Finite Group Databasebeta

← #13252956

Subgroups — #13252956

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740736C211yesmaximal, characteristic
2C7×#1980C211yesmaximal, characteristic, complemented
3#11729565C211yesmaximal, characteristic, complemented
4#11727621C211yesmaximal, characteristic, complemented
5M16×C7⋊C8C211yesmaximal, characteristic, complemented
6C4×#6381C211yesmaximal, characteristic, complemented
7#11727223C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9#90168C2211yescharacteristic
10C2×C7×#104—22no—
11C4×C7×M16C2211yescharacteristic
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6399—22no—
14#89860C411yescharacteristic, complemented
15#89850C411yescharacteristic, complemented
16#89845C411yescharacteristic, complemented
17#89842C411yescharacteristic, complemented
18#89810—21no—
19#89810—21no—
20C4×#1141—21no—
21C4×#1141—21no—
22C2×#6381—21no—
23C4×C4⋊Dic7C2211yescharacteristic
24#89697C2211yescharacteristic
25C2×C4×C7⋊C8C2211yescharacteristic
26#89332—22no—
27#89324—21no—
28#89250—21no—
29C8×C7⋊C8—21no—
30#89246—22no—
31#9719—71nomaximal, Sylow
32C2×C7×#41—21no—
33C2×#1157—21no—
34C2×C7×M16D412yes—
35C7×C4⋊Q8C2×C411yescharacteristic
36C7×#127C2×C411yescharacteristic
37C7×#124—21no—
38C7×#121—21no—
39C7×#121—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C42×C7—21no—
44C2×C42×C7C2311yescharacteristic
45C2×#1141—21no—
46C2×#1141—21no—
47#6493—21no—
48#6493—21no—
49C2×C4⋊Dic7—21no—
50C2×C4×Dic7—21no—
51C22×C7⋊C8D411yescharacteristic
52C22×C7⋊C8D411yescharacteristic
53#6438—21no—
54#6437C2×C411yescharacteristic
55C4×D28—21no—
56C4×D28—21no—
57#6432C2×C411yescharacteristic
58C7×#104—44no—
59C7×#97—21no—
60C4×C7×C8—21no—
61C7×#95—21no—
62#6406—21no—
63#6406—21no—
64C7⋊#95—21no—
65C22⋊(C7⋊C8)D412yes—
66#6399—44no—
67#6383—21no—
68#6383—21no—
69#6381—24no—
70#6381—21no—
71#6381—21no—
72C4⋊(C7⋊C8)—21no—
73C4×C7⋊C8—21no—
74C4×C7⋊C8C2×C411yescharacteristic
75C4×C7⋊C8C2×C411yescharacteristic
76C4×C7⋊C8D411yescharacteristic
77C4×C7⋊C8D411yescharacteristic
78#3169—71no—
79#1980—71no—
80#1980D711yescharacteristic, complemented
81C4×#107—71no—
82#1502—71no—
83#1502—71no—
84C8×M16—71no—
85C7×#41—42no—
86C7×#41—42no—
87C2×C7×Q8—21no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C2×C22⋊D7—41no—
98C22×Dic7—41no—
99#1157—42no—
100#1157—42no—
101C2×D28—21no—
102C2×D28—21no—
103C2×D28—21no—
104C2×C4×D7—41no—
105C2×Dic14—21no—
106C7×M16—22no—
107C7×M16D814yescomplemented
108C2×C7×C8—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C42×C7—42no—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7#4111yescharacteristic
119C42×C7#4111yescharacteristic
120#1141—42no—
121#1141—42no—
122(C2×D7)⋊C4—41no—
123(C2×D7)⋊C4—41no—
124C4⋊Dic7—22no—
125C4⋊Dic7—21no—
126C4⋊Dic7—21no—
127C4⋊Dic7—41no—
128C4×Dic7—42no—
129C2×C7⋊C8—22no—
130C2×C7⋊C8—22no—
131C2×C7⋊C8—41no—
132C2×C7⋊C8C22⋊C411yescharacteristic
133C2×C7⋊C8C22⋊C411yescharacteristic
134C2×C7⋊C8C22⋊C411yescharacteristic
135C2×C7⋊C8C22⋊C411yescharacteristic
136#532—71no—
137#532C2×D711yescharacteristic
138#500—71no—
139#493—71no—
140#486—71no—
141#486—71no—
142#466—141no—
143#466—141no—
144C4×D8—141no—
145C4×D8—141no—
146#431—71no—
147C2×#107—141no—
148C2×#104—22no—
149C2×#104—142no—
150C4×M16C2×D711yescharacteristic
151C2×C4×C8—71no—
152C4×C4⋊C4—71no—
153C4×C4⋊C4C2×D711yescharacteristic
154#340—141no—
155#327—142no—
156#327—142no—
157#322—141no—
158C82—141no—
159C23×C7—41no—
160C23×C7#12711yescharacteristic
161C22×D7—41no—
162C7×Q8—42no—
163C7×D4—41no—
164C7×D4—41no—
165C7×D4—42no—
166C7×D4—81no—
167C7×D4—81no—
168C7×D4—81no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—41no—
172C2×C4×C7—42no—
173C2×C4×C7—41no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—82no—
177C2×C4×C7C2×C22⋊C411yescharacteristic
178C2×C4×C7C4×D411yescharacteristic
179C2×C4×C7C4×D411yescharacteristic
180C2×C4×C7#12111yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7#12411yescharacteristic
183C2×C4×C7C2×D812yes—
184C2×C4×C7#13512yes—
185C22⋊D7—81no—
186C22⋊D7—81no—
187C2×Dic7—42no—
188C2×Dic7—41no—
189C2×Dic7—81no—
190C2×Dic7—81no—
191D28—41no—
192D28—41no—
193D28—42no—
194D28—81no—
195C4×D7—82no—
196Dic14—42no—
197C7×C8—24no—
198C7⋊C8—21no—
199C7⋊C8—21no—
200C7⋊C8—21no—
201C7⋊C8—42no—
202C7⋊C8#10412yescomplemented
