Finite Group Databasebeta

← #13337296

Subgroups — #13337296

Order 1792.

652 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3165C211yesmaximal, characteristic, complemented
2#11740808C211yesmaximal, characteristic
3#11740658C211yesmaximal, characteristic
4#11735967C211yesmaximal, characteristic
5#11729582C211yesmaximal, characteristic, complemented
6C2×#89342C211yesmaximal, characteristic, complemented
7#11729127C212yesmaximal, complemented
8#11729125C212yesmaximal, complemented
9#11729099C212yesmaximal, complemented
10#11729098C212yesmaximal, complemented
11#11727604C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×#6470—21no—
15#90168C2211yescharacteristic
16C7×#501—22no—
17C7×#500C2212yes—
18C7×#498C2212yes—
19C7×#428—21no—
20C7×#428—21no—
21C4×C7×M16C2211yescharacteristic
22#89999—22no—
23#89901—21no—
24#89857—21no—
25#89856C2212yes—
26#89855C2212yes—
27#89855C2212yes—
28#89855C2212yes—
29#89853—21no—
30#89820—22no—
31#89799—21no—
32#89792—21no—
33#89697C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36#89478—22no—
37#89460C2212yes—
38C8×D28C2212yes—
39#89342C2212yes—
40#89342C2212yes—
41#89342C2211yescharacteristic
42#89342C2211yescharacteristic
43#89341C2211yescharacteristic
44#89341C2211yescharacteristic
45#89253C2212yes—
46#89252C2212yes—
47#19232—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C2×#1157—21no—
50C2×C7×M16D411yescharacteristic
51C2×C7×M16D411yescharacteristic
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8C2311yescharacteristic
56C7×C4⋊Q8C2312yes—
57C7×C4⋊Q8C2311yescharacteristic
58C7×#125—21no—
59C7×C22⋊Q8—21no—
60C7×C22⋊Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8—21no—
63C7×#117—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C42×C7C2311yescharacteristic
67C2×#1143—22no—
68C2×#1143—22no—
69C2×#1143—22no—
70#6493—21no—
71#6493D412yes—
72C22×C7⋊C8—21no—
73C22×C7⋊C8—21no—
74#6470—44no—
75C2×#1131—22no—
76C2×C8×D7—22no—
77#6438—21no—
78#6437D411yescharacteristic
79C4×D28—21no—
80C4×D28C2312yes—
81#6432D411yescharacteristic
82C7×#107—22no—
83C7×#107—22no—
84C7×#106—22no—
85C7×#106—22no—
86C7×#97D412yes—
87C4×C7×C8C2312yes—
88#6409—22no—
89#6409—22no—
90#6409—22no—
91C22⋊(C7⋊C8)—21no—
92(C2×D7)⋊C8—22no—
93(C2×D7)⋊C8—22no—
94#6383—22no—
95#6382—22no—
96C4⋊(C7⋊C8)—21no—
97C4⋊(C7⋊C8)—21no—
98C4⋊(C7⋊C8)C2312yes—
99C4⋊(C7⋊C8)C2311yescharacteristic
100C4⋊(C7⋊C8)C2311yescharacteristic
101C4⋊(C7⋊C8)C2311yescharacteristic
102C4⋊(C7⋊C8)C2311yescharacteristic
103C4×C7⋊C8—21no—
104#3246—71no—
105#3206—71no—
106#3165D711yescharacteristic, complemented
107#2935—71no—
108#1693—72no—
109#1691—72no—
110#1673—72no—
111#1671—72no—
112#1565—71no—
113#1542—71no—
114C2×#332—71no—
115C2×C7×Q8—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—21no—
118C2×C7×Q8—41no—
119C2×C7×Q8—81no—
120C2×C7×Q8—81no—
121C22×C4×C7—41no—
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7#4111yescharacteristic
125C2×C22⋊D7—22no—
126#1157—42no—
127#1157—42no—
128C2×D28—21no—
129C2×D28—21no—
130C2×D28C2×D412yes—
131C2×C4×D7—22no—
132C2×Dic14—21no—
133C7×M16—22no—
