Finite Group Databasebeta

← #13254561

Subgroups — #13254561

Order 1792.

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

normal subgroups only

This group has 568 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×#2536C211yesmaximal, characteristic
2#11739876C211yesmaximal, characteristic, complemented
3#11739759C211yesmaximal, characteristic
4#11729375C211yesmaximal, characteristic, complemented
5#11729352C211yesmaximal, characteristic
6#11727300C211yesmaximal, characteristic, complemented
7#11725437C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8—21no—
9C2×C4×C7×Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11#90161C2211yescharacteristic
12C7×#398—21no—
13C7×#394—21no—
14C7×#388—22no—
15C7×#382C2211yescharacteristic
16C7×#380—21no—
17#89997—21no—
18#89990—21no—
19#89794—22no—
20#89793—21no—
21#89792—21no—
22#89791—21no—
23C4×#1142—22no—
24#89780—21no—
25#89743—21no—
26#89736—21no—
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89696C2211yescharacteristic
30Q8⋊(C7⋊C8)—22no—
31D4⋊(C7⋊C8)—22no—
32#89247—22no—
33#10001—71nomaximal, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C7×C22⋊Q8—42no—
38C7×C22⋊Q8—42no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C7×C4⋊C4D411yescharacteristic
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—21no—
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57C2×#1142—22no—
58#6489—21no—
59#6487—21no—
60#6487—21no—
61C2×#1136—21no—
62#6433—21no—
63#6432D411yescharacteristic
64#6432D411yescharacteristic
65C4×Dic14—21no—
66C4×Dic14—21no—
67C7×#95—22no—
68C7×#95—21no—
69C7×#95D412yes—
70#6409—22no—
71#6409—22no—
72#6406—22no—
73C22⋊(C7⋊C8)—21no—
74C22⋊(C7⋊C8)—21no—
75#6384—22no—
76#6382—22no—
77#6382—22no—
78#6381—22no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)—21no—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)D412yes—
83C4×C7⋊C8—21no—
84C4×C7⋊C8D412yes—
85#3101—71no—
86#3078—71no—
87#2536D711yescharacteristic
88#1518—71no—
89#1486—71no—
90#1479—71no—
91#1286—71no—
92C24×C7—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8SD1612yescomplemented
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—42no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C2×Dic14—21no—
116C2×Dic14—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—44no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—44no—
127C7×C22⋊C4—44no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7—41no—
131C42×C7—41no—
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7#4111yescharacteristic
136C22⋊Dic7—41no—
137#1142—44no—
138C4⋊Dic7—21no—
139C4⋊Dic7—21no—
140C4⋊Dic7—21no—
141C4⋊Dic7—21no—
142Dic7⋊C4—41no—
143Dic7⋊C4—41no—
144C4×Dic7—41no—
145#1136—41no—
146#1136—41no—
147C2×C7⋊C8—22no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—41no—
150#533—71no—
151C2×C22⋊Q8—21no—
152C2×C4×Q8C2×D711yescharacteristic
153C2×C4×D4C2×D711yescharacteristic
154#483—141no—
155#479—142no—
156#477—141no—
157#476—141no—
158#475—141no—
159#461—141no—
160#450—141no—
161C4×SD16—142no—
162#431—71no—
163#431—71no—
164#428—141no—
165#423—71no—
166#419—141no—
167#398—21no—
168#394—21no—
169#388—22no—
170#382C2×D711yescharacteristic
171#380—21no—
172#328—142no—
173Q8⋊C8—142no—
174D4⋊C8—142no—
175C23×C7—22no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7#12011yescharacteristic
183C7×Q8—22no—
184C7×Q8—22no—
185C7×Q8—41no—
186C7×Q8—41no—
187C7×D4—22no—
188C7×D4—22no—
189C7×D4—41no—
190C7×D4—41no—
191C2×C4×C7—22no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—42no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—81no—
205C2×C4×C7—81no—
206C2×C4×C7—81no—
207C2×C4×C7—81no—
208C2×C4×C7#12011yescharacteristic
209C2×C4×C7#12011yescharacteristic
210C2×C4×C7#12111yescharacteristic
211C2×C4×C7#12111yescharacteristic
212C2×C4×C7#12111yescharacteristic
213C2×C4×C7#12711yescharacteristic
214C2×C4×C7C2×SD1611yescharacteristic
215C2×C4×C7#13511yescharacteristic
216C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
217C2×C4×C7#13711yescharacteristic
218C2×Dic7—41no—
