Finite Group Databasebeta

← #13256799

Subgroups — #13256799

Order 1792.

500 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731401C211yesmaximal, characteristic
2C7×#1824C211yesmaximal, characteristic
3#11729323C211yesmaximal, characteristic
4#11729308C211yesmaximal, characteristic
5#11725481C211yesmaximal, characteristic
6#11725480C211yesmaximal, characteristic
7#11725430C211yesmaximal, characteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×C7×C22⋊C8C2211yescharacteristic
10C7×#380C2211yescharacteristic
11#89984C2211yescharacteristic
12C2×C7⋊#95—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×C7⋊#95C2211yescharacteristic
16C2×C22⋊(C7⋊C8)C2211yescharacteristic
17#89758—21no—
18#89756—21no—
19#89722—21no—
20#89718—21no—
21#89433—21no—
22#89432—21no—
23C7×#340—21no—
24C7×#340—21no—
25#89324—21no—
26#89324—21no—
27#10347—71nomaximal, Sylow
28C23×C4×C7—21no—
29C23×C4×C7C2311yescharacteristic
30C23×Dic7—21no—
31C23×Dic7D411yescharacteristic
32C23×Dic7D411yescharacteristic
33C22×C7×C8—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4Q812yes—
42C2×C7×C22⋊C4—21no—
43C2×C22⋊Dic7—21no—
44C2×C4⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×C4⋊Dic7D411yescharacteristic
47C2×C4⋊Dic7D411yescharacteristic
48C2×Dic7⋊C4—21no—
49C2×Dic7⋊C4—21no—
50C22×C7⋊C8—21no—
51C7×C22⋊C8—21no—
52C7×C22⋊C8—21no—
53C7×#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62C7⋊#95—21no—
63C7⋊#95—42no—
64C7⋊#95—42no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67#2646—71no—
68#1824—71no—
69#1824D711yescharacteristic
70#1445—71no—
71#1294—71no—
72#1294—71no—
73#1280—71no—
74C24×C7C2×D411yescharacteristic
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—22no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×Q811yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C22×C4×C7#4111yescharacteristic
92C22×Dic7—22no—
93C22×Dic7—21no—
94C22×Dic7—21no—
95C22×Dic7—21no—
96C22×Dic7—21no—
97C22×Dic7—21no—
98C22×Dic7—21no—
99C22×Dic7—21no—
100C22×Dic7—21no—
101C22×Dic7—41no—
102C22×Dic7—41no—
103C22×Dic7—41no—
104C22×Dic7—41no—
105C22×Dic7—41no—
106C22×Dic7—41no—
107C22×Dic7—41no—
108C22×Dic7—41no—
109C22×Dic7—41no—
110C22×Dic7—41no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C22⋊Dic7—41no—
123C22⋊Dic7—41no—
124C4⋊Dic7—41no—
125C4⋊Dic7—41no—
126C4⋊Dic7—81no—
127C4⋊Dic7—81no—
128Dic7⋊C4—81no—
129Dic7⋊C4—81no—
130C2×C7⋊C8—41no—
131C2×C7⋊C8—41no—
132C2×C7⋊C8—41no—
133C22×C4⋊C4C2×D711yescharacteristic
134C2×C22⋊C8—71no—
135C2×C22⋊C8C2×D711yescharacteristic
136#400—141no—
137#400—141no—
138#398—141no—
139#398—141no—
140#385—141no—
141#385—141no—
142#380—71no—
143#380C2×D711yescharacteristic
144C2×#95—71no—
145C2×#95—71no—
146C2×#95—141no—
147C2×#95—141no—
148#340—21no—
149#340—21no—
150#340—141no—
151#340—141no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7#12011yescharacteristic
157C23×C7C22⋊Q811yescharacteristic
158C23×C7#12411yescharacteristic
159C23×C7#13511yescharacteristic
160C23×C7#13511yescharacteristic
161C23×C7#13711yescharacteristic
162C23×C7#13711yescharacteristic
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7C22⋊Q811yescharacteristic
185C2×C4×C7C22⋊Q811yescharacteristic
186C2×C4×C7#12411yescharacteristic
187C2×C4×C7#12411yescharacteristic
188C2×Dic7—42no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7—42no—
192C2×Dic7—42no—
193C2×Dic7—41no—
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—
203C2×Dic7—41no—
204C2×Dic7—41no—
205C2×Dic7—81no—
206C2×Dic7—81no—
207C2×Dic7—81no—
208C2×Dic7—81no—
209C2×Dic7—81no—
210C2×Dic7—81no—
211C7×C8—81no—
212C7⋊C8—81no—
213C23×C4—21no—
214C23×C4—71no—
215C23×C4—71no—
216C23×C4—141no—
217C23×C4C22×D711yescharacteristic
218C22×C8—21no—
219C22×C8—141no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—71no—
226C2×C4⋊C4—71no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—141no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4Dic1412yes—
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C4⋊C4C22⋊D711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—141no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—141no—
