Finite Group Databasebeta

← #13321546

Subgroups — #13321546

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743326C211yesmaximal, characteristic
2#11736708C211yesmaximal, characteristic
3#11731082C211yesmaximal, characteristic
4C7×#1503C211yesmaximal, characteristic
5#11729592C211yesmaximal, characteristic
6#11728547C211yesmaximal, characteristic
7#11727225C211yesmaximal, characteristic
8#11726068C214yesmaximal
9#11726063C214yesmaximal
10C7×#533C2211yescharacteristic
11#90413C412yescomplemented
12#90412C412yescomplemented
13#90376—22no—
14#90314—22no—
15#90313C2214yes—
16#90258—22no—
17#90258C412yescomplemented
18Q8×Dic14C412yescomplemented
19#90210—21no—
20#90207—21no—
21#90161C2211yescharacteristic
22C2×C4×Dic14—21no—
23C2×C4×Dic14C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26#89744—22no—
27#89708C2211yescharacteristic
28#89700—21no—
29#89697C2211yescharacteristic
30#89602—22no—
31#89490C2214yes—
32C8×Dic14—22no—
33#89422C2212yes—
34#89421C2212yes—
35C7×#328C2212yes—
36C7×#328C2212yes—
37#89333C2214yes—
38#89262C2214yes—
39#89247C2212yes—
40#89247C2212yes—
41#17619—71nomaximal, Sylow
42C22×Dic14D411yescharacteristic
43C7×C4⋊Q8C2×C411yescharacteristic
44C7×C4⋊Q8C2×C411yescharacteristic
45C7×#125—21no—
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8C2312yes—
48C4×C7×Q8—21no—
49C4×C7×Q8C2×C412yes—
50C7×#117—21no—
51C2×C7×C4⋊C4—21no—
52C2×C42×C7—21no—
53C2×C42×C7C2311yescharacteristic
54Q8×Dic7C2×C414yes—
55#6507—24no—
56#6507C2×C414yes—
57#6489—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7D411yescharacteristic
60#6487—21no—
61#6487C2312yes—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4D412yes—
65C2×C4×Dic7—22no—
66C2×C4×Dic7C2312yes—
67#6453—22no—
68#6452—24no—
69#6452—24no—
70#6452—24no—
71#6451—22no—
72#6451—24no—
73#6451C2×C414yes—
74#6450—22no—
75#6450—22no—
76#6450—24no—
77#6450C2×C414yes—
78#6441—24no—
79#6440—24no—
80#6433—21no—
81#6433—22no—
82#6432—22no—
83#6432C2×C411yescharacteristic
84#6432C2×C411yescharacteristic
85C4×Dic14—22no—
86C4×Dic14—22no—
87C4×Dic14—21no—
88C4×Dic14—44no—
89C4×Dic14C2×C412yes—
90C4×Dic14C2×C412yes—
91C4×Dic14C2×C412yes—
92C4×Dic14D412yes—
93C4×Dic14D412yes—
94C7×C4⋊C8—21no—
95C7×C22⋊C8C2312yes—
96C4×C7×C8—21no—
97C7×#95C2311yescharacteristic
98C7×#95C2311yescharacteristic
99C7⋊#95—21no—
100C7⋊#95—22no—
101C7⋊#95C2312yes—
102C22⋊(C7⋊C8)C2312yes—
103Dic7⋊C8—24no—
104C8×Dic7—24no—
105C4⋊(C7⋊C8)—21no—
106C4×C7⋊C8—21no—
107#3502—71no—
108#2982—71no—
109#2351—71no—
110#1518—71no—
111#1518—74no—
112#1518—71no—
113#1503—71no—
114#1503—74no—
115#1503D711yescharacteristic
116C2×C7×Q8—21no—
117C2×C7×Q8C22×C412yes—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2411yescharacteristic
123C22×Dic7—22no—
124C22×Dic7C2×D412yes—
125C22×Dic7C2×D412yes—
126C2×Dic14—21no—
127C2×Dic14—24no—
128C2×Dic14C22⋊C412yes—
129C2×Dic14C22⋊C412yes—
130C2×Dic14SD1614yes—
131C2×Dic14Q1614yes—
132C2×Dic14C22×C412yes—
133C2×C7×C8—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4C22×C412yes—
141C7×C22⋊C4—22no—
142C42×C7—22no—
143C42×C7—42no—
