no—
207C22×D7—41no—
208C7×Q8—48no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—24no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7C4×D412yes—
221C2×C4×C7C4×D412yes—
222C2×C4×C7C2×SD1612yes—
223C2×C4×C7#13512yes—
224C2×C4×C7C23×C411yescharacteristic
225C2×C4×C7C2×#4111yescharacteristic
226C2×C4×C7H(5,𝔽2)11yescharacteristic
227C2×C4×C72−1+411yescharacteristic
228C2×C4×C72−1+411yescharacteristic
229C2×C4×C72−1+411yescharacteristic
230C2×Dic7—21no—
231C2×Dic7—21no—
232C2×Dic7—21no—
233C2×Dic7—21no—
234C2×Dic7—21no—
235C2×Dic7—21no—
236C2×Dic7—22no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7C2×SD1612yes—
246C2×Dic7C2×SD1612yes—
247C2×Dic7AGL(1,ℤ/8)12yes—
248D28—22no—
249D28—22no—
250D28—41no—
251D28—41no—
252C4×D7—48no—
253C7×C8—42no—
254C7⋊C8—24no—
255C22×Q8—21no—
256C22×D4—71no—
257C23×C4—141no—
258C22×C8—72no—
259C22×C8C22×D712yes—
260C4⋊Q8—22no—
261C4⋊Q8C4×D714yes—
262C4×Q8—284no—
263C4×D4—74no—
264C4×D4—142no—
265#117—284no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—71no—
268C2×C4⋊C4—72no—
269C2×C4⋊C4—141no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—142no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4C22×D711yescharacteristic
275C2×C4⋊C4C22×D711yescharacteristic
276C2×C4⋊C4C22×D712yes—
277C2×C22⋊C4—141no—
278C2×C22⋊C4—141no—
279C2×C22⋊C4—141no—
280C2×C22⋊C4—141no—
281C2×C22⋊C4—141no—
282C2×C42—21no—
283C2×C42—71no—
284C2×C42—71no—
285C2×C42—71no—
286C2×C42—71no—
287C2×C42—71no—
288C2×C42—71no—
289C2×C42—142no—
290C2×C42C22×D711yescharacteristic
291#106—78no—
292#106—282no—
293C4⋊C8—22no—
294C4⋊C8—74no—
295#102—78no—
296#102—282no—
297C4×C8—282no—
298#95—21no—
299#95—141no—
300#95—141no—
301#95—142no—
302#95—142no—
303C22×C7C2×C4×D411yescharacteristic
304C22×C7#53611yescharacteristic
305C22×C7#53711yescharacteristic
306C22×C7C22×SD1611yescharacteristic
307C22×C7C2×#13511yescharacteristic
308C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C7#57511yescharacteristic
310C2×D7—42no—
311C2×D7—42no—
312C2×D7—42no—
313C2×D7—81no—
314C2×D7—81no—
315C2×D7—81no—
316C2×D7—81no—
317C4×C7—22no—
318C4×C7—44no—
319C4×C7—81no—
320C4×C7—81no—
321C4×C7C4×SD1614yes—
322C4×C7#51911yescharacteristic
323C4×C7#52011yescharacteristic
324C4×C7#54111yescharacteristic
325C4×C7#54211yescharacteristic
326Dic7—22no—
327Dic7—22no—
328Dic7—22no—
329Dic7—42no—
330Dic7—41no—
331Dic7—82no—
332C24—141no—
333C2×Q8—24no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×D4—72no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—141no—
340C2×D4—141no—
341C2×D4—141no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—71no—
346C22×C4—71no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—141no—
350C22×C4—144no—
351C22×C4—142no—
352C22×C4—282no—
353C22×C4—282no—
354C22×C4D4×D711yescharacteristic
355C22×C4D4×D711yescharacteristic
356C22×C4C23×D711yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—74no—
361C2×C8—142no—
362C2×C8—142no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—72no—
370C4⋊C4—141no—
371C4⋊C4—142no—
