Finite Group Databasebeta

← #13250049

Subgroups — #13250049

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3101C211yesmaximal, characteristic
2C4×D4×Dic7C211yesmaximal, characteristic
3#11729593C211yesmaximal, characteristic, complemented
4#11729589C211yesmaximal, characteristic, complemented
5#11729352C211yesmaximal, characteristic
6#11725552C211yesmaximal, characteristic, complemented
7#11725542C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C2×D4×Dic7—21no—
11C22×C4×Dic7—21no—
12C7×#483—21no—
13C7×#476C412yescomplemented
14C7×#461C412yescomplemented
15C7×#450—21no—
16C4×C7×SD16—22no—
17C7×#431C2211yescharacteristic
18C4×C22⋊Dic7—21no—
19C4⋊C4×Dic7—21no—
20C22⋊C4×Dic7—22no—
21C4×C4⋊Dic7C2211yescharacteristic
22C42×Dic7C2211yescharacteristic
23#89697C2211yescharacteristic
24#89696C2211yescharacteristic
25#89333—22no—
26D4⋊(C7⋊C8)—22no—
27#89327—21no—
28#89327—21no—
29#9231—71nomaximal, Sylow
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C23×Dic7—22no—
33C2×C7×SD16—22no—
34C7×C4⋊Q8C2×C411yescharacteristic, complemented
35C7×C4⋊Q8C2×C411yescharacteristic, complemented
36C7×#125—21no—
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4C2×C412yes—
44C4×C7×D4D412yes—
45C7×#117—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C22⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—22no—
50D4×Dic7—28no—
51C2×C4⋊Dic7—21no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—22no—
54C2×C4×Dic7—21no—
55C2×C4×Dic7—21no—
56C2×C4×Dic7—42no—
57C2×C4×Dic7—42no—
58C2×C4×Dic7D411yescharacteristic
59C2×C4×Dic7D411yescharacteristic
60C2×#1136—21no—
61C7×#106—22no—
62C7×C4⋊C8C2×C412yescomplemented
63C7×#103—22no—
64C7×#102—22no—
65C7×C22⋊C8—21no—
66C4×C7×C8—21no—
67C7⋊#95—22no—
68C7⋊#95D412yes—
69C22⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)—21no—
73C4×C7⋊C8—21no—
74#3101D711yescharacteristic
75C42×D4—71no—
76#1994—71no—
77#1503—71no—
78#1486—71no—
79#1303—71no—
80#1276—71no—
81C24×C7—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×D4—24no—
85C2×C7×D4—21no—
86C2×C7×D4C22⋊C412yes—
87C22×C4×C7—21no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×Dic7—21no—
98C22×Dic7—28no—
99C22×Dic7—21no—
100C22×Dic7—22no—
101C22×Dic7—44no—
102C22×Dic7—44no—
103C7×SD16—44no—
104C2×C7×C8—22no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—41no—
116C42×C7—41no—
117C42×C7C22×C411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7#4111yescharacteristic
120C42×C7#4111yescharacteristic
121C22⋊Dic7—28no—
122C4⋊Dic7—24no—
123C4×Dic7—24no—
124C4×Dic7—21no—
125C4×Dic7—21no—
126C4×Dic7—21no—
127C4×Dic7—42no—
128C4×Dic7—44no—
129C4×Dic7—41no—
130C4×Dic7—41no—
131C4×Dic7SD1612yescomplemented
132#1136—41no—
133#1136—41no—
134C2×C7⋊C8—41no—
135C2×C7⋊C8—41no—
136#533C2×D711yescharacteristic
137C2×C4×D4—141no—
138C2×C4×D4C2×D711yescharacteristic
139C22×C42—141no—
140#483—21no—
141#476Dic712yescomplemented
142#461Dic712yescomplemented
143#450—21no—
144C4×SD16—22no—
145#431—71no—
146#431C2×D711yescharacteristic
147#423—71no—
148C4×C4⋊C4—71no—
149C4×C4⋊C4—141no—
150C4×C22⋊C4—142no—
151C4×C22⋊C4—141no—
152C43—71no—
153#339—141no—
154#339—141no—
155#328—142no—
156D4⋊C8—142no—
157C23×C7—24no—
158C23×C7—22no—
159C23×C7—42no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7#12011yescharacteristic
163C7×Q8—42no—
164C7×Q8—81no—
165C7×D4—24no—
166C7×D4—22no—
167C7×D4#10414yes—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7C2×C22⋊C411yescharacteristic
180C2×C4×C7C4×D411yescharacteristic
181C2×C4×C7C4×D411yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12311yescharacteristic
185C2×C4×C7C2×SD1611yescharacteristic
186C2×C4×C7#13511yescharacteristic
187C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C7#13711yescharacteristic
189C2×Dic7—22no—
190C2×Dic7—22no—
191C2×Dic7—22no—
192C2×Dic7—22no—
193C2×Dic7—24no—
194C2×Dic7—416no—
195C2×Dic7—41no—
196C2×Dic7—42no—
197C2×Dic7—42no—
198C2×Dic7—41no—
199C7×C8—42no—
200C7⋊C8—81no—
201C7⋊C8—81no—
202C22×D4C22⋊D711yescharacteristic
203C23×C4—21no—
204C23×C4—142no—
205C2×SD16—22no—
206C2×M16—141no—
207C4⋊Q8C2×Dic711yescharacteristic, complemented
208C4⋊Q8C2×Dic711yescharacteristic, complemented
209#125—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—148no—
