Finite Group Databasebeta

← #13244346

Subgroups — #13244346

Order 1792.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89502C211yesmaximal, characteristic
2#11727702C211yesmaximal, characteristic, complemented
3#11727673C211yesmaximal, characteristic
4C4×#6380C211yesmaximal, characteristic
5C22⋊C4×C7⋊C8C211yesmaximal, characteristic, complemented
6C7×#1282C211yesmaximal, characteristic, complemented
7#11725483C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D7C2211yescharacteristic
9C23×C7⋊C8C2211yescharacteristic
10C22×#1131C411yescharacteristic, complemented
11C2×C7×C22⋊C8C2211yescharacteristic
12C2×C7×C22⋊C8C2211yescharacteristic
13C4×C7×C22⋊C4C2211yescharacteristic
14C4×C22⋊Dic7C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2211yescharacteristic
16C2×(C2×D7)⋊C8C411yescharacteristic, complemented
17C2×#6389—21no—
18C2×#6389C411yescharacteristic, complemented
19C2×Dic7⋊C8C411yescharacteristic, complemented
20C2×#6380C412yes—
21C2×C4×C7⋊C8—21no—
22#89502—21no—
23#89502—21no—
24#89502—21no—
25#89502—21no—
26#89345—21no—
27#89322C412yes—
28#89319—21no—
29#89264C412yescomplemented
30#89263C412yescomplemented
31#8161—71nomaximal, Sylow
32C23×C4×C7C2311yescharacteristic
33C22×C22⋊D7C2×C411yescharacteristic
34C22×C4×D7C2×C411yescharacteristic
35C22×C7×C8—21no—
36C22×C7×C8C2×C411yescharacteristic
37C22×C7×C8C2×C411yescharacteristic
38C2×C7×C22⋊C4C2×C412yes—
39C2×C42×C7—21no—
40C2×C22⋊Dic7C2×C412yes—
41C2×C22⋊Dic7C2×C411yescharacteristic
42C4×C22⋊D7—21no—
43C4×C22⋊D7—21no—
44C4×C22⋊D7—21no—
45C4×C22⋊D7—21no—
46C2×(C2×D7)⋊C4C2×C411yescharacteristic
47C2×Dic7⋊C4C2×C411yescharacteristic
48C2×C4×Dic7—21no—
49C2×C4×Dic7C2×C411yescharacteristic
50C22×C7⋊C8—21no—
51C22×C7⋊C8—42no—
52C22×C7⋊C8C2×C411yescharacteristic
53C22×C7⋊C8C2×C411yescharacteristic
54C22×C7⋊C8D411yescharacteristic
55C22×C7⋊C8D411yescharacteristic
56C22×C7⋊C8D411yescharacteristic
57C22×C7⋊C8D411yescharacteristic
58C2×#1131—21no—
59C2×#1131—21no—
60C2×#1131—42no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×C22⋊C8—21no—
64C7×C22⋊C8C2×C412yes—
65C7×#95—21no—
66C7⋊#95—21no—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)C2×C412yes—
69(C2×D7)⋊C8—21no—
70(C2×D7)⋊C8—21no—
71#6389—21no—
72#6389—21no—
73#6389—42no—
74Dic7⋊C8—21no—
75Dic7⋊C8—21no—
76#6380—24no—
77C4×C7⋊C8—42no—
78C2×#435—71no—
79C4×#99—71no—
80#1752—71no—
81C8×C22⋊C4—71no—
82#1459—71no—
83#1282—71no—
84#1282D711yescharacteristic, complemented
85C24×C7C22×C411yescharacteristic
86C23×D7C4211yescharacteristic
87C22×C4×C7—22no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C22×C411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C22×C4×C7#4111yescharacteristic
96C2×C22⋊D7—21no—
97C2×C22⋊D7—21no—
98C2×C22⋊D7—22no—
99C2×C22⋊D7—22no—
100C22×Dic7—22no—
101C22×Dic7C4211yescharacteristic
102C22×Dic7C4211yescharacteristic
103C22×Dic7C4211yescharacteristic
104C2×C4×D7—21no—
105C2×C4×D7—21no—
106C2×C4×D7—41no—
107C2×C4×D7—41no—
108C2×C7×C8—21no—
109C2×C7×C8—21no—
110C2×C7×C8—22no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C7×C22⋊C4—22no—
116C7×C22⋊C4M1614yescomplemented
117C42×C7—41no—
118C42×C7—41no—
119C22⋊Dic7—21no—
120C22⋊Dic7—21no—
121C22⋊Dic7—22no—
122C22⋊Dic7—22no—