203C2×#41—21no—
204C2×#41—141no—
205C2×D8—141no—
206C2×D8—141no—
207C2×M16C22⋊D712yes—
208C22×C8—71no—
209C22×C8—71no—
210C4⋊Q8—71no—
211C4⋊Q8C4×D711yescharacteristic
212#127—71no—
213#127C4×D711yescharacteristic
214#124—21no—
215#124—141no—
216#121—21no—
217#121—21no—
218#121—141no—
219#121—141no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—141no—
223C4×D4—141no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—141no—
226C2×C42—21no—
227C2×C42—141no—
228C2×C42C22×D711yescharacteristic
229#107—141no—
230#107—141no—
231#107—144no—
232C4⋊C8—141no—
233#104—44no—
234#104—284no—
235#102—141no—
236#102—141no—
237#102—141no—
238#102—141no—
239C22⋊C8—72no—
240#97—21no—
241C4×C8—21no—
242C4×C8—71no—
243C4×C8—71no—
244C4×C8—71no—
245C4×C8—71no—
246C4×C8—141no—
247#95—21no—
248#95—141no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7#39011yescharacteristic
255C22×C7#49311yescharacteristic
256C22×C7#49511yescharacteristic
257C2×D7—81no—
258C2×D7—81no—
259C2×D7—81no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—82no—
264C4×C7—81no—
265C4×C7C2×#10411yescharacteristic
266C4×C7#42111yescharacteristic
267C4×C7C4×D812yes—
268C4×C7#46612yes—
269Dic7—82no—
270Dic7—81no—
271#41—42no—
272#41—42no—
273#41—282no—
274#41—282no—
275C2×Q8—21no—
276C2×Q8—141no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—141no—
282C2×D4—141no—
283C2×D4—141no—
284C2×D4—281no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—281no—
288C22×C4—281no—
289C22×C4D4×D711yescharacteristic
290C22×C4D4×D711yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292D8—282no—
293D8—282no—
294M16—22no—
295M16#114114yescomplemented
296C2×C8—22no—
297C2×C8—71no—
298C2×C8—71no—
299C2×C8—71no—
300C2×C8—71no—
301C2×C8—142no—
302C2×C8—142no—
303C2×C8—281no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—142no—
309C4⋊C4—141no—
310C4⋊C4—141no—
311C4⋊C4—281no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—281no—
315C22⋊C4—281no—
316C42—42no—
317C42—282no—
318C42C2×C4×D711yescharacteristic
319C42#115711yescharacteristic
320C42D4×D711yescharacteristic
321C42#115911yescharacteristic
322C2×C7—21no—
323C2×C7—21no—
324C2×C7—161no—
325C2×C7#197011yescharacteristic
326C2×C7#197611yescharacteristic
327C2×C7#198011yescharacteristic
328D7—161no—
329C23—41no—
330C23—281no—
331C23#650311yescharacteristic
332Q8—42no—
333Q8—282no—
334D4—42no—
335D4—41no—
336D4—41no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—282no—
341D4—281no—
342D4—281no—
343D4—561no—
344D4—561no—
345D4—561no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—282no—
355C2×C4—281no—
356C2×C4—561no—
357C2×C4—561no—
358C2×C4—562no—
359C2×C4D7×C22⋊C411yescharacteristic
360C2×C4(C22⋊D7)⋊C411yescharacteristic
361C2×C4#644711yescharacteristic
362C2×C4#644811yescharacteristic
363C2×C4C4×C22⋊D711yescharacteristic
364C2×C4C2×#114112yes—
365C2×C4#650211yescharacteristic
366C2×C4#651312yes—
367C8—24no—
368C8—72no—
369C8—141no—
370C8—141no—
371C8—141no—
372C8—282no—
373C7#971911yescharacteristic, complemented, Sylow
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—561no—
380C22—561no—
381C22—561no—
382C22#8972911yescharacteristic
383C22#8983111yescharacteristic
384C22#8984511yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—562no—
391C4—561no—
392C4D7×#10411yescharacteristic
393C4#8959311yescharacteristic
394C4C4×#114112yes—
395C4#8981012yes—
396C2—21no—
397C2—21no—
398C2—161no—
399C2—1121no—
400C2#1172740111yescharacteristic
401C2#1172811211yescharacteristic
402C2#1172815111yescharacteristic
subgroupG/H
centerC2×C4#6502
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1980C2
socleC22×C7#390
prime radical#11740736C2
cosocleC2×C42×C7C23
hypercenter#1980D7
hypocenterC7#9719
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. C7×#1980C2semidirect
3. #11729565C2semidirect
4. #11727621C2semidirect
5. M16×C7⋊C8C2semidirect
6. C4×#6381C2semidirect
7. #11727223C2semidirect
14. #89860C4semidirect
15. #89850C4semidirect
16. #89845C4semidirect
17. #89842C4semidirect
80. #1980D7semidirect
107. C7×M16D8semidirect
202. C7⋊C8#104semidirect
295. M16#1141semidirect
373. C7#9719semidirect

Maximal subgroups

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

subgroupindex
#117407362
C7×#19802
#117295652
#117276212
M16×C7⋊C82
C4×#63812
#117272232
#97197