Finite Group Databasebeta

← #13247356

Subgroups — #13247356

Order 1792.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11730781C211yesmaximal, characteristic
2#11727673C211yesmaximal, characteristic
3#11727284C211yesmaximal, characteristic
4C7×#1294C211yesmaximal, characteristic, complemented
5#11725482C211yesmaximal, characteristic, complemented
6#11725431C211yesmaximal, characteristic, complemented
7#11725418C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D7C2211yescharacteristic
9C2×C7×C22⋊C8C2211yescharacteristic
10C7×#380C2211yescharacteristic
11C2×C7×#95C2211yescharacteristic
12#89985—21no—
13#89984C2211yescharacteristic
14#89984C2211yescharacteristic
15C4×C22⋊Dic7—21no—
16C2×C22⋊(C7⋊C8)C2211yescharacteristic
17C2×#6380—21no—
18#89442—21no—
19#89438C411yescharacteristic, complemented
20#89436C411yescharacteristic, complemented
21#89427—21no—
22#89423C411yescharacteristic, complemented
23#89420C411yescharacteristic, complemented
24#89322—21no—
25#89264—21no—
26#89263—21no—
27#8763—71nomaximal, Sylow
28C23×C4×C7—21no—
29C23×C4×C7C2311yescharacteristic
30C22×C22⋊D7D411yescharacteristic
31C22×C4×D7C2×C411yescharacteristic
32C22×C7×C8—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7D411yescharacteristic
38C4×C22⋊D7—21no—
39C4×C22⋊D7—21no—
40C4×C22⋊D7—21no—
41C4×C22⋊D7—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×(C2×D7)⋊C4D411yescharacteristic
45C2×C4⋊Dic7—21no—
46C2×C4⋊Dic7—21no—
47C2×Dic7⋊C4D411yescharacteristic
48C2×C4×Dic7—21no—
49C2×C4×Dic7—21no—
50C2×C4×Dic7C2×C411yescharacteristic
51C22×C7⋊C8—21no—
52C7×C22⋊C8—21no—
53C7×C22⋊C8—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62C7⋊#95—21no—
63C7⋊#95C2×C411yescharacteristic
64C7⋊#95C2×C411yescharacteristic
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67#6380—42no—
68#2353—71no—
69#1799—71no—
70#1459—71no—
71#1294—71no—
72#1294D711yescharacteristic, complemented
73#1289—71no—
74#1289—71no—
75C24×C7C2×D411yescharacteristic
76C23×D7C22⋊C411yescharacteristic
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—42no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—81no—
84C22×C4×C7—81no—
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C2×C22⋊D7—21no—
92C2×C22⋊D7—21no—
93C2×C22⋊D7—21no—
94C2×C22⋊D7—21no—
95C2×C22⋊D7—21no—
96C2×C22⋊D7—21no—
97C22×Dic7—21no—
98C22×Dic7—21no—
99C22×Dic7—41no—
100C22×Dic7C22⋊C411yescharacteristic
101C22×Dic7C22⋊C411yescharacteristic
102C22×Dic7C22⋊C411yescharacteristic
103C2×C4×D7—21no—
104C2×C4×D7—21no—
105C2×C4×D7—81no—
106C2×C7×C8—41no—
107C2×C7×C8—81no—
108C7×C4⋊C4—81no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C22⋊Dic7—21no—
112C22⋊Dic7—21no—
113C22⋊Dic7—41no—
114C22⋊Dic7—41no—
115C22⋊Dic7—41no—
116C22⋊Dic7—41no—
117(C2×D7)⋊C4—21no—
118(C2×D7)⋊C4—21no—
119(C2×D7)⋊C4—42no—
120(C2×D7)⋊C4—81no—
121C4⋊Dic7—81no—
122C4⋊Dic7—81no—
123Dic7⋊C4—21no—
124Dic7⋊C4—21no—
125C4×Dic7—21no—
126C4×Dic7—21no—
127C4×Dic7—42no—
128C4×Dic7—81no—
129C2×C7⋊C8—41no—
130C2×C7⋊C8—81no—
131C2×C4×D4—71no—
132C2×#99—141no—
133C2×C22⋊C8—71no—
134C2×C22⋊C8C2×D711yescharacteristic
135#388—141no—
136#387—141no—
137#386—141no—
138#383—71no—
139#381—71no—
140#380—71no—
141#380—71no—
142#380C2×D711yescharacteristic
143C4×C22⋊C4—71no—
144C4×C22⋊C4—141no—
145#376—71no—
146C2×#95C2×D711yescharacteristic
147#342—141no—
148#324—141no—
149#323—141no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7C2×C22⋊C411yescharacteristic
153C23×C7#12311yescharacteristic
154C23×C7#12311yescharacteristic
155C23×C7#13511yescharacteristic
156C23×C7#13511yescharacteristic
157C23×C7AGL(1,ℤ/8)11yescharacteristic
158C23×C7#13711yescharacteristic
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—21no—
162C22×D7—41no—
163C22×D7—41no—
164C2×C4×C7—22no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—82no—
177C2×C4×C7—82no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7#11711yescharacteristic
182C2×C4×C7#11711yescharacteristic
183C2×C4×C7#12311yescharacteristic
184C2×C4×C7#12311yescharacteristic
185C22⋊D7—41no—
186C22⋊D7—41no—
187C22⋊D7—41no—
188C22⋊D7—41no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
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—81no—
203C2×Dic7—81no—
204C2×Dic7—81no—
205C4×D7—81no—
206C4×D7—81no—
207C7×C8—81no—
208C7⋊C8—81no—
209C22×D4—71no—
210C23×C4—21no—
211C23×C4—71no—
212C23×C4C22×D711yescharacteristic