123(C2×D7)⋊C4—21no—
124(C2×D7)⋊C4—21no—
125Dic7⋊C4—21no—
126Dic7⋊C4—21no—
127C4×Dic7—21no—
128C4×Dic7—21no—
129C4×Dic7—41no—
130C4×Dic7—41no—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—22no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8—41no—
136C2×C7⋊C8—41no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C2×C7⋊C8C22⋊C411yescharacteristic
141C2×C7⋊C8C22⋊C411yescharacteristic
142C2×C7⋊C8C22⋊C411yescharacteristic
143C2×C7⋊C8C22⋊C411yescharacteristic
144#1131—42no—
145#1131—42no—
146C22×M16—71no—
147C23×C8—71no—
148C2×C4×D4—71no—
149#435—141no—
150#435—141no—
151#435—141no—
152#435—141no—
153C2×C4⋊C8—71no—
154C2×#99—72no—
155C2×C22⋊C8—71no—
156C2×C22⋊C8—71no—
157C2×C22⋊C8C2×D711yescharacteristic
158C2×C22⋊C8C2×D711yescharacteristic
159C2×#97—71no—
160C2×#97—141no—
161C2×C4×C8—141no—
162C4×C22⋊C4—71no—
163C4×C22⋊C4C2×D711yescharacteristic
164#342—72no—
165#336—141no—
166#336—141no—
167#324—72no—
168#323—72no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7C2×C4211yescharacteristic
172C23×C7C4×D411yescharacteristic
173C23×C7C4×D411yescharacteristic
174C23×C7C2×M1612yes—
175C23×C7#13112yes—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—41no—
180C22×D7—41no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—24no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7C2×C22⋊C411yescharacteristic
194C2×C4×C7#11711yescharacteristic
195C2×C4×C7C4×D411yescharacteristic
196C2×C4×C7C4×D411yescharacteristic
197C22⋊D7—42no—
198C22⋊D7—42no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—21no—
204C2×Dic7—41no—
205C2×Dic7—41no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C4×D7—42no—
210C4×D7—41no—
211C4×D7—41no—
212C7×C8—42no—
213C7×C8—41no—
214C7×C8—41no—
215C7⋊C8—21no—
216C7⋊C8—21no—
217C7⋊C8—21no—
218C7⋊C8—42no—
219C7⋊C8#9912yescomplemented
220C22×D4—71no—
221C23×C4—71no—
222C23×C4C22×D711yescharacteristic
223C2×M16—141no—
224C2×M16—141no—
225C2×M16—282no—
226C22×C8—21no—
227C22×C8—71no—
228C22×C8—71no—
229C22×C8—71no—
230C22×C8—71no—
231C22×C8—71no—
232C22×C8—71no—
233C22×C8—141no—
234C22×C8—282no—
235C22×C8C4×D711yescharacteristic
236C22×C8C4×D711yescharacteristic
237C4×D4—141no—
238C4×D4—141no—
239C4×D4—141no—
240C4×D4—141no—
241C2×C4⋊C4—71no—
242C2×C22⋊C4—71no—
243C2×C22⋊C4—71no—
244C2×C22⋊C4—72no—
245C2×C22⋊C4C4×D712yes—
246C2×C42—21no—
247C2×C42—71no—
248C2×C42—141no—
249C4⋊C8—141no—
250C4⋊C8—141no—
251#99—144no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—72no—
256C22⋊C8—141no—
257C22⋊C8—141no—
258C22⋊C8—141no—
259C22⋊C8C4×D712yes—
260#97—141no—
261#97—141no—
262#97—282no—
263C4×C8—282no—
264#95—21no—
265#95—141no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—42no—
273C22×C7—42no—
274C22×C7C4×C22⋊C411yescharacteristic
275C22×C7C4×M1611yescharacteristic
276C22×C7#40511yescharacteristic
277C22×C7#43512yes—
278C22×C7#43612yes—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7—81no—
283C2×D7—81no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—44no—
288C4×C7C2×#9911yescharacteristic
289C4×C7#41011yescharacteristic
290Dic7—41no—
291Dic7—41no—
292Dic7—41no—
293Dic7—82no—
294C24—71no—
295C24C2×C4×D711yescharacteristic
296C2×D4—141no—
297C2×D4—141no—
298C2×D4—142no—
