Finite Group Databasebeta

← #13249384

Subgroups — #13249384

Order 1792.

674 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
1#11739525C211yesmaximal, characteristic
2C2×#89698C211yesmaximal, characteristic, complemented
3C7×#1434C211yesmaximal, characteristic, complemented
4C2×#89246C212yesmaximal
5#11725418C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×#6493C2211yescharacteristic
8C2×#6492—21no—
9C2×C7×#95—21no—
10C2×C7×#95—21no—
11C2×C7×#95C2212yes—
12#89985—21no—
13#89984C2211yescharacteristic
14C2×C22⋊(C7⋊C8)C2212yes—
15#89712—21no—
16#89712—21no—
17#89711C411yescharacteristic, complemented
18#89711C411yescharacteristic, complemented
19#89700C411yescharacteristic, complemented
20#89700C411yescharacteristic, complemented
21#89698—24no—
22C2×C4⋊(C7⋊C8)—21no—
23C2×#6377—21no—
24#89246—24no—
25#9118—71nomaximal, Sylow
26C23×C4×C7—22no—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C22×C22⋊D7—21no—
31C22×D28C2×C411yescharacteristic
32C2×C42×C7—22no—
33C2×C42×C7—22no—
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C22⋊Dic7—21no—
39#6493—22no—
40#6493—22no—
41#6492—44no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×(C2×D7)⋊C4—21no—
45C2×(C2×D7)⋊C4—21no—
46C2×(C2×D7)⋊C4—21no—
47C2×C4⋊Dic7C2×C411yescharacteristic
48C2×C4⋊Dic7C2×C411yescharacteristic
49C2×C4⋊Dic7C2×C411yescharacteristic
50C2×Dic7⋊C4—21no—
51C2×Dic7⋊C4—21no—
52C22×C7⋊C8—22no—
53C7×#95—24no—
54C7×#95—22no—
55C7×#95—22no—
56C7×#95—42no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)D414yes—
62C4⋊(C7⋊C8)—24no—
63#6377—24no—
64C2×#432—71no—
65#2854—71no—
66C2×#327—72no—
67#1434D711yescharacteristic, complemented
68#1294—72no—
69C24×C7C2×D411yescharacteristic
70C23×D7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—24no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22×C411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D412yes—
94C22×C4×C7#4112yes—
95C2×C22⋊D7—42no—
96C2×C22⋊D7—42no—
97C2×C22⋊D7—41no—
98C2×C22⋊D7—41no—
99C22×Dic7—21no—
100C22×Dic7—21no—
101C22×Dic7—21no—
102C2×D28—22no—
103C2×D28—41no—
104C2×D28—41no—
105C42×C7—22no—
106C42×C7—22no—
107C42×C7—21no—
108C42×C7—21no—
109C42×C7—21no—
110C42×C7—21no—
111C22⋊Dic7—42no—
112(C2×D7)⋊C4—42no—
113(C2×D7)⋊C4—42no—
114(C2×D7)⋊C4—42no—
115(C2×D7)⋊C4—42no—
116(C2×D7)⋊C4—41no—
117(C2×D7)⋊C4—41no—
118C4⋊Dic7—22no—
119C4⋊Dic7—41no—
120C4⋊Dic7—41no—
121Dic7⋊C4—42no—
122Dic7⋊C4—42no—
123C2×C7⋊C8—24no—
124C2×C7⋊C8—42no—
125C2×C7⋊C8—42no—
126C2×#123—141no—
127C2×#121—71no—
128C22×C42C2×D711yescharacteristic
129#432—144no—
130C2×C4⋊C8—141no—
131C2×C22⋊C8—72no—
132C2×#97—141no—
133#390—71no—
134#390—71no—
135#389—71no—
136#389—71no—
137#388—141no—
138#388—141no—
139#386—141no—
140#380—71no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95C2×D712yes—
144#327—144no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7C2×C22⋊C411yescharacteristic
150C23×C7#12112yes—
151C23×C7AGL(1,ℤ/8)11yescharacteristic
152C23×C7AGL(1,ℤ/8)12yes—
153C23×C7#13711yescharacteristic
154C22×D7—41no—
155C22×D7—41no—
156C22×D7—41no—
157C22×D7—41no—
158C22×D7—41no—
159C22×D7—41no—
160C22×D7—41no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—44no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—44no—
182C2×C4×C7—44no—
183C2×C4×C7—44no—
184C2×C4×C7—82no—
185C2×C4×C7—82no—
186C2×C4×C7—82no—
187C2×C4×C7—82no—
188C2×C4×C7#10412yes—
189C2×C4×C7C4×D412yes—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12311yescharacteristic
192C22⋊D7—82no—
193C22⋊D7—82no—
194C2×Dic7—41no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—41no—
200C2×Dic7—41no—
201C2×Dic7—41no—
202C2×Dic7—41no—
203D28—82no—
204C7⋊C8—44no—
205C22×D4—71no—
206C22×D4—141no—
207C23×C4—22no—
208C23×C4C22⋊D711yescharacteristic
209C23×C4C22⋊D711yescharacteristic
210C23×C4C22×D711yescharacteristic
211C22×C8—142no—
212#123—284no—
213#121—142no—
214#121—142no—
215C2×C4⋊C4—71no—
216C2×C4⋊C4—71no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—141no—
219C2×C4⋊C4—141no—
220C2×C22⋊C4—141no—
221C2×C22⋊C4—141no—
222C2×C22⋊C4—141no—