219C2×Dic7—41no—
220Dic14—42no—
221Dic14—81no—
222C7⋊C8—42no—
223C7⋊C8—42no—
224C7⋊C8—81no—
225C22×Q8C22⋊D711yescharacteristic
226C22×D4C22⋊D711yescharacteristic
227C23×C4—21no—
228C2×SD16—142no—
229C2×M16—141no—
230C4⋊Q8—71no—
231C4⋊Q8—71no—
232#125—141no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—141no—
236C22⋊Q8—141no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—141no—
242C4×Q8—141no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247#117—141no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22⋊D711yescharacteristic
252C2×C4⋊C4C22⋊D711yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42C22×D711yescharacteristic
258#107—142no—
259#106—142no—
260#106—142no—
261C4⋊C8—72no—
262C4⋊C8—141no—
263C4⋊C8—141no—
264C4⋊C8—141no—
265#103—142no—
266#103—142no—
267#103—142no—
268#102—142no—
269C22⋊C8—141no—
270C22⋊C8—141no—
271C4×C8—72no—
272C4×C8—141no—
273#95—22no—
274#95—21no—
275#95C22⋊D712yes—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—42no—
279C22×C7—42no—
280C22×C7—42no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7—81no—
284C22×C7—81no—
285C22×C7#39211yescharacteristic
286C22×C7#45011yescharacteristic
287C22×C7#45211yescharacteristic
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—21no—
291C4×C7—21no—
292C4×C7—42no—
293C4×C7—82no—
294C4×C7—81no—
295C4×C7—81no—
296C4×C7#44811yescharacteristic
297C4×C7#44911yescharacteristic
298C4×C7#46511yescharacteristic
299C4×C7#46811yescharacteristic
300Dic7—81no—
301Dic7—81no—
302C24—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—141no—
309C2×Q8—141no—
310C2×Q8#114212yescomplemented
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4C2×C22⋊D711yescharacteristic
325C22×C4C2×C22⋊D711yescharacteristic
326C22×C4C2×C22⋊D711yescharacteristic
327SD16—284no—
328M16—281no—
329M16—281no—
330C2×C8—142no—
331C2×C8—142no—
332C2×C8—281no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—141no—
343C4⋊C4—141no—
344C4⋊C4—141no—
345C4⋊C4—141no—
346C4⋊C4—281no—
347C4⋊C4—281no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—281no—
353C42—41no—
354C42—41no—
355C42—281no—
356C42D4×D711yescharacteristic
357C42D4×D711yescharacteristic
358C42D4×D711yescharacteristic
359C42#115911yescharacteristic
360C2×C7—21no—
361C2×C7—21no—
362C2×C7—82no—
363C2×C7#200011yescharacteristic
364C2×C7#200911yescharacteristic
365C2×C7#201011yescharacteristic
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23#651611yescharacteristic
374Q8—22no—
375Q8—22no—
376Q8—41no—
377Q8—41no—
378Q8—282no—
379Q8—561no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—281no—
402C2×C4—281no—
403C2×C4#649511yescharacteristic
404C2×C4C2×#114211yescharacteristic
405C2×C4#650011yescharacteristic
406C2×C4#650011yescharacteristic
407C2×C4#650111yescharacteristic
408C2×C4#650211yescharacteristic
409C2×C4#650211yescharacteristic
410C2×C4#650311yescharacteristic
411C2×C4#651311yescharacteristic
412C2×C4#651411yescharacteristic
413C8—282no—
414C8—282no—
415C8—561no—
416C7#1000111yescharacteristic, complemented, Sylow
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#8999011yescharacteristic
427C22#8999511yescharacteristic
428C22#9001111yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—561no—
438C4—561no—
439C4#8980911yescharacteristic
440C4#8981211yescharacteristic
441C4#8981311yescharacteristic
442C4#8981411yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2#1172943911yescharacteristic
447C2#1172946911yescharacteristic
448C2#1172962311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2536C2
socleC22×C7#392
prime radicalC22×C7×D4D4
hypercenter#2536D7
hypocenterC7#10001
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #11739876C2semidirect
4. #11729375C2semidirect
6. #11727300C2semidirect
7. #11725437C2semidirect
98. C2×C7×Q8SD16semidirect
310. C2×Q8#1142semidirect
416. C7#10001semidirect

Maximal subgroups

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

subgroupindex
C7×#25362
#117398762
#117397592
#117293752
#117293522
#117273002
#117254372
#100017