Finite Group Databasebeta

← #13249357

Subgroups — #13249357

Order 1792.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2855C211yesmaximal, characteristic
2C2×#89697C211yesmaximal, characteristic
3C2×#89333C212yesmaximal
4#11729288C211yesmaximal, characteristic
5#11725430C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C42×C7C2211yescharacteristic
8C7×#391C411yescharacteristic, complemented
9C7×#391C411yescharacteristic, complemented
10C7×#389C411yescharacteristic, complemented
11C7×#389C411yescharacteristic, complemented
12C7×#387—21no—
13C7×#387—21no—
14C7×#385—21no—
15C7×#381—21no—
16C7×#380C2211yescharacteristic
17C2×C7⋊#95—21no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21#89697—24no—
22C2×C4⋊(C7⋊C8)—21no—
23C2×C4×C7⋊C8—21no—
24#89333—24no—
25#9112—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C23×Dic7—22no—
31C7×C22⋊Q8—22no—
32C7×C22⋊Q8—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C42×C7—22no—
42C2×C42×C7—22no—
43C2×C42×C7C2×C411yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C22×C7⋊C8—22no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7⋊#95—24no—
56C7⋊#95—22no—
57C7⋊#95—22no—
58C7⋊#95—42no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)D414yes—
61C4⋊(C7⋊C8)—24no—
62C4×C7⋊C8—24no—
63C2×#431—71no—
64#2855D711yescharacteristic
65C2×#328—72no—
66#1434—71no—
67#1294—72no—
68C24×C7C2×D411yescharacteristic
69C2×C7×Q8—22no—
70C2×C7×Q8—41no—
71C2×C7×Q8—41no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7SD1612yes—
88C22×C4×C7Q1612yes—
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7#4112yes—
93C22×Dic7—24no—
94C22×Dic7—42no—
95C22×Dic7—42no—
96C22×Dic7—42no—
97C22×Dic7—42no—
98C22×Dic7—42no—
99C22×Dic7—42no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—42no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C42×C7—22no—
110C42×C7—22no—
111C42×C7—21no—
112C42×C7—21no—
113C42×C7—21no—
114C42×C7—21no—
115C2×C7⋊C8—24no—
116C2×C7⋊C8—42no—
117C2×C7⋊C8—42no—
118C2×C22⋊Q8C2×D711yescharacteristic
119C22×C42C2×D711yescharacteristic
120#431—144no—
121C2×C4⋊C8—141no—
122C2×C22⋊C8—72no—
123C2×C4×C8—141no—
124#391Dic711yescharacteristic, complemented
125#391Dic711yescharacteristic, complemented
126#389Dic711yescharacteristic, complemented
127#389Dic711yescharacteristic, complemented
128#387—21no—
129#387—21no—
130#385—21no—
131#381—21no—
132#380C2×D711yescharacteristic
133C2×#95—72no—
134C2×#95—141no—
135C2×#95—141no—
136#328—144no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7C2×C22⋊C411yescharacteristic
142C23×C7#12112yes—
143C23×C7C2×SD1611yescharacteristic
144C23×C7C2×Q1611yescharacteristic
145C23×C7#13512yes—
146C7×Q8—82no—
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—24no—
158C2×C4×C7—24no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—44no—
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—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7#10312yes—
178C2×C4×C7#10312yes—
179C2×C4×C7#10412yes—
180C2×C4×C7C4×D412yes—
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12311yescharacteristic
183C2×Dic7—44no—
184C2×Dic7—44no—
185C2×Dic7—44no—
186C2×Dic7—82no—
187C2×Dic7—82no—
188C2×Dic7—82no—
189C2×Dic7—82no—
190C7⋊C8—44no—
191C22×Q8C2×Dic711yescharacteristic
192C23×C4—142no—
193C23×C4C22⋊D711yescharacteristic
194C23×C4C22⋊D711yescharacteristic
195C23×C4C22×D711yescharacteristic
196C22×C8—142no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic711yescharacteristic
203C2×C4⋊C4C2×Dic711yescharacteristic
204C2×C4⋊C4C2×Dic711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C2×Dic711yescharacteristic