223C2×C22⋊C4—141no—
224C2×C22⋊C4—141no—
225C2×C22⋊C4—141no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C4×D711yescharacteristic
229C2×C42C4×D711yescharacteristic
230C2×C42D2811yescharacteristic
231C2×C42D2811yescharacteristic
232C4⋊C8—144no—
233C22⋊C8—74no—
234C22⋊C8—142no—
235#97—144no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—42no—
240#95—141no—
241#95—141no—
242#95—141no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—22no—
251C22×C7#38611yescharacteristic
252C22×C7#41011yescharacteristic
253C22×C7#41011yescharacteristic
254C22×C7#41711yescharacteristic
255C22×C7#41811yescharacteristic
256C22×C7C2×#10411yescharacteristic
257C22×C7#42111yescharacteristic
258C22×C7#46812yes—
259C22×C7#46912yes—
260C2×D7—81no—
261C2×D7—81no—
262C2×D7—81no—
263C2×D7—81no—
264C2×D7—81no—
265C2×D7—81no—
266C2×D7—81no—
267C4×C7—22no—
268C4×C7—21no—
269C4×C7—22no—
270C4×C7—21no—
271C4×C7—41no—
272C4×C7—41no—
273C4×C7—41no—
274C4×C7—84no—
275Dic7—81no—
276Dic7—81no—
277Dic7—81no—
278C24—141no—
279C24C2×C22⋊D711yescharacteristic
280C2×D4—142no—
281C2×D4—281no—
282C2×D4—281no—
283C2×D4—281no—
284C2×D4—282no—
285C2×D4—281no—
286C2×D4—282no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—141no—
304C22×C4—141no—
305C22×C4—141no—
306C22×C4(C2×D7)⋊C411yescharacteristic
307C22×C4(C2×D7)⋊C411yescharacteristic
308C22×C4(C2×D7)⋊C411yescharacteristic
309C22×C4(C2×D7)⋊C411yescharacteristic
310C22×C4C2×C4×D711yescharacteristic
311C22×C4C2×D2811yescharacteristic
312C22×C4D4×D712yes—
313C22×C4#115912yes—
314C2×C8—144no—
315C2×C8—282no—
316C2×C8—282no—
317C4⋊C4—142no—
318C4⋊C4—281no—
319C4⋊C4—281no—
320C4⋊C4—282no—
321C4⋊C4—282no—
322C22⋊C4—282no—
323C22⋊C4—282no—
324C22⋊C4—282no—
325C22⋊C4—282no—
326C22⋊C4—281no—
327C22⋊C4—281no—
328C22⋊C4—282no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C2×C7—22no—
336C2×C7—22no—
337C2×C7#150511yescharacteristic
338C2×C7#150911yescharacteristic
339C2×C7#151211yescharacteristic
340C2×C7#151411yescharacteristic
341C2×C7#187111yescharacteristic
342C2×C7#189511yescharacteristic
343C2×C7#189911yescharacteristic
344D7—161no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—281no—
350C23—281no—
351C23—281no—
352C23—281no—
353C23—281no—
354C23—281no—
355C23—281no—
356C23C2×(C2×D7)⋊C411yescharacteristic
357C23#649511yescharacteristic
358C23#650212yes—
359C23#650511yescharacteristic
360C23#651212yes—
361D4—562no—
362D4—562no—
363D4—562no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4—281no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—281no—
399C2×C4—281no—
400C2×C4#637912yes—
401C2×C4(C22⋊D7)⋊C412yes—
402C2×C4#644511yescharacteristic
403C2×C4#644911yescharacteristic
404C8—284no—
405C7#911811yescharacteristic, complemented, Sylow
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—561no—
415C22—561no—
416C22—561no—
417C22—561no—
418C22—561no—
419C22—561no—
420C22—561no—
421C22C2×#637911yescharacteristic
422C22#8973011yescharacteristic
423C22#8973811yescharacteristic
424C22#8977311yescharacteristic
425C22#8977711yescharacteristic
426C22#8977811yescharacteristic
427C22#8977811yescharacteristic
428C22#8981112yes—
429C22#8982112yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—561no—
439C4—561no—
440C4—561no—
441C2—22no—
442C2—22no—
443C2—1121no—
444C2#1172718611yescharacteristic
445C2#1172720711yescharacteristic
446C2#1172722411yescharacteristic
447C2#1172727111yescharacteristic
448C2#1172728911yescharacteristic
449C2#1172729811yescharacteristic
450C2#1172731811yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1434C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#1434D7
hypocenterC7#9118
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#89698C2semidirect
3. C7×#1434C2semidirect
5. #11725418C2semidirect
17. #89711C4semidirect
18. #89711C4semidirect
19. #89700C4semidirect
20. #89700C4semidirect
67. #1434D7semidirect
405. C7#9118semidirect

Maximal subgroups

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

subgroupindex
#117395252
C2×#896982
C7×#14342
C2×#892462
#117254182
#91187