213C22×C8—21no—
214C22×C8—141no—
215C4×D4—141no—
216C4×D4—141no—
217C4×D4—141no—
218C4×D4—141no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4—141no—
222C2×C4⋊C4—141no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—141no—
227C2×C22⋊C4—141no—
228C2×C22⋊C4—141no—
229C2×C22⋊C4—141no—
230C2×C42—71no—
231C2×C42—141no—
232C2×C42—141no—
233#99—282no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—141no—
237C22⋊C8—141no—
238#95—21no—
239#95—21no—
240#95—71no—
241#95—71no—
242#95—141no—
243#95—141no—
244#95—141no—
245#95—141no—
246#95—141no—
247#95C4×D711yescharacteristic
248#95C4×D711yescharacteristic
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7#38111yescharacteristic
260C22×C7#41611yescharacteristic
261C22×C7#41711yescharacteristic
262C22×C7#48211yescharacteristic
263C22×C7#48211yescharacteristic
264C22×C7#48511yescharacteristic
265C22×C7#48511yescharacteristic
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—81no—
270C2×D7—81no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—42no—
276C4×C7—81no—
277C4×C7—161no—
278Dic7—41no—
279Dic7—41no—
280Dic7—41no—
281Dic7—81no—
282Dic7—81no—
283Dic7—161no—
284C24—71no—
285C24D4×D711yescharacteristic
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—141no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—71no—
299C22×C4—71no—
300C22×C4—71no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—141no—
304C22×C4—141no—
305C22×C4—141no—
306C22×C4—141no—
307C22×C4—281no—
308C22×C4—561no—
309C22×C4C2×C4×D711yescharacteristic
310C22×C4#115711yescharacteristic
311C22×C4#115711yescharacteristic
312C22×C4D4×D711yescharacteristic
313C22×C4#115911yescharacteristic
314C22×C4#116111yescharacteristic
315C2×C8—41no—
316C2×C8—81no—
317C2×C8—281no—
318C2×C8—561no—
319C4⋊C4—81no—
320C4⋊C4—141no—
321C4⋊C4—141no—
322C4⋊C4—561no—
323C4⋊C4—561no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—141no—
327C22⋊C4—141no—
328C22⋊C4—141no—
329C22⋊C4—141no—
330C22⋊C4—282no—
331C22⋊C4—281no—
332C22⋊C4—281no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C22⋊C4—561no—
336C42—141no—
337C42—141no—
338C42—282no—
339C42—561no—
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—21no—
343C2×C7—21no—
344C2×C7—21no—
345C2×C7—21no—
346C2×C7#182511yescharacteristic
347C2×C7#182711yescharacteristic
348C2×C7#182811yescharacteristic
349D7—81no—
350C23—41no—
351C23—41no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—281no—
356C23—281no—
357C23D7×C22⋊C411yescharacteristic
358C23#644611yescharacteristic
359C23#645711yescharacteristic
360C23#647511yescharacteristic
361C23#647711yescharacteristic
362C23#647911yescharacteristic
363C23#648111yescharacteristic
364D4—281no—
365D4—281no—
366D4—281no—
367D4—281no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4—281no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—561no—
399C2×C4—561no—
400C2×C4—561no—
401C2×C4—561no—
402C2×C4—561no—
403C2×C4#643511yescharacteristic
404C2×C4#644611yescharacteristic
405C2×C4#645511yescharacteristic
406C2×C4#645711yescharacteristic
407C8—81no—
408C8—561no—
409C7#876311yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—281no—
421C22—281no—
422C22—281no—
423C22—561no—
424C22—561no—
425C22#8943611yescharacteristic
426C22#8954411yescharacteristic
427C22#8955611yescharacteristic
428C22#8957511yescharacteristic
429C22#8958711yescharacteristic
430C22#8963811yescharacteristic
431C22#8966011yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—161no—
439C4—281no—
440C4—281no—
441C4—281no—
442C4—561no—
443C4—561no—
444C4—1121no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—561no—
452C2#1172649111yescharacteristic
453C2#1172662711yescharacteristic
454C2#1172671411yescharacteristic
subgroupG/H
centerC22#89436
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1294C2
socleC22×C7#381
prime radicalC22×C22⋊D7D4
cosocleC23×C4×C7C23
hypercenter#1294D7
hypocenterC7#8763
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
4. C7×#1294C2semidirect
5. #11725482C2semidirect
6. #11725431C2semidirect
7. #11725418C2semidirect
19. #89438C4semidirect
20. #89436C4semidirect
22. #89423C4semidirect
23. #89420C4semidirect
72. #1294D7semidirect
409. C7#8763semidirect

Maximal subgroups

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

subgroupindex
#117307812
#117276732
#117272842
C7×#12942
#117254822
#117254312
#117254182
#87637