210C2×C42C2×Dic711yescharacteristic
211C2×C42C22⋊D711yescharacteristic
212C2×C42C22⋊D711yescharacteristic
213C4⋊C8—144no—
214C22⋊C8—74no—
215C22⋊C8—142no—
216C4×C8—144no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—144no—
225#95—142no—
226#95—142no—
227#95—282no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—22no—
236C22×C7#38611yescharacteristic
237C22×C7#41011yescharacteristic
238C22×C7#41011yescharacteristic
239C22×C7C2×#10311yescharacteristic
240C22×C7#41611yescharacteristic
241C22×C7C2×#10411yescharacteristic
242C22×C7#42111yescharacteristic
243C22×C7#46512yes—
244C22×C7#46712yes—
245C4×C7—22no—
246C4×C7—22no—
247C4×C7—22no—
248C4×C7—21no—
249C4×C7—22no—
250C4×C7—21no—
251C4×C7—41no—
252C4×C7—81no—
253C4×C7—81no—
254C4×C7—81no—
255C4×C7—81no—
256Dic7—84no—
257C24C2×C22⋊D711yescharacteristic
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—144no—
273C22×C4—282no—
274C22×C4—282no—
275C22×C4—282no—
276C22×C4—282no—
277C22×C4—282no—
278C22×C4—282no—
279C22×C4#114312yes—
280C22×C4#114412yes—
281C22×C4C22⋊Dic711yescharacteristic
282C22×C4C22⋊Dic711yescharacteristic
283C22×C4C22⋊Dic711yescharacteristic
284C22×C4C22⋊Dic711yescharacteristic
285C22×C4D4×D712yes—
286C22×C4#115912yes—
287C22×C4C22×Dic711yescharacteristic
288C22×C4C2×C22⋊D711yescharacteristic
289C2×C8—144no—
290C2×C8—282no—
291C2×C8—282no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C2×C7—22no—
308C2×C7—22no—
309C2×C7#150011yescharacteristic
310C2×C7#150311yescharacteristic
311C2×C7#151811yescharacteristic
312C2×C7#152111yescharacteristic
313C2×C7#187111yescharacteristic
314C2×C7#189211yescharacteristic
315C2×C7#189911yescharacteristic
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#650212yes—
321C23C2×#114311yescharacteristic
322C23C2×#114411yescharacteristic
323C23#651312yes—
324C23C2×C22⋊Dic711yescharacteristic
325Q8—82no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
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—284no—
357C2×C4—284no—
358C2×C4—284no—
359C2×C4—562no—
360C2×C4—562no—
361C2×C4—562no—
362C2×C4—562no—
363C2×C4#640912yes—
364C2×C4#640912yes—
365C2×C4#641112yes—
366C2×C4D4×Dic712yes—
367C2×C4#649811yescharacteristic
368C2×C4#650011yescharacteristic
369C8—284no—
370C7#911211yescharacteristic, complemented, Sylow
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22#8982012yes—
380C22#8982312yes—
381C22#8999211yescharacteristic
382C22C2×#640911yescharacteristic
383C22#9000511yescharacteristic
384C22C2×#641111yescharacteristic
385C22#9000811yescharacteristic
386C22#9001211yescharacteristic
387C22#9001211yescharacteristic
388C4—22no—
389C4—22no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—564no—
400C2—22no—
401C2—22no—
402C2#1172945011yescharacteristic
403C2#1172945911yescharacteristic
404C2#1172946711yescharacteristic
405C2#1172950811yescharacteristic
406C2#1172951111yescharacteristic
407C2#1172958911yescharacteristic
408C2#1172959211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2855C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2855D7
hypocenterC7#9112
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
8. C7×#391C4semidirect
9. C7×#391C4semidirect
10. C7×#389C4semidirect
11. C7×#389C4semidirect
124. #391Dic7semidirect
125. #391Dic7semidirect
126. #389Dic7semidirect
127. #389Dic7semidirect
370. C7#9112semidirect

Maximal subgroups

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

subgroupindex
C7×#28552
C2×#896972
C2×#893332
#117292882
#117254302
#91127