Finite Group Databasebeta

← #13339359

Subgroups — #13339359

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3209C211yesmaximal, characteristic, complemented
2#11740795C211yesmaximal, characteristic
3C2×#89859C211yesmaximal, characteristic
4#11736733C211yesmaximal, characteristic
5#11729582C211yesmaximal, characteristic, complemented
6#11729529C211yesmaximal, characteristic, complemented
7#11729060C214yesmaximal, complemented
8#11728989C214yesmaximal, complemented
9#11728503C211yesmaximal, characteristic, complemented
10C7×#533C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×C4×D28C2211yescharacteristic
13C7×#494C2212yescomplemented
14C7×#487C2212yescomplemented
15C7×#462—22no—
16C7×#451—22no—
17C7×#431C2211yescharacteristic
18#89859—22no—
19#89859C2214yes—
20#89859C2212yes—
21#89856C2212yes—
22#89838—22no—
23#89773—21no—
24#89767—21no—
25C2×#6383—21no—
26C2×#6381—21no—
27#89697C2211yescharacteristic
28#89696C2211yescharacteristic
29C2×C4⋊(C7⋊C8)C2211yescharacteristic
30#89610—22no—
31#89498—22no—
32#89460C2214yes—
33#89342C2214yescomplemented
34#89342C2212yescomplemented
35#89341C2212yescomplemented
36#89252C2214yescomplemented
37#19423—71nomaximal, Sylow
38C22×C7×Q8—21no—
39C22×D28D411yescharacteristic
40C22×C4×D7—21no—
41C2×C7×Q16—24no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8C2311yescharacteristic
46C7×C4⋊Q8C2312yes—
47C7×C4⋊Q8C2311yescharacteristic
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C2×(C2×D7)⋊C4—21no—
58C2×C4⋊Dic7D411yescharacteristic
59C2×#1136—21no—
60C22×C7⋊C8—21no—
61C2×#1131—24no—
62C4×D28—21no—
63C4×D28—21no—
64C4×D28D412yes—
65C4×D28C2312yes—
66C7×C4⋊C8D412yes—
67C7×#103—24no—
68C7×C22⋊C8D412yes—
69C4×C7×C8C2312yes—
70C22⋊(C7⋊C8)—21no—
71(C2×D7)⋊C8—24no—
72#6383—22no—
73#6383—24no—
74#6383—22no—
75#6382—22no—
76#6381—24no—
77#6381—22no—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)C2312yes—
81C4⋊(C7⋊C8)C2312yes—
82C4⋊(C7⋊C8)C2312yes—
83C4×C7⋊C8—21no—
84#3209D711yescharacteristic, complemented
85#3077—71no—
86C2×#474—71no—
87#2975—71no—
88#1703—74no—
89#1689—74no—
90#1570—71no—
91#1549—71no—
92#1542—71no—
93C23×D7—21no—
94C2×C7×Q8—22no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—41no—
98C2×C7×Q8—82no—
99C22×C4×C7—41no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7#4111yescharacteristic
103C22×Dic7—21no—
104C2×D28—22no—
105C2×D28—22no—
106C2×D28—21no—
107C2×D28C2×D412yes—
108C2×C4×D7—22no—
109C2×C4×D7—22no—
110C2×C4×D7—82no—
111C7×Q16—84no—
112C2×C7×C8C2×D414yes—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C22⋊C4—41no—
124C42×C7—41no—
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C42×C7C2411yescharacteristic
129(C2×D7)⋊C4—22no—
130(C2×D7)⋊C4—22no—
131C4⋊Dic7—21no—
132C4⋊Dic7C2×D412yes—
133#1136—42no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8—22no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—42no—
138#1131—84no—
139#533C2×D711yescharacteristic
140C2×C4⋊Q8C2×D711yescharacteristic
141C2×C4×D4—71no—
142#494C2×D712yescomplemented
143#487C2×D712yescomplemented
144#478—142no—
145#476—72no—
146#474—72no—
147#474—74no—
148#474—142no—
149#462—22no—
150#451—22no—
151#436—74no—
152#436—142no—
153#431—71no—
154#431C2×D711yescharacteristic
155#428—141no—
156C2×#107—141no—
157#423—71no—
158C2×C4⋊C8—71no—
159#418—141no—
160C2×#102—141no—
161#407—142no—
162#334—74no—
163#333—72no—
164#332—72no—
165#332—74no—
166C23×C7#12111yescharacteristic
167C22×D7—22no—
168C22×D7—42no—
169C22×D7—81no—
170C22×D7—81no—
171C7×Q8—42no—
172C7×Q8—44no—
173C7×Q8—81no—
174C7×Q8—82no—
175C2×C4×C7—22no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—82no—
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#13712yes—
185C2×C4×C7#13712yes—
186C2×C4×C7C22×D411yescharacteristic
187C2×C4×C7C22×D411yescharacteristic
188C2×C4×C7C2×#4111yescharacteristic
189C2×Dic7—22no—
190C2×Dic7—81no—
191D28—22no—
192D28—22no—
193D28—42no—
194C4×D7—44no—
195C4×D7—82no—
