Finite Group Databasebeta

← #13254568

Subgroups — #13254568

Order 1792.

548 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C7×#2580C211yesmaximal, characteristic
2#11739892C211yesmaximal, characteristic, complemented
3#11739741C211yesmaximal, characteristic
4#11729376C211yesmaximal, characteristic, complemented
5#11729351C211yesmaximal, characteristic
6#11727300C211yesmaximal, characteristic, complemented
7#11725437C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10#90161C2211yescharacteristic
11C7×#399—21no—
12C7×#393—21no—
13C7×#390—21no—
14C7×#389—21no—
15C7×#385—22no—
16C7×#382C2211yescharacteristic
17#90000—21no—
18#89987—21no—
19#89803—21no—
20C4×#1144—22no—
21#89793—21no—
22#89781—22no—
23#89780—21no—
24#89779—21no—
25#89742—21no—
26#89736—21no—
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89696C2211yescharacteristic
30Q8⋊(C7⋊C8)—22no—
31D4⋊(C7⋊C8)—22no—
32#89247—22no—
33#10002—71nomaximal, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C4×C7×Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—22no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—22no—
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C2×#1144—22no—
56#6489—21no—
57#6487—21no—
58#6487—21no—
59C2×#1136—21no—
60#6433—21no—
61#6432D411yescharacteristic
62#6432D411yescharacteristic
63C4×Dic14—21no—
64C4×Dic14—21no—
65C7×#95—22no—
66C7×#95—21no—
67C7×#95D412yes—
68#6409—22no—
69#6406—22no—
70#6406—22no—
71C22⋊(C7⋊C8)—21no—
72C22⋊(C7⋊C8)—21no—
73#6384—22no—
74#6382—22no—
75#6381—22no—
76#6381—22no—
77C4⋊(C7⋊C8)—21no—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4×C7⋊C8—21no—
82C4×C7⋊C8D412yes—
83#3104—71no—
84#3077—71no—
85#2580D711yescharacteristic
86#1518—71no—
87#1487—71no—
88#1478—71no—
89#1286—71no—
90C24×C7—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—42no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4Q1612yescomplemented
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7—82no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C2×Dic14—21no—
111C2×Dic14—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—82no—
121C7×C22⋊C4—44no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—82no—
125C42×C7—41no—
126C42×C7—41no—
127C42×C7C2×D411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C22⋊Dic7—41no—
132#1144—44no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137Dic7⋊C4—41no—
138Dic7⋊C4—41no—
139C4×Dic7—41no—
140#1136—41no—
141#1136—41no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8—22no—
144C2×C7⋊C8—41no—
145#533—71no—
146C2×C4×Q8C2×D711yescharacteristic
147C2×C4×D4C2×D711yescharacteristic
148#484—141no—
149#478—142no—
150#477—141no—
151#476—141no—
152#474—141no—
153#462—141no—
154#451—141no—
155C4×Q16—142no—
156#431—71no—
157#431—71no—
158#428—141no—
159#423—71no—
160#418—141no—
161#399—21no—
162#393—21no—
163#390—21no—
164#389—21no—
165#385—22no—
166#382C2×D711yescharacteristic
167#328—142no—
168Q8⋊C8—142no—
169D4⋊C8—142no—
170C23×C7—22no—
171C23×C7—42no—
172C23×C7—42no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7#12011yescharacteristic
176C7×Q8—22no—
177C7×Q8—22no—
178C7×Q8—41no—
179C7×Q8—41no—
180C7×D4—22no—
181C7×D4—22no—
182C7×D4—41no—
183C7×D4—41no—
184C2×C4×C7—22no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—82no—
198C2×C4×C7—82no—
199C2×C4×C7#12011yescharacteristic
200C2×C4×C7#12011yescharacteristic
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7#12711yescharacteristic
205C2×C4×C7C2×Q1611yescharacteristic
206C2×C4×C7#13511yescharacteristic
207C2×C4×C7#13711yescharacteristic
208C2×C4×C7#13711yescharacteristic
209C2×Dic7—41no—
210C2×Dic7—41no—
211Dic14—42no—
212Dic14—81no—
213C7⋊C8—42no—
214C7⋊C8—42no—
215C7⋊C8—81no—
216C22×Q8C22⋊D711yescharacteristic
217C22×D4C22⋊D711yescharacteristic
218C23×C4—21no—
219C2×Q16—142no—