217C4×D4C2×Dic712yes—
218C4×D4C22⋊D712yes—
219#117—21no—
220C2×C4⋊C4—141no—
221C2×C4⋊C4C22⋊D711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—142no—
224C2×C42—71no—
225C2×C42—71no—
226C2×C42—141no—
227C2×C42—142no—
228C2×C42—141no—
229C2×C42—141no—
230C2×C42—282no—
231C2×C42—282no—
232C2×C42C22×D711yescharacteristic
233#106—22no—
234C4⋊C8—141no—
235C4⋊C8—141no—
236C4⋊C8—141no—
237C4⋊C8C2×Dic712yescomplemented
238#103—22no—
239#102—22no—
240C22⋊C8—21no—
241C22⋊C8—141no—
242C4×C8—21no—
243C4×C8—141no—
244#95—72no—
245#95—142no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—24no—
249C22×C7—44no—
250C22×C7—44no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7#38611yescharacteristic
254C22×C7#45011yescharacteristic
255C22×C7#45211yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—42no—
260C4×C7—81no—
261C4×C7—81no—
262C4×C7C2×#10412yes—
263C4×C7C4×SD1611yescharacteristic
264C4×C7#44011yescharacteristic
265C4×C7#46111yescharacteristic
266C4×C7#46311yescharacteristic
267Dic7—24no—
268Dic7—42no—
269Dic7—44no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×D4—24no—
274C2×D4—21no—
275C2×D4C22⋊Dic712yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—141no—
284C22×C4—148no—
285C22×C4—142no—
286C22×C4—141no—
287C22×C4—284no—
288C22×C4—284no—
289C22×C4D4×D711yescharacteristic
290C22×C4D4×D711yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292SD16—44no—
293M16—281no—
294M16—281no—
295C2×C8—22no—
296C2×C8—281no—
297C2×C8—281no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—144no—
306C4⋊C4C22⋊Dic712yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—148no—
311C42—41no—
312C42—72no—
313C42—144no—
314C42—141no—
315C42—141no—
316C42—141no—
317C42—282no—
318C42—284no—
319C42—281no—
320C42—281no—
321C42#115911yescharacteristic
322C42#115911yescharacteristic
323C42C22×Dic711yescharacteristic
324C42C2×C22⋊D711yescharacteristic
325C2×C7—21no—
326C2×C7—21no—
327C2×C7—44no—
328C2×C7#187711yescharacteristic
329C2×C7#189611yescharacteristic
330C2×C7#190611yescharacteristic
331C23—24no—
332C23—22no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23#650011yescharacteristic
337Q8—42no—
338Q8—81no—
339D4—24no—
340D4—22no—
341D4#641114yes—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—142no—
354C2×C4—142no—
355C2×C4—142no—
356C2×C4—142no—
357C2×C4—144no—
358C2×C4—2816no—
359C2×C4—281no—
360C2×C4—282no—
361C2×C4—282no—
362C2×C4—281no—
363C2×C4D7×SD1611yescharacteristic
364C2×C4#647711yescharacteristic
365C2×C4#647811yescharacteristic
366C2×C4#647911yescharacteristic
367C2×C4D4×Dic711yescharacteristic
368C2×C4D4×Dic711yescharacteristic
369C2×C4#649811yescharacteristic
370C2×C4#650211yescharacteristic
371C2×C4#650211yescharacteristic
372C2×C4C2×C22⋊Dic711yescharacteristic
373C8—42no—
374C8—561no—
375C8—561no—
376C7#923111yescharacteristic, complemented, Sylow
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22#8994111yescharacteristic
385C22#8994511yescharacteristic
386C22#8999211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—144no—
394C4—284no—
395C4—282no—
396C4SD16×Dic711yescharacteristic
397C4#8993411yescharacteristic
398C4#8993611yescharacteristic
399C4#8993711yescharacteristic
400C4C2×#641112yes—
401C2—21no—
402C2—21no—
403C2—44no—
404C2#1172946511yescharacteristic
405C2#1172980111yescharacteristic
406C2#1172986811yescharacteristic
subgroupG/H
centerC2×C4#6502
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3101C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#3101D7
hypocenterC7#9231
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
3. #11729593C2semidirect
4. #11729589C2semidirect
6. #11725552C2semidirect
7. #11725542C2semidirect
13. C7×#476C4semidirect
14. C7×#461C4semidirect
34. C7×C4⋊Q8C2×C4semidirect
35. C7×C4⋊Q8C2×C4semidirect
62. C7×C4⋊C8C2×C4semidirect
131. C4×Dic7SD16semidirect
141. #476Dic7semidirect
142. #461Dic7semidirect
207. C4⋊Q8C2×Dic7semidirect
208. C4⋊Q8C2×Dic7semidirect
237. C4⋊C8C2×Dic7semidirect
376. C7#9231semidirect

Maximal subgroups

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

subgroupindex
C7×#31012
C4×D4×Dic72
#117295932
#117295892
#117293522
#117255522
#117255422
#92317