144C42×C7C22×C411yescharacteristic
145C42×C7C22×C411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7C2×D411yescharacteristic
148C22⋊Dic7—22no—
149C4⋊Dic7—21no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—24no—
153C4⋊Dic7—42no—
154C4⋊Dic7C22⋊C412yes—
155C4⋊Dic7C22⋊C412yes—
156C4⋊Dic7C22×C412yes—
157Dic7⋊C4—22no—
158Dic7⋊C4—22no—
159Dic7⋊C4—24no—
160Dic7⋊C4—24no—
161Dic7⋊C4—24no—
162Dic7⋊C4—24no—
163Dic7⋊C4—44no—
164Dic7⋊C4—44no—
165Dic7⋊C4C22⋊C414yes—
166Dic7⋊C4C22⋊C414yes—
167C4×Dic7—22no—
168C4×Dic7—24no—
169C4×Dic7—24no—
170C4×Dic7—44no—
171C4×Dic7C22×C414yes—
172C4×Dic7C2×D414yes—
173C2×C7⋊C8—22no—
174Q82—72no—
175Q82—72no—
176#557—72no—
177#557—72no—
178#557—142no—
179#557—142no—
180#544—141no—
181#544—142no—
182#533—74no—
183#533—71no—
184#533C2×D711yescharacteristic
185#519—141no—
186C2×C4×Q8—71no—
187C2×C4×Q8—141no—
188C8×Q8—142no—
189C8×Q8—142no—
190#431—71no—
191#431—74no—
192#431C2×D711yescharacteristic
193#389—141no—
194#389—142no—
195#382—71no—
196#382—72no—
197C4×C4⋊C4—72no—
198C4×C4⋊C4C2×D711yescharacteristic
199#328—74no—
200#328—74no—
201#328—72no—
202#328—72no—
203#328C2×D712yes—
204#328C2×D712yes—
205C23×C7C22×D411yescharacteristic
206C7×Q8—42no—
207C7×Q8—81no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7C2×C22⋊C411yescharacteristic
219C2×C4×C7C2×C22⋊C411yescharacteristic
220C2×C4×C7C2×C22⋊C411yescharacteristic
221C2×C4×C7C2×C22⋊C411yescharacteristic
222C2×C4×C7C2×SD1612yes—
223C2×C4×C7C2×Q1612yes—
224C2×C4×C7C23×C411yescharacteristic
225C2×C4×C7C22×D411yescharacteristic
226C2×Dic7—22no—
227C2×Dic7—22no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—42no—
231C2×Dic7—42no—
232C2×Dic7C2×C22⋊C414yes—
233C2×Dic7C2×C22⋊C414yes—
234C2×Dic7C2×SD1614yes—
235C2×Dic7C2×Q1614yes—
236Dic14—42no—
237Dic14—84no—
238Dic14—81no—
239Dic14#10318yes—
240Dic14#10318yes—
241C7×C8—42no—
242C7⋊C8—42no—
243C22×Q8—71no—
244C4⋊Q8—74no—
245C4⋊Q8—74no—
246C4⋊Q8—71no—
247C4⋊Q8—71no—
248C4⋊Q8—142no—
249C4⋊Q8—144no—
250C4⋊Q8—142no—
251C4⋊Q8—144no—
252C4⋊Q8C4×D711yescharacteristic
253C4⋊Q8C4×D711yescharacteristic
254#125—21no—
255#125—142no—
256#125—144no—
257#125—144no—
258#125—144no—
259#125—142no—
260#125—141no—
261C22⋊Q8—21no—
262C22⋊Q8—72no—
263C22⋊Q8—144no—
264C22⋊Q8—141no—
265C22⋊Q8C22×D712yes—
266C4×Q8—21no—
267C4×Q8—72no—
268C4×Q8—72no—
269C4×Q8—72no—
270C4×Q8—72no—
271C4×Q8—74no—
272C4×Q8—74no—
273C4×Q8—72no—
274C4×Q8—142no—
275C4×Q8—142no—
276C4×Q8—144no—
277C4×Q8—142no—
278C4×Q8—142no—
279C4×Q8—141no—
280C4×Q8—284no—
281C4×Q8C4×D712yes—
282#117—21no—
283#117—144no—
284#117—141no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—72no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—142no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4—141no—
291C2×C42—21no—
292C2×C42—72no—
293C2×C42—142no—
294C2×C42C22×D711yescharacteristic
295C4⋊C8—21no—
296C4⋊C8—144no—
297C4⋊C8—141no—
298C22⋊C8—72no—
299C22⋊C8C22×D712yes—
300C4×C8—21no—
301C4×C8—144no—
302C4×C8—141no—
303#95—72no—
304#95—142no—
305#95—141no—
306#95C22×D711yescharacteristic
307#95C22×D711yescharacteristic
