Finite Group Databasebeta

← #13253876

Subgroups — #13253876

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2851C211yesmaximal, characteristic
2#11729289C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725430C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C22×C42×C7C2211yescharacteristic
8C22×C4⋊Dic7—21no—
9C7×#389C411yescharacteristic, complemented
10C7×#389C411yescharacteristic, complemented
11C7×#387—21no—
12C7×#387—21no—
13C7×#385—21no—
14C7×#385—21no—
15C7×#382C411yescharacteristic, complemented
16C7×#380C2211yescharacteristic
17C7×#380C2211yescharacteristic
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21C2×#6377—21no—
22#89319—21no—
23#9887—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—22no—
28C7×#125—42no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7—42no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C4⋊Dic7—24no—
45C2×C4⋊Dic7—42no—
46C2×C4⋊Dic7—42no—
47C22×C7⋊C8—22no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7⋊#95—24no—
55C7⋊#95—42no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)Q814yes—
58#6377—42no—
59#2851D711yescharacteristic
60#1749—71no—
61C4⋊#95—71no—
62#1294—72no—
63#1294—72no—
64C24×C7C2×D411yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—41no—
76C22×C4×C7—41no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×Q812yes—
86C22×C4×C7#4112yes—
87C22×Dic7—24no—
88C22×Dic7—42no—
89C22×Dic7—42no—
90C22×Dic7—42no—
91C22×Dic7—42no—
92C22×Dic7—42no—
93C22×Dic7—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4—42no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C42×C7—22no—
107C42×C7—22no—
108C42×C7—41no—
109C42×C7—41no—
110C4⋊Dic7—48no—
111C2×C7⋊C8—24no—
112C2×C7⋊C8—42no—
113C2×C7⋊C8—42no—
114C2×#125Dic711yescharacteristic, complemented
115C22×C4⋊C4—141no—
116C22×C42C2×D711yescharacteristic
117C2×C22⋊C8—72no—
118C2×#97—141no—
119#389Dic711yescharacteristic, complemented
120#389Dic711yescharacteristic, complemented
121#387—21no—
122#387—21no—
123#385—21no—
124#385—21no—
125#382Dic711yescharacteristic, complemented
126#380C2×D711yescharacteristic
127#380C2×D711yescharacteristic
128C2×#95—72no—
129C2×#95—141no—
130#336—141no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7C2×C22⋊C411yescharacteristic
136C23×C7#12411yescharacteristic
137C23×C7C4⋊Q811yescharacteristic
138C23×C7AGL(1,ℤ/8)11yescharacteristic
139C23×C7AGL(1,ℤ/8)11yescharacteristic
140C23×C7#13711yescharacteristic
141C23×C7#13711yescharacteristic
142C2×C4×C7—24no—
143C2×C4×C7—24no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7#9912yes—
175C2×C4×C7#10412yes—
176C2×C4×C7C4×Q812yes—
177C2×C4×C7C22⋊Q812yes—
178C2×Dic7—44no—
179C2×Dic7—44no—
180C2×Dic7—44no—
181C2×Dic7—82no—
182C2×Dic7—82no—
183C2×Dic7—82no—
184C2×Dic7—82no—
185C7⋊C8—44no—
186C23×C4—142no—
187C23×C4C22⋊D711yescharacteristic
188C23×C4C22⋊D711yescharacteristic
189C23×C4C22×D711yescharacteristic
190C22×C8—142no—
191#125—42no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—144no—
197C2×C4⋊C4—282no—
198C2×C4⋊C4—282no—
199C2×C4⋊C4C2×Dic711yescharacteristic
200C2×C4⋊C4C2×Dic711yescharacteristic
201C2×C4⋊C4C2×Dic711yescharacteristic
202C2×C4⋊C4C2×Dic711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—42no—
206C2×C42C2×Dic711yescharacteristic
207C2×C42C2×Dic711yescharacteristic
208C2×C42C22⋊D711yescharacteristic
209C2×C42C22⋊D711yescharacteristic
210C22⋊C8—74no—
211C22⋊C8—142no—
212#97—282no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—144no—
220#95—282no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7#39111yescharacteristic
234C22×C7C2×#9911yescharacteristic
235C22×C7#41011yescharacteristic
236C22×C7#41711yescharacteristic
237C22×C7#41711yescharacteristic
238C22×C7C2×#10411yescharacteristic
239C22×C7#42111yescharacteristic
240C22×C7#49011yescharacteristic
241C22×C7#49111yescharacteristic
242C22×C7#50111yescharacteristic
243C22×C7#50111yescharacteristic
244C4×C7—22no—
245C4×C7—22no—
246C4×C7—22no—
247C4×C7—22no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—81no—
253C4×C7—81no—
254Dic7—84no—
255C24C2×C22⋊D711yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—144no—
271C22×C4—282no—
272C22×C4—282no—
273C22×C4—282no—
274C22×C4—282no—
275C22×C4—282no—
276C22×C4—282no—
277C22×C4C22⋊Dic711yescharacteristic
278C22×C4C22⋊Dic711yescharacteristic
279C22×C4C22⋊Dic711yescharacteristic
280C22×C4C22⋊Dic711yescharacteristic
281C22×C4Q8×D712yes—
282C22×C4#116112yes—
283C22×C4C22×Dic711yescharacteristic
284C22×C4C2×C22⋊D711yescharacteristic
285C2×C8—144no—
286C2×C8—282no—
287C2×C8—282no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—288no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C2×C7—21no—
306C2×C7—21no—
307C2×C7—21no—
308C2×C7—21no—
309C2×C7#150411yescharacteristic
310C2×C7#150911yescharacteristic
311C2×C7#151211yescharacteristic
312C2×C7#151311yescharacteristic
313C2×C7#198611yescharacteristic
314C2×C7#199211yescharacteristic
315C2×C7#199611yescharacteristic
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#650711yescharacteristic
321C23#651011yescharacteristic
322C23#651211yescharacteristic
323C23#651211yescharacteristic
324C23#651411yescharacteristic
325C23#651411yescharacteristic
326C23C2×C22⋊Dic711yescharacteristic
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—284no—
360C2×C4—284no—
361C2×C4—284no—
362C2×C4—562no—
363C2×C4—562no—
364C2×C4—562no—
365C2×C4—562no—
366C2×C4#640812yes—
367C2×C4#641112yes—
368C2×C4Q8×Dic712yes—
369C2×C4#650912yes—
370C8—284no—
371C7#988711yescharacteristic, complemented, Sylow
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#8983711yescharacteristic
385C22#8983711yescharacteristic
386C22#8983911yescharacteristic
387C22#8984111yescharacteristic
388C22C2×#640811yescharacteristic
389C22#9000211yescharacteristic
390C22#9000411yescharacteristic
391C22#9000411yescharacteristic
392C22C2×#641111yescharacteristic
393C22#9000811yescharacteristic
394C22#9001211yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—564no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#1172945511yescharacteristic
411C2#1172946211yescharacteristic
412C2#1172946311yescharacteristic
413C2#1172949511yescharacteristic
414C2#1172950911yescharacteristic
415C2#1172951011yescharacteristic
416C2#1172959511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2851C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2851D7
hypocenterC7#9887
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
9. C7×#389C4semidirect
10. C7×#389C4semidirect
15. C7×#382C4semidirect
114. C2×#125Dic7semidirect
119. #389Dic7semidirect
120. #389Dic7semidirect
125. #382Dic7semidirect
371. C7#9887semidirect

Maximal subgroups

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

subgroupindex
C7×#28512
#117292892
#117292832
#117254302
#98877