Finite Group Databasebeta

← #13249031

Subgroups — #13249031

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
1C7×#2935C211yesmaximal, characteristic
2#11740735C211yesmaximal, characteristic
3#11729565C211yesmaximal, characteristic
4#11728183C211yesmaximal, characteristic, complemented
5#11727622C211yesmaximal, characteristic, complemented
6#11725510C211yesmaximal, characteristic, complemented
7#11725473C211yesmaximal, characteristic, complemented
8C2×C7×#131—21no—
9C7×#532C2211yescharacteristic
10#90161C2211yescharacteristic
11C7×#436—21no—
12C7×#435—22no—
13C7×C8×D4—21no—
14C2×C7×C4⋊C8C2211yescharacteristic
15C7×#408—21no—
16C4×C7×M16C2211yescharacteristic
17#89865C411yescharacteristic, complemented
18#89851C411yescharacteristic, complemented
19#89848C411yescharacteristic, complemented
20#89842C411yescharacteristic, complemented
21#89813—22no—
22C4×#1142—22no—
23C4×C4⋊Dic7C2211yescharacteristic
24#89697C2211yescharacteristic
25C2×C4×C7⋊C8C2211yescharacteristic
26#89350—21no—
27#89350—21no—
28#89332—22no—
29#89279—21no—
30#89279—21no—
31#9055—71nomaximal, Sylow
32C2×C7×#41—21no—
33C7×#131—42no—
34C7×#131—42no—
35C2×C7×M16—22no—
36C2×C7×M16D411yescharacteristic
37C2×C7×M16D411yescharacteristic
38C22×C7×C8—22no—
39C22×C7×C8—21no—
40C7×C4⋊Q8C2×C411yescharacteristic
41C7×#127C2×C411yescharacteristic
42C7×#124—21no—
43C7×#121—21no—
44C7×#121D412yes—
45C4×C7×D4—21no—
46C4×C7×D4D412yes—
47C2×C42×C7C2311yescharacteristic
48C2×#1142—22no—
49#6489—21no—
50C2×C4⋊Dic7—21no—
51#6487—21no—
52#6487—21no—
53C2×C4×Dic7—21no—
54C22×C7⋊C8—21no—
55#6433—21no—
56#6432C2×C411yescharacteristic
57#6432C2×C411yescharacteristic
58C4×Dic14—21no—
59C4×Dic14—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8D411yescharacteristic
62C7×C4⋊C8D411yescharacteristic
63C7×C22⋊C8—22no—
64C7×C22⋊C8—22no—
65C7×#97—21no—
66C4×C7×C8—21no—
67#6406—22no—
68C7⋊#95—21no—
69C22⋊(C7⋊C8)—21no—
70#6384—22no—
71#6382—22no—
72C4⋊(C7⋊C8)—21no—
73C4×C7⋊C8—21no—
74C4×C7⋊C8—21no—
75C4×C7⋊C8C2×C411yescharacteristic
76C4×C7⋊C8C2×C411yescharacteristic
77#3167—71no—
78#2935D711yescharacteristic
79#1986—71no—
80#1502—71no—
81#1481—71no—
82#1279—71no—
83#1278—71no—
84C7×#41—41no—
85C7×#41—41no—
86C7×#41—41no—
87C7×#41—41no—
88C2×C7×Q8—21no—
89C2×C7×D4—22no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4C22⋊C412yes—
93C22×C4×C7—22no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7#4111yescharacteristic
97C22×Dic7—41no—
98C2×Dic14—21no—
99C2×Dic14—21no—
100C7×M16—21no—
101C7×M16—21no—
102C7×M16—21no—
103C7×M16—44no—
104C7×M16SD1612yescomplemented
105C2×C7×C8—21no—
106C2×C7×C8—21no—
107C2×C7×C8—21no—
108C2×C7×C8—21no—
109C2×C7×C8—44no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4C22⋊C412yes—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—22no—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7#4111yescharacteristic
120C22⋊Dic7—41no—
121#1142—44no—
122C4⋊Dic7—22no—
123C4⋊Dic7—21no—
124C4⋊Dic7—21no—
125C4⋊Dic7—21no—
126C4⋊Dic7—21no—
127C4⋊Dic7—41no—
128Dic7⋊C4—41no—
129Dic7⋊C4—41no—
130C4×Dic7—41no—
131C4×Dic7—41no—
132C4×Dic7—41no—
133C2×C7⋊C8—21no—
134C2×C7⋊C8—21no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—41no—
137C2×#131—21no—
138#533—71no—
139#532C2×D711yescharacteristic
140#498—71no—
141#492—71no—
142#487—71no—
143#486—71no—
144#465—142no—
145C4×SD16—142no—
146#436—21no—
147#435—22no—
148C8×D4—21no—
149#431—71no—
150C2×C4⋊C8C2×D711yescharacteristic
151#408—21no—
152C4×M16C2×D711yescharacteristic
153C2×C4×C8—71no—
154C4×C4⋊C4—71no—
155#339—141no—
156#339—141no—
157#327—142no—
158Q8⋊C8—141no—
159Q8⋊C8—141no—
160C23×C7—22no—
161C23×C7#12111yescharacteristic
162C7×Q8—41no—
163C7×Q8—41no—
164C7×D4—42no—
165C7×D4—42no—
166C7×D4—42no—
167C7×D4—41no—
168C7×D4—41no—
169C7×D4—41no—
170C7×D4—41no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—22no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7C2×C22⋊C411yescharacteristic
178C2×C4×C7C4×D411yescharacteristic
179C2×C4×C7C4×D411yescharacteristic
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7#12311yescharacteristic
183C2×C4×C7C2×SD1611yescharacteristic
184C2×C4×C7#13511yescharacteristic
185C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C7#13711yescharacteristic
187C2×Dic7—42no—
188C2×Dic7—41no—
189C2×Dic7—41no—
190C2×Dic7—81no—
191C2×Dic7—81no—
192Dic14—42no—
193Dic14—41no—
194Dic14—41no—
195C7×C8—22no—
196C7×C8—41no—
197C7×C8—41no—
198C7×C8—41no—