134C7×M16SD1614yescomplemented
135C2×C7×C8C2×D412yes—
136C2×C7×C8C2×D412yes—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—21no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4—41no—
146C7×C4⋊C4—41no—
147C7×C22⋊C4—41no—
148C42×C7—41no—
149C42×C7C2×D411yescharacteristic
150C42×C7C2×D411yescharacteristic
151C42×C7#4111yescharacteristic
152C42×C7C2411yescharacteristic
153#1143—82no—
154#1143—82no—
155#1143—82no—
156(C2×D7)⋊C4—22no—
157(C2×D7)⋊C4—22no—
158C4⋊Dic7—21no—
159C4⋊Dic7C2×D412yes—
160C2×C7⋊C8—21no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—42no—
166C2×C7⋊C8—81no—
167#1131—44no—
168C8×D7—44no—
169C2×#131—141no—
170#533C2×D711yescharacteristic
171#532—71no—
172C2×C4⋊Q8C2×D711yescharacteristic
173#501—22no—
174#500C2×D712yes—
175#498C2×D712yes—
176#492—141no—
177#487—141no—
178#476—72no—
179#475—141no—
180#465—142no—
181#461—72no—
182#461—72no—
183#461—72no—
184#460—141no—
185#448—142no—
186#436—72no—
187#435—142no—
188C8×D4—72no—
189#431—71no—
190#428—21no—
191#428—21no—
192C2×C4⋊C8—71no—
193C2×C4⋊C8—71no—
194#408—141no—
195C4×M16C2×D711yescharacteristic
196#335—72no—
197#334—72no—
198#333—71no—
199#333—71no—
200#332—72no—
201#332—72no—
202#332—71no—
203#332—71no—
204C23×C7#12111yescharacteristic
205C22×D7—22no—
206C7×Q8—41no—
207C7×Q8—42no—
208C7×Q8—42no—
209C7×Q8—41no—
210C7×Q8—81no—
211C7×Q8—82no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—42no—
218C2×C4×C7—82no—
219C2×C4×C7#12111yescharacteristic
220C2×C4×C7#12111yescharacteristic
221C2×C4×C7#12111yescharacteristic
222C2×C4×C7C2×SD1612yes—
223C2×C4×C7#13712yes—
224C2×C4×C7C22×D411yescharacteristic
225C2×C4×C7C22×D411yescharacteristic
226C2×C4×C7C2×#4111yescharacteristic
227C22⋊D7—44no—
228C2×Dic7—22no—
229D28—22no—
230D28—22no—
231D28—42no—
232D28—81no—
233C4×D7—42no—
234C4×D7—42no—
235Dic14—42no—
236C7×C8—22no—
237C7×C8C2×SD1614yes—
238C7⋊C8—41no—
239C7⋊C8—41no—
240C7⋊C8—42no—
241C7⋊C8—41no—
242C7⋊C8—41no—
243C2×#41—141no—
244C22×Q8—21no—
245C2×SD16—142no—
246C2×SD16—142no—
247C2×SD16—142no—
248#131—284no—
249C2×M16—142no—
250C2×M16C22⋊D711yescharacteristic
251C2×M16C22⋊D711yescharacteristic
252C22×C8—141no—
253C22×C8—141no—
254C22×C8—142no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—71no—
259C4⋊Q8C22×D712yes—
260C4⋊Q8C22×D711yescharacteristic
261C4⋊Q8C22×D711yescharacteristic
262#127—71no—
263#125—21no—
264#124—141no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267#121—72no—
268#121—141no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×D4—72no—
272C4×D4—141no—
273#117—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C42C22×D711yescharacteristic
277#107—22no—
278#107—22no—
279#106—22no—
280#106—22no—
281#106—142no—
282C4⋊C8—71no—
283C4⋊C8—71no—
284C4⋊C8—72no—
285C4⋊C8—71no—
286C4⋊C8—71no—
287C4⋊C8—141no—
288C4⋊C8—141no—
289#103—142no—
290#103—142no—
291#103—142no—
292#102—142no—
293C22⋊C8—141no—
294C22⋊C8—142no—
295C22⋊C8—142no—
296#97C22⋊D712yes—
297C4×C8—141no—
298C4×C8C22×D712yes—
299C22×C7—21no—
300C22×C7—21no—
301C22×C7#46112yes—
302C22×C7C2×#12111yescharacteristic
303C2×D7—42no—
304C2×D7—42no—
305C2×D7—42no—
