Finite Group Databasebeta

← #13249028

Subgroups — #13249028

Order 1792.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740788C211yesmaximal, characteristic
2#11735967C211yesmaximal, characteristic
3C7×#1986C211yesmaximal, characteristic, complemented
4#11729578C211yesmaximal, characteristic, complemented
5#11727223C211yesmaximal, characteristic
6#11725473C211yesmaximal, characteristic, complemented
7#11725425C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×#6470—21no—
10#90168C2211yescharacteristic
11C4×C7×M16C2211yescharacteristic
12C4×C7×C4⋊C4C2211yescharacteristic
13#89901—21no—
14#89860C411yescharacteristic, complemented
15#89857C411yescharacteristic, complemented
16#89853C411yescharacteristic, complemented
17#89851C411yescharacteristic, complemented
18#89820—22no—
19C4×#1143—22no—
20#89697C2211yescharacteristic
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C2×C4×C7⋊C8C2211yescharacteristic
23#89478—22no—
24#89460—21no—
25C8×D28—21no—
26C7×#339—21no—
27C7×#339—21no—
28Q8⋊(C7⋊C8)—21no—
29Q8⋊(C7⋊C8)—21no—
30#89246—22no—
31#9055—71nomaximal, Sylow
32C2×#1157—21no—
33C2×C7×M16D411yescharacteristic
34C2×C7×M16D411yescharacteristic
35C7×C4⋊Q8C2×C411yescharacteristic
36C7×C4⋊Q8C2×C411yescharacteristic
37C7×#125—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C7×#117—21no—
43C2×C7×C4⋊C4—21no—
44C2×C42×C7—21no—
45C2×C42×C7C2311yescharacteristic
46C2×#1143—22no—
47#6493—21no—
48#6493D412yes—
49C22×C7⋊C8—21no—
50C22×C7⋊C8—21no—
51#6470—42no—
52#6470—42no—
53C2×#1131—22no—
54C2×C8×D7—22no—
55#6438—21no—
56#6437C2×C411yescharacteristic
57C4×D28—21no—
58C4×D28D412yes—
59#6432C2×C411yescharacteristic
60C7×#97—21no—
61C4×C7×C8—21no—
62C7×#95—21no—
63#6409—22no—
64C22⋊(C7⋊C8)—21no—
65(C2×D7)⋊C8—22no—
66(C2×D7)⋊C8—22no—
67#6383—22no—
68#6382—22no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)D411yescharacteristic
72C4⋊(C7⋊C8)D411yescharacteristic
73C4×C7⋊C8—21no—
74C4×C7⋊C8—21no—
75C4×C7⋊C8C2×C411yescharacteristic
76C4×C7⋊C8C2×C411yescharacteristic
77#3167—71no—
78#2935—71no—
79#1986D711yescharacteristic, complemented
80#1502—71no—
81#1481—71no—
82#1279—71no—
83#1278—71no—
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C22×C4×C7—41no—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C2×C22⋊D7—22no—
91#1157—41no—
92#1157—41no—
93#1157—41no—
94#1157—41no—
95C2×D28—21no—
96C2×D28—21no—
97C2×D28C22⋊C412yes—
98C2×C4×D7—22no—
99C2×Dic14—21no—
100C7×M16—21no—
101C7×M16—21no—
102C7×M16—21no—
103C7×M16SD1612yescomplemented
104C2×C7×C8—21no—
105C2×C7×C8—21no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C22⋊C4—41no—
115C42×C7—41no—
116C42×C7—41no—
117C42×C7—41no—
118C42×C7C22×C411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7C2×D411yescharacteristic
121C42×C7#4111yescharacteristic
122#1143—44no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4—22no—
125C4⋊Dic7—21no—
126C4⋊Dic7C22⋊C412yes—
127C2×C7⋊C8—21no—
128C2×C7⋊C8—21no—
129C2×C7⋊C8—21no—
130C2×C7⋊C8—21no—
131C2×C7⋊C8—42no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C7⋊C8—41no—
135#1131—44no—
136C8×D7—44no—
137C2×#131—141no—
138#533C2×D711yescharacteristic
139#532—71no—
140#498—71no—
141#492—71no—
142#487—71no—
143#486—71no—
144#465—142no—
145C4×SD16—142no—
146#436—141no—
147#435—142no—
148C8×D4—141no—
149#431—71no—
150C2×C4⋊C8—71no—
151#408—141no—
152C4×M16C2×D711yescharacteristic
153C2×C4×C8—71no—
154C4×C4⋊C4C2×D711yescharacteristic
155#339—21no—
156#339—21no—
157#327—142no—
158Q8⋊C8—141no—
159Q8⋊C8—141no—
160C23×C7#12111yescharacteristic
161C22×D7—22no—
162C7×Q8—42no—
163C7×Q8—41no—
164C7×Q8—41no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—81no—
172C2×C4×C7—81no—
173C2×C4×C7C2×C22⋊C411yescharacteristic
174C2×C4×C7C4×D411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7#12011yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12311yescharacteristic
179C2×C4×C7C2×SD1611yescharacteristic
180C2×C4×C7#13511yescharacteristic
181C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C7#13711yescharacteristic
183C22⋊D7—42no—
184C22⋊D7—42no—
185C2×Dic7—22no—
186D28—42no—
187D28—41no—
188D28—41no—
189D28—41no—
190D28—41no—
191C4×D7—42no—
192C4×D7—42no—
193Dic14—41no—
194Dic14—41no—
195C7×C8—22no—
196C7×C8—41no—
197C7⋊C8—22no—
198C7⋊C8—41no—
199C7⋊C8—41no—
200C7⋊C8—41no—
201C7⋊C8—81no—
202C2×#41—141no—
203C2×SD16—142no—