199C7⋊C8—22no—
200C7⋊C8—41no—
201C7⋊C8—81no—
202C2×#41—21no—
203C2×SD16—142no—
204#131—42no—
205#131—42no—
206C2×M16—22no—
207C2×M16C22⋊D711yescharacteristic
208C2×M16C22⋊D711yescharacteristic
209C22×C8—22no—
210C22×C8—21no—
211C22×C8—141no—
212C4⋊Q8—71no—
213C4⋊Q8—71no—
214C4⋊Q8C4×D711yescharacteristic
215#127C4×D711yescharacteristic
216#125—141no—
217#124—21no—
218C22⋊Q8—141no—
219C22⋊Q8—141no—
220#121—21no—
221#121D2812yes—
222C4×Q8—141no—
223C4×Q8—141no—
224C4×D4—21no—
225C4×D4C22⋊D712yes—
226#117—141no—
227C2×C4⋊C4—141no—
228C2×C42—141no—
229C2×C42C22×D711yescharacteristic
230#106—142no—
231C4⋊C8—21no—
232C4⋊C8—141no—
233C4⋊C8C22⋊D711yescharacteristic
234C4⋊C8C22⋊D711yescharacteristic
235#103—142no—
236#102—142no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—141no—
240#97—21no—
241C4×C8—21no—
242C4×C8—71no—
243C4×C8—71no—
244C4×C8—141no—
245C4×C8—141no—
246#95—141no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7#38611yescharacteristic
253C22×C7#46111yescharacteristic
254C22×C7#46311yescharacteristic
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—42no—
259C4×C7C4×SD1611yescharacteristic
260C4×C7#44011yescharacteristic
261C4×C7#44311yescharacteristic
262C4×C7#44411yescharacteristic
263C4×C7#45011yescharacteristic
264C4×C7#45211yescharacteristic
265Dic7—82no—
266Dic7—81no—
267Dic7—81no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—141no—
274C2×Q8—141no—
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4(C2×D7)⋊C412yes—
279C22×C4—22no—
280C22×C4—281no—
281C22×C4C2×D2811yescharacteristic
282C22×C4#115711yescharacteristic
283C22×C4C2×C22⋊D711yescharacteristic
284SD16—284no—
285M16—21no—
286M16—21no—
287M16—21no—
288M16—44no—
289M16#114212yescomplemented
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—44no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—141no—
298C2×C8—141no—
299C2×C8—282no—
300C2×C8—281no—
301C4⋊C4—21no—
302C4⋊C4—142no—
303C4⋊C4—141no—
304C4⋊C4—141no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C4⋊C4—281no—
308C4⋊C4—281no—
309C4⋊C4—281no—
310C4⋊C4(C2×D7)⋊C412yes—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—281no—
314C42—281no—
315C42—281no—
316C42—281no—
317C42C2×C4×D711yescharacteristic
318C42#115711yescharacteristic
319C42C2×C22⋊D711yescharacteristic
320C42C2×C22⋊D711yescharacteristic
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—82no—
324C2×C7#187611yescharacteristic
325C2×C7#187711yescharacteristic
326C2×C7#188411yescharacteristic
327C23—22no—
328C23#649311yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—282no—
332Q8—281no—
333Q8—281no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—282no—
348C2×C4—281no—
349C2×C4—281no—
350C2×C4—561no—
351C2×C4—561no—
352C2×C4C2×(C2×D7)⋊C411yescharacteristic
353C2×C4C4×C22⋊D711yescharacteristic
354C2×C4C4×C22⋊D711yescharacteristic
355C2×C4#649211yescharacteristic
356C2×C4#649311yescharacteristic
357C2×C4#649511yescharacteristic
358C2×C4C2×#114211yescharacteristic
359C2×C4#651311yescharacteristic
360C2×C4#651411yescharacteristic
361C2×C4#651611yescharacteristic
362C8—22no—
363C8—41no—
364C8—41no—
365C8—41no—
366C8—142no—
367C8—281no—
368C8—561no—
369C7#905511yescharacteristic, complemented, Sylow
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#8978811yescharacteristic
376C22#8979111yescharacteristic
377C22#8998511yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—562no—
383C4—561no—
384C4—561no—
385C4C4×#114211yescharacteristic
386C4#8979011yescharacteristic
387C4#8991611yescharacteristic
388C4#8991711yescharacteristic
389C4#8999011yescharacteristic
390C4#9001111yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#1172936611yescharacteristic
395C2#1172976411yescharacteristic
396C2#1172977511yescharacteristic
subgroupG/H
centerC2×C4#6516
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2935C2
socleC22×C7#386
prime radicalC7×#532C22
cosocleC2×C42×C7C23
hypercenter#2935D7
hypocenterC7#9055
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
4. #11728183C2semidirect
5. #11727622C2semidirect
6. #11725510C2semidirect
7. #11725473C2semidirect
17. #89865C4semidirect
18. #89851C4semidirect
19. #89848C4semidirect
20. #89842C4semidirect
104. C7×M16SD16semidirect
289. M16#1142semidirect
369. C7#9055semidirect

Maximal subgroups

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

subgroupindex
C7×#29352
#117407352
#117295652
#117281832
#117276222
#117255102
#117254732
#90557