306C4×C7—21no—
307C4×C7—21no—
308C4×C7—21no—
309C4×C7—82no—
310C4×C7—81no—
311C4×C7—81no—
312C4×C7#46112yes—
313C4×C7C22×SD1611yescharacteristic
314C4×C7C2×#13711yescharacteristic
315C4×C7#57611yescharacteristic
316C4×C7#57711yescharacteristic
317Dic7—42no—
318#41—282no—
319#41—282no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8—141no—
327C2×D4—72no—
328C2×D4—142no—
329C2×D4—141no—
330C2×D4—141no—
331C22×C4—41no—
332C22×C4—142no—
333C22×C4D4×D711yescharacteristic
334C22×C4#115911yescharacteristic
335C22×C4C2×C22⋊D711yescharacteristic
336SD16—562no—
337SD16—562no—
338SD16—562no—
339M16—22no—
340M16—284no—
341M16#114314yescomplemented
342C2×C8—141no—
343C2×C8—141no—
344C2×C8—142no—
345C2×C8—141no—
346C2×C8—141no—
347C2×C8—282no—
348C2×C8—284no—
349C2×C8—561no—
350C2×C8C2×C22⋊D712yes—
351C2×C8C2×C22⋊D712yes—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—72no—
363C4⋊C4—141no—
364C22⋊C4—41no—
365C22⋊C4—142no—
366C22⋊C4—142no—
367C42—41no—
368C42D4×D711yescharacteristic
369C42#115911yescharacteristic
370C42C2×C22⋊D711yescharacteristic
371C42C23×D711yescharacteristic
372C2×C7—22no—
373C2×C7C2×#46111yescharacteristic
374C2×C7#306211yescharacteristic
375C2×C7#306911yescharacteristic
376D7—82no—
377C23—142no—
378C23#650111yescharacteristic
379Q8—41no—
380Q8—42no—
381Q8—42no—
382Q8—41no—
383Q8—81no—
384Q8—82no—
385Q8—282no—
386D4—142no—
387D4—142no—
388D4—284no—
389D4—282no—
390D4—561no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—142no—
399C2×C4—282no—
400C2×C4—282no—
401C2×C4#650111yescharacteristic
402C2×C4#650111yescharacteristic
403C2×C4#650111yescharacteristic
404C2×C4C2×#114312yes—
405C2×C4#650512yes—
406C2×C4C2×D4×D711yescharacteristic
407C2×C4C2×#115911yescharacteristic
408C2×C4C22×C22⋊D711yescharacteristic
409C8—22no—
410C8—282no—
411C8—281no—
412C8—281no—
413C8—281no—
414C8—281no—
415C8C2×#114314yes—
416C7#1923211yescharacteristic, complemented, Sylow
417C22—21no—
418C22—21no—
419C22—282no—
420C22—282no—
421C22—282no—
422C22#8985512yes—
423C22C2×#650111yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—82no—
429C4—81no—
430C4—282no—
431C4#8985512yes—
432C4#9046511yescharacteristic
433C4#9046611yescharacteristic
434C4C22×#114311yescharacteristic
435C4C2×#650511yescharacteristic
436C2—22no—
437C2—562no—
438C2C2×#8985511yescharacteristic
439C2#1174139611yescharacteristic
440C2#1174144311yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3165C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#3165D7
hypocenterC7#19232
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. C7×#3165C2semidirect
5. #11729582C2semidirect
6. C2×#89342C2semidirect
7. #11729127C2semidirect
8. #11729125C2semidirect
9. #11729099C2semidirect
10. #11729098C2semidirect
11. #11727604C2semidirect
106. #3165D7semidirect
134. C7×M16SD16semidirect
341. M16#1143semidirect
416. C7#19232semidirect

Maximal subgroups

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

subgroupindex
C7×#31652
#117408082
#117406582
#117359672
#117295822
C2×#893422
#117291272
#117291252
#117290992
#117290982
#117276042
#192327