299C2×D4—142no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—71no—
304C22×C4—71no—
305C22×C4—71no—
306C22×C4—141no—
307C22×C4—141no—
308C22×C4—142no—
309C22×C4—281no—
310C22×C4—281no—
311C22×C4C2×C4×D711yescharacteristic
312C22×C4C2×C4×D711yescharacteristic
313C22×C4D4×D711yescharacteristic
314C22×C4D4×D711yescharacteristic
315C22×C4#115911yescharacteristic
316C22×C4#116111yescharacteristic
317M16—282no—
318M16—282no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—71no—
327C2×C8—71no—
328C2×C8—71no—
329C2×C8—71no—
330C2×C8—142no—
331C2×C8—142no—
332C2×C8—142no—
333C2×C8—142no—
334C2×C8—281no—
335C2×C8—281no—
336C2×C8—281no—
337C2×C8—281no—
338C2×C8—281no—
339C4⋊C4—141no—
340C4⋊C4—141no—
341C22⋊C4—22no—
342C22⋊C4—141no—
343C22⋊C4—141no—
344C22⋊C4—141no—
345C22⋊C4—141no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4#113114yescomplemented
349C42—41no—
350C42—41no—
351C42—141no—
352C42—141no—
353C42—281no—
354C42—281no—
355C2×C7—22no—
356C2×C7—22no—
357C2×C7—21no—
358C2×C7—21no—
359C2×C7#175311yescharacteristic
360C2×C7C4×#9911yescharacteristic
361C2×C7#175711yescharacteristic
362D7—81no—
363C23—41no—
364C23—41no—
365C23—141no—
366C23—141no—
367C23—141no—
368C23—281no—
369C23—281no—
370C23C42×D711yescharacteristic
371C23(C22⋊D7)⋊C411yescharacteristic
372C23#645611yescharacteristic
373C23C2×#113112yes—
374C23#647012yes—
375D4—282no—
376D4—282no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—141no—
394C2×C4—282no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—281no—
399C2×C4—282no—
400C2×C4—282no—
401C2×C4—282no—
402C2×C4D7×C22⋊C411yescharacteristic
403C2×C4(C22⋊D7)⋊C411yescharacteristic
404C2×C4#645511yescharacteristic
405C2×C4#645611yescharacteristic
406C8—42no—
407C8—41no—
408C8—41no—
409C8—72no—
410C8—141no—
411C8—141no—
412C8—141no—
413C8—282no—
414C7#816111yescharacteristic, complemented, Sylow
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—281no—
424C22—281no—
425C22—281no—
426C22—561no—
427C22—561no—
428C22#8943811yescharacteristic
429C22C4×#113111yescharacteristic
430C22#8948011yescharacteristic
431C22#8950212yes—
432C22#8961012yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—281no—
438C4—281no—
439C4—281no—
440C4—562no—
441C4#8951211yescharacteristic
442C4D7×#9911yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—561no—
448C2#1172640311yescharacteristic
449C2#1172670611yescharacteristic
450C2#1172676811yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC23×C7C2×C42
FrattiniC23×C4C22×D7
FittingC7×#1282C2
socleC22×C7C4×C22⋊C4
prime radicalC22×C22⋊D7C2×C4
cosocleC23×C4×C7C23
hypercenter#1282D7
hypocenterC7#8161
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. #11727702C2semidirect
5. C22⋊C4×C7⋊C8C2semidirect
6. C7×#1282C2semidirect
7. #11725483C2semidirect
10. C22×#1131C4semidirect
16. C2×(C2×D7)⋊C8C4semidirect
18. C2×#6389C4semidirect
19. C2×Dic7⋊C8C4semidirect
29. #89264C4semidirect
30. #89263C4semidirect
84. #1282D7semidirect
116. C7×C22⋊C4M16semidirect
219. C7⋊C8#99semidirect
348. C22⋊C4#1131semidirect
414. C7#8161semidirect

Maximal subgroups

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

subgroupindex
C2×#895022
#117277022
#117276732
C4×#63802
C22⋊C4×C7⋊C82
C7×#12822
#117254832
#81617