204#131—282no—
205#131—282no—
206C2×M16—142no—
207C2×M16C22⋊D711yescharacteristic
208C2×M16C22⋊D711yescharacteristic
209C22×C8—142no—
210C22×C8—141no—
211C22×C8—141no—
212C4⋊Q8—71no—
213C4⋊Q8C4×D711yescharacteristic
214C4⋊Q8C4×D711yescharacteristic
215#127—71no—
216#125—21no—
217#124—141no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220#121—72no—
221#121—141no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×D4—72no—
225C4×D4—141no—
226#117—21no—
227C2×C4⋊C4—21no—
228C2×C42—21no—
229C2×C42C22×D711yescharacteristic
230#106—142no—
231C4⋊C8—71no—
232C4⋊C8—71no—
233C4⋊C8—141no—
234C4⋊C8—141no—
235#103—142no—
236#102—142no—
237C22⋊C8—142no—
238C22⋊C8—142no—
239C22⋊C8—141no—
240#97—21no—
241C4×C8—21no—
242C4×C8—71no—
243C4×C8—71no—
244C4×C8—141no—
245C4×C8—141no—
246#95—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7#38611yescharacteristic
250C22×C7#46111yescharacteristic
251C22×C7#46311yescharacteristic
252C2×D7—42no—
253C2×D7—42no—
254C2×D7—42no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—82no—
259C4×C7—81no—
260C4×C7—81no—
261C4×C7C4×SD1611yescharacteristic
262C4×C7#44011yescharacteristic
263C4×C7#44311yescharacteristic
264C4×C7#44411yescharacteristic
265C4×C7#45011yescharacteristic
266C4×C7#45211yescharacteristic
267Dic7—42no—
268#41—281no—
269#41—281no—
270#41—281no—
271#41—281no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—141no—
275C2×D4—72no—
276C2×D4—142no—
277C2×D4—141no—
278C2×D4—141no—
279C22×C4—41no—
280C22×C4—142no—
281C22×C4D4×D711yescharacteristic
282C22×C4#115911yescharacteristic
283C22×C4C2×C22⋊D711yescharacteristic
284SD16—284no—
285M16—21no—
286M16—21no—
287M16—21no—
288M16—284no—
289M16#114312yescomplemented
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—141no—
293C2×C8—141no—
294C2×C8—141no—
295C2×C8—141no—
296C2×C8—284no—
297C2×C8—282no—
298C2×C8—281no—
299C2×C8—281no—
300C2×C8—281no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—72no—
310C4⋊C4—141no—
311C22⋊C4—41no—
312C22⋊C4—142no—
313C22⋊C4—142no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42C2×C4×D711yescharacteristic
318C42#115711yescharacteristic
319C42D4×D711yescharacteristic
320C42D4×D711yescharacteristic
321C2×C7—21no—
322C2×C7—21no—
323C2×C7#187611yescharacteristic
324C2×C7#187711yescharacteristic
325C2×C7#188411yescharacteristic
326D7—82no—
327C23—142no—
328C23#650111yescharacteristic
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—281no—
333Q8—281no—
334D4—282no—
335D4—282no—
336D4—282no—
337D4—281no—
338D4—281no—
339D4—281no—
340D4—281no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—142no—
350C2×C4—282no—
351C2×C4—282no—
352C2×C4D7×C22⋊C411yescharacteristic
353C2×C4(C22⋊D7)⋊C411yescharacteristic
354C2×C4#644611yescharacteristic
355C2×C4#644711yescharacteristic
356C2×C4C4×C22⋊D711yescharacteristic
357C2×C4#650011yescharacteristic
358C2×C4C2×#114311yescharacteristic
359C2×C4#650511yescharacteristic
360C2×C4#651211yescharacteristic
361C2×C4#651311yescharacteristic
362C8—22no—
363C8—41no—
364C8—142no—
365C8—281no—
366C8—281no—
367C8—281no—
368C8—561no—
369C7#905511yescharacteristic, complemented, Sylow
370C22—21no—
371C22—21no—
372C22—282no—
373C22—282no—
374C22—282no—
375C22#8972811yescharacteristic
376C22#8982911yescharacteristic
377C22#8985511yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—282no—
385C4#8959511yescharacteristic
386C4#8959611yescharacteristic
387C4C4×#114311yescharacteristic
388C4#8979811yescharacteristic
389C4#8981811yescharacteristic
390C4#8981911yescharacteristic
391C2—21no—
392C2—21no—
393C2—562no—
394C2#1172744111yescharacteristic
395C2#1172814411yescharacteristic
396C2#1172817111yescharacteristic
subgroupG/H
centerC2×C4#6500
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1986C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#1986D7
hypocenterC7#9055
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. C7×#1986C2semidirect
4. #11729578C2semidirect
6. #11725473C2semidirect
7. #11725425C2semidirect
14. #89860C4semidirect
15. #89857C4semidirect
16. #89853C4semidirect
17. #89851C4semidirect
79. #1986D7semidirect
103. C7×M16SD16semidirect
289. M16#1143semidirect
369. C7#9055semidirect

Maximal subgroups

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

subgroupindex
#117407882
#117359672
C7×#19862
#117295782
#117272232
#117254732
#117254252
#90557