372C4⋊C4—142no—
373C4⋊C4—282no—
374C4⋊C4—282no—
375C4⋊C4—282no—
376C4⋊C4—284no—
377C4⋊C4—282no—
378C4⋊C4C2×C4×D714yes—
379C22⋊C4—144no—
380C22⋊C4—282no—
381C22⋊C4—282no—
382C22⋊C4—282no—
383C22⋊C4—282no—
384C42—42no—
385C42—74no—
386C42—142no—
387C42—142no—
388C42—141no—
389C42—141no—
390C42—141no—
391C42—141no—
392C42—281no—
393C42—284no—
394C42C2×C4×D712yes—
395C42#115912yes—
396C2×C7#232311yescharacteristic
397C2×C7C2×C4×SD1611yescharacteristic
398C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
399C2×C7#322711yescharacteristic
400C2×C7#323311yescharacteristic
401C2×C7#324611yescharacteristic
402C2×C7#324711yescharacteristic
403D7—82no—
404C23—142no—
405C23—281no—
406C23—281no—
407C23—281no—
408C23—281no—
409C23—281no—
410C23—281no—
411C23C2×D4×D711yescharacteristic
412Q8—48no—
413D4—142no—
414D4—142no—
415D4—281no—
416D4—281no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—72no—
429C2×C4—72no—
430C2×C4—72no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—142no—
438C2×C4—141no—
439C2×C4—141no—
440C2×C4—141no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—288no—
444C2×C4—282no—
445C2×C4—282no—
446C2×C4—282no—
447C2×C4(C22⋊D7)⋊C412yes—
448C2×C4(C22⋊D7)⋊C412yes—
449C2×C4D7×SD1612yes—
450C2×C4#648212yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4C2×#115911yescharacteristic
453C2×C4#655111yescharacteristic
454C2×C4#655111yescharacteristic
455C2×C4#655311yescharacteristic
456C2×C4#655411yescharacteristic
457C8—42no—
458C8—144no—
459C7#2211111yescharacteristic, complemented, Sylow
460C22—282no—
461C22—282no—
462C22—282no—
463C22—561no—
464C22—561no—
465C22—561no—
466C22—561no—
467C22C2×(C22⋊D7)⋊C411yescharacteristic
468C22#9031911yescharacteristic
469C22#9032011yescharacteristic
470C22C2×D7×SD1611yescharacteristic
471C22C2×#648211yescharacteristic
472C22D7×AGL(1,ℤ/8)11yescharacteristic
473C22#9046411yescharacteristic
474C4—22no—
475C4—44no—
476C4—81no—
477C4—81no—
478C4—142no—
479C4—142no—
480C4—142no—
481C4—282no—
482C4—281no—
483C4—562no—
484C4#8956014yes—
485C4#9026311yescharacteristic
486C4#9026511yescharacteristic
487C4#9037311yescharacteristic
488C4#9042211yescharacteristic
489C2—562no—
490C2#1173405611yescharacteristic
491C2C2×#8956011yescharacteristic
492C2#1173635911yescharacteristic
493C2#1173850911yescharacteristic
494C2#1173851511yescharacteristic
495C2#1173855811yescharacteristic
496C2#1173857611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1987C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1987D7
hypocenterC7#22111
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. Dic7×C4⋊Q8C2semidirect
4. C7×#1987C2semidirect
6. Dic7×C4⋊C8C2semidirect
7. #11727221C2semidirect
8. #11726612C2semidirect
11. #11726542C2semidirect
21. #89856C4semidirect
96. #1987D7semidirect
459. C7#22111semidirect

Maximal subgroups

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

subgroupindex
Dic7×C4⋊Q82
C2×#898562
#117310622
C7×#19872
#117282332
Dic7×C4⋊C82
#117272212
#117266122
#117266072
#117265462
#117265422
#221117