220C2×M16—141no—
221C4⋊Q8—71no—
222C4⋊Q8—71no—
223#125—141no—
224C22⋊Q8—141no—
225C22⋊Q8—141no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—141no—
231C4×Q8—141no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—141no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C4⋊C4C22⋊D711yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C42—21no—
246C2×C42C22×D711yescharacteristic
247#107—142no—
248#107—142no—
249#106—142no—
250C4⋊C8—72no—
251C4⋊C8—141no—
252C4⋊C8—141no—
253C4⋊C8—141no—
254#103—142no—
255#103—142no—
256#102—142no—
257#102—142no—
258C22⋊C8—141no—
259C22⋊C8—141no—
260C4×C8—72no—
261C4×C8—141no—
262#95—22no—
263#95—21no—
264#95C22⋊D712yes—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—44no—
268C22×C7—42no—
269C22×C7—82no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7#39211yescharacteristic
273C22×C7#45111yescharacteristic
274C22×C7#45211yescharacteristic
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—42no—
280C4×C7—82no—
281C4×C7—81no—
282C4×C7—81no—
283C4×C7#45111yescharacteristic
284C4×C7#45211yescharacteristic
285C4×C7#46711yescharacteristic
286C4×C7#47011yescharacteristic
287Dic7—81no—
288Dic7—81no—
289C24—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—42no—
293C2×Q8—141no—
294C2×Q8—141no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4#114412yescomplemented
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4C2×C22⋊D711yescharacteristic
309C22×C4C2×C22⋊D711yescharacteristic
310C22×C4C2×C22⋊D711yescharacteristic
311Q16—284no—
312M16—281no—
313M16—281no—
314C2×C8—142no—
315C2×C8—142no—
316C2×C8—281no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—82no—
326C4⋊C4—141no—
327C4⋊C4—141no—
328C4⋊C4—141no—
329C4⋊C4—141no—
330C4⋊C4—281no—
331C4⋊C4—281no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—82no—
336C22⋊C4—281no—
337C42—41no—
338C42—41no—
339C42—281no—
340C42D4×D711yescharacteristic
341C42D4×D711yescharacteristic
342C42D4×D711yescharacteristic
343C42#115911yescharacteristic
344C2×C7—21no—
345C2×C7—21no—
346C2×C7—82no—
347C2×C7#200111yescharacteristic
348C2×C7#200811yescharacteristic
349C2×C7#201011yescharacteristic
350C23—22no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23#651611yescharacteristic
356Q8—22no—
357Q8—22no—
358Q8—41no—
359Q8—41no—
360Q8—282no—
361Q8—561no—
362D4—22no—
363D4—22no—
364D4—41no—
365D4—41no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—281no—
382C2×C4—281no—
383C2×C4#650011yescharacteristic
384C2×C4#650011yescharacteristic
385C2×C4#650111yescharacteristic
386C2×C4#650211yescharacteristic
387C2×C4#650211yescharacteristic
388C2×C4#650311yescharacteristic
389C2×C4#650511yescharacteristic
390C2×C4C2×#114411yescharacteristic
391C2×C4#651311yescharacteristic
392C2×C4#651411yescharacteristic
393C8—282no—
394C8—282no—
395C8—561no—
396C7#1000211yescharacteristic, complemented, Sylow
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22#8999511yescharacteristic
405C22#9000011yescharacteristic
406C22#9001111yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—561no—
416C4—561no—
417C4#8981911yescharacteristic
418C4#8982211yescharacteristic
419C4#8982311yescharacteristic
420C4#8982411yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2#1172945411yescharacteristic
425C2#1172946911yescharacteristic
426C2#1172961211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2580C2
socleC22×C7#392
prime radicalC22×C7×D4D4
hypercenter#2580D7
hypocenterC7#10002
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #11739892C2semidirect
4. #11729376C2semidirect
6. #11727300C2semidirect
7. #11725437C2semidirect
99. C2×C7×D4Q16semidirect
300. C2×D4#1144semidirect
396. C7#10002semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117398922
#117397412
#117293762
#117293512
#117273002
#117254372
#100027