308C22×C7—21no—
309C22×C7—21no—
310C22×C7C22×C22⋊C411yescharacteristic
311C22×C7C22×SD1611yescharacteristic
312C22×C7C22×Q1611yescharacteristic
313C4×C7—21no—
314C4×C7—21no—
315C4×C7—21no—
316C4×C7—21no—
317C4×C7—42no—
318C4×C7—42no—
319C4×C7—42no—
320C4×C7—81no—
321C4×C7C2×#10312yes—
322C4×C7C2×#10312yes—
323Dic7—24no—
324Dic7—42no—
325Dic7—42no—
326Dic7—82no—
327Dic7C2×#10318yes—
328C2×Q8—21no—
329C2×Q8—72no—
330C2×Q8—72no—
331C2×Q8—74no—
332C2×Q8—74no—
333C2×Q8—72no—
334C2×Q8—144no—
335C2×Q8—141no—
336C2×Q8C2×C4×D712yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—72no—
340C22×C4—72no—
341C22×C4—142no—
342C22×C4D4×D711yescharacteristic
343C22×C4D4×D711yescharacteristic
344C22×C4C23×D711yescharacteristic
345C2×C8—22no—
346C2×C8—142no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—74no—
354C4⋊C4—74no—
355C4⋊C4—72no—
356C4⋊C4—72no—
357C4⋊C4—72no—
358C4⋊C4—144no—
359C4⋊C4—144no—
360C4⋊C4—144no—
361C4⋊C4—144no—
362C4⋊C4—144no—
363C4⋊C4—142no—
364C4⋊C4—141no—
365C4⋊C4—141no—
366C4⋊C4—141no—
367C4⋊C4—142no—
368C4⋊C4—284no—
369C4⋊C4—284no—
370C4⋊C4—282no—
371C4⋊C4C2×C4×D712yes—
372C22⋊C4—22no—
373C22⋊C4—142no—
374C42—22no—
375C42—42no—
376C42—74no—
377C42—74no—
378C42—144no—
379C42—144no—
380C42—142no—
381C42—284no—
382C42C2×C4×D711yescharacteristic
383C42C2×C4×D711yescharacteristic
384C42D4×D711yescharacteristic
385C42D4×D711yescharacteristic
386C2×C7—21no—
387C2×C7—21no—
388C2×C7#288411yescharacteristic
389C2×C7C22×#10311yescharacteristic
390C2×C7#290111yescharacteristic
391C23C2×D4×D711yescharacteristic
392Q8—42no—
393Q8—78no—
394Q8—78no—
395Q8—81no—
396Q8—282no—
397Q8—561no—
398Q8—564no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—74no—
410C2×C4—74no—
411C2×C4—74no—
412C2×C4—74no—
413C2×C4—142no—
414C2×C4—142no—
415C2×C4—282no—
416C2×C4—282no—
417C2×C4—282no—
418C2×C4—282no—
419C2×C4D7×C22⋊C411yescharacteristic
420C2×C4D7×C22⋊C411yescharacteristic
421C2×C4D7×C22⋊C411yescharacteristic
422C2×C4D7×C22⋊C411yescharacteristic
423C2×C4D7×SD1612yes—
424C2×C4D7×Q1612yes—
425C2×C4C22×C4×D711yescharacteristic
426C2×C4C2×D4×D711yescharacteristic
427C8—42no—
428C8—282no—
429C7#1761911yescharacteristic, complemented, Sylow
430C22—21no—
431C22—21no—
432C22C2×D7×C22⋊C411yescharacteristic
433C22C2×D7×SD1611yescharacteristic
434C22C2×D7×Q1611yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—78no—
443C4—81no—
444C4—144no—
445C4—282no—
446C4—282no—
447C4—562no—
448C4D7×#10312yes—
449C4D7×#10312yes—
450C2—21no—
451C2—21no—
452C2#1173613411yescharacteristic
453C2C2×D7×#10311yescharacteristic
454C2#1173642811yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1503C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1503D7
hypocenterC7#17619
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
11. #90413C4semidirect
12. #90412C4semidirect
17. #90258C4semidirect
18. Q8×Dic14C4semidirect
429. C7#17619semidirect

Maximal subgroups

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

subgroupindex
#117433262
#117367082
#117310822
C7×#15032
#117295922
#117285472
#117272252
#117260682
#117260632
#176197