239C22⋊C8—141no—
240#95—21no—
241#95—141no—
242#95—141no—
243#95—141no—
244#95—141no—
245#95—141no—
246#95—141no—
247#95—141no—
248#95—141no—
249#95—141no—
250#95—282no—
251#95—282no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7#39311yescharacteristic
265C22×C7#45211yescharacteristic
266C22×C7#45211yescharacteristic
267C22×C7#45311yescharacteristic
268C22×C7#45411yescharacteristic
269C22×C7#45811yescharacteristic
270C22×C7#45811yescharacteristic
271C22×C7#47911yescharacteristic
272C22×C7#47911yescharacteristic
273C22×C7#48811yescharacteristic
274C22×C7#49111yescharacteristic
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—81no—
280C4×C7—81no—
281C4×C7—82no—
282Dic7—82no—
283Dic7—81no—
284Dic7—81no—
285Dic7—81no—
286Dic7—81no—
287C24D4×D711yescharacteristic
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—142no—
300C22×C4—141no—
301C22×C4—141no—
302C22×C4—141no—
303C22×C4—141no—
304C22×C4—141no—
305C22×C4—141no—
306C22×C4—141no—
307C22×C4—141no—
308C22×C4—281no—
309C22×C4—281no—
310C22×C4—281no—
311C22×C4—281no—
312C22×C4—281no—
313C22×C4—281no—
314C22×C4—281no—
315C22×C4—281no—
316C22×C4—281no—
317C22×C4—281no—
318C22×C4C2×Dic1411yescharacteristic
319C22×C4#115711yescharacteristic
320C22×C4D4×D711yescharacteristic
321C22×C4#115911yescharacteristic
322C22×C4#115911yescharacteristic
323C22×C4C2×C22⋊D711yescharacteristic
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—281no—
328C2×C8—281no—
329C2×C8—281no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—281no—
337C4⋊C4—281no—
338C4⋊C4—561no—
339C4⋊C4—561no—
340C4⋊C4—561no—
341C4⋊C4—561no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—281no—
345C22⋊C4—281no—
346C2×C7—21no—
347C2×C7—21no—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7#161511yescharacteristic
351C2×C7#161511yescharacteristic
352C2×C7#161811yescharacteristic
353C2×C7#161811yescharacteristic
354C2×C7#202611yescharacteristic
355C2×C7#202711yescharacteristic
356C2×C7#203011yescharacteristic
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23#644111yescharacteristic
362C23#644811yescharacteristic
363C23#647811yescharacteristic
364C23#647911yescharacteristic
365C23#648111yescharacteristic
366C23#648211yescharacteristic
367C23#650011yescharacteristic
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—282no—
390C2×C4—282no—
391C2×C4—282no—
392C2×C4—282no—
393C2×C4—282no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—281no—
399C2×C4—281no—
400C2×C4—281no—
401C2×C4—281no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—281no—
405C2×C4—281no—
406C2×C4—561no—
407C2×C4—561no—
408C2×C4—561no—
409C2×C4—561no—
410C2×C4—561no—
411C2×C4—561no—
412C2×C4#644111yescharacteristic
413C2×C4#644811yescharacteristic
414C2×C4#648711yescharacteristic
415C2×C4#649911yescharacteristic
416C8—81no—
417C8—561no—
418C7#1034711yescharacteristic, complemented, Sylow
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22#8956811yescharacteristic
432C22#8956911yescharacteristic
433C22#8957111yescharacteristic
434C22#8957211yescharacteristic
435C22#8971911yescharacteristic
436C22#8982511yescharacteristic
437C22#8982511yescharacteristic
438C22#8994511yescharacteristic
439C22#8995911yescharacteristic
440C22#8997011yescharacteristic
441C22#8997311yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—562no—
450C4—561no—
451C4—561no—
452C4—561no—
453C4—561no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1172729011yescharacteristic
459C2#1172798611yescharacteristic
460C2#1172806711yescharacteristic
461C2#1172892311yescharacteristic
462C2#1172892411yescharacteristic
463C2#1172893111yescharacteristic
464C2#1172893211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1824C2
socleC23×C7#120
prime radicalC24×C7C2×D4
hypercenter#1824D7
hypocenterC7#10347
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
418. C7#10347semidirect

Maximal subgroups

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

subgroupindex
#117314012
C7×#18242
#117293232
#117293082
#117254812
#117254802
#117254302
#103477