196C7×C8—24no—
197C7⋊C8—42no—
198C7⋊C8—42no—
199C7⋊C8—42no—
200C22×Q8—21no—
201C22×D4—71no—
202C23×C4—141no—
203C2×Q16—24no—
204C2×M16—141no—
205C2×M16—144no—
206C22×C8—141no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8C22×D712yes—
211C4⋊Q8C22×D711yescharacteristic
212C4⋊Q8C22×D711yescharacteristic
213#125—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×D4—72no—
219C4×D4—72no—
220C4×D4—141no—
221C4×D4—141no—
222#117—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—71no—
226C2×C22⋊C4—141no—
227C2×C42C22×D711yescharacteristic
228#107—144no—
229#107—142no—
230#106—142no—
231C4⋊C8—72no—
232C4⋊C8—72no—
233C4⋊C8—72no—
234C4⋊C8—141no—
235C4⋊C8—141no—
236C4⋊C8C22⋊D712yes—
237#103—24no—
238#102—144no—
239#102—142no—
240#102—142no—
241C22⋊C8—141no—
242C22⋊C8—144no—
243C22⋊C8C22⋊D712yes—
244C4×C8—141no—
245C4×C8C22×D712yes—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7#47012yes—
249C22×C7C2×#12111yescharacteristic
250C2×D7—42no—
251C2×D7—42no—
252C2×D7—42no—
253C2×D7—84no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—82no—
258C4×C7—81no—
259C4×C7—81no—
260C4×C7#47012yes—
261C4×C7C2×#13711yescharacteristic
262C4×C7C2×#13711yescharacteristic
263C4×C7#57612yes—
264Dic7—42no—
265C24—141no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—41no—
270C2×Q8—82no—
271C2×D4—72no—
272C2×D4—142no—
273C2×D4—142no—
274C2×D4—141no—
275C22×C4—41no—
276C22×C4—141no—
277C22×C4—142no—
278C22×C4—142no—
279C22×C4—562no—
280C22×C4D4×D711yescharacteristic
281C22×C4#115911yescharacteristic
282C22×C4C2×C22⋊D711yescharacteristic
283Q16—84no—
284M16—282no—
285M16—564no—
286C2×C8—142no—
287C2×C8—142no—
288C2×C8—142no—
289C2×C8—282no—
290C2×C8C2×C22⋊D714yes—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—72no—
302C4⋊C4—141no—
303C22⋊C4—41no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C42—41no—
307C42D4×D711yescharacteristic
308C42#115911yescharacteristic
309C42C2×C22⋊D711yescharacteristic
310C42C23×D711yescharacteristic
311C2×C7—22no—
312C2×C7#304111yescharacteristic
313C2×C7C2×#47011yescharacteristic
314C2×C7#306911yescharacteristic
315D7—82no—
316C23—142no—
317C23—282no—
318C23—561no—
319C23—561no—
320C23#650111yescharacteristic
321Q8—42no—
322Q8—44no—
323Q8—81no—
324Q8—82no—
325D4—142no—
326D4—142no—
327D4—282no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—142no—
335C2×C4—284no—
336C2×C4—562no—
337C2×C4—561no—
338C2×C4#648112yes—
339C2×C4#648112yes—
340C2×C4#650111yescharacteristic
341C2×C4#650111yescharacteristic
342C2×C4#650111yescharacteristic
343C2×C4C2×D4×D711yescharacteristic
344C2×C4C2×#115911yescharacteristic
345C2×C4C22×C22⋊D711yescharacteristic
346C8—24no—
347C8—282no—
348C8—282no—
349C8—282no—
350C7#1942311yescharacteristic, complemented, Sylow
351C22—21no—
352C22—21no—
353C22—282no—
354C22—282no—
355C22—282no—
356C22—564no—
357C22#8996112yes—
358C22C2×#650111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—282no—
366C4#8996112yes—
367C4C2×#648111yescharacteristic
368C4C2×#648111yescharacteristic
369C4#9052612yes—
370C2—22no—
371C2—562no—
372C2#1174082911yescharacteristic
373C2C2×#8996111yescharacteristic
374C2#1174144311yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3209C2
socleC22×C7C2×#121
prime radicalC22×D28D4
hypercenter#3209D7
hypocenterC7#19423
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C7×#3209C2semidirect
5. #11729582C2semidirect
6. #11729529C2semidirect
7. #11729060C2semidirect
8. #11728989C2semidirect
9. #11728503C2semidirect
13. C7×#494C22semidirect
14. C7×#487C22semidirect
33. #89342C22semidirect
34. #89342C22semidirect
35. #89341C22semidirect
36. #89252C22semidirect
84. #3209D7semidirect
142. #494C2×D7semidirect
143. #487C2×D7semidirect
350. C7#19423semidirect

Maximal subgroups

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

subgroupindex
C7×#32092
#117407952
C2×#898592
#117367332
#117295822
#117295292
#117290602
#117289892
#117285032
#194237