Finite Group Databasebeta

← #13249232

Subgroups — #13249232

Order 1792.

504 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89798C211yesmaximal, characteristic
2#11736733C211yesmaximal, characteristic
3C7×#1479C211yesmaximal, characteristic, complemented
4#11729378C211yesmaximal, characteristic, complemented
5#11728511C211yesmaximal, characteristic
6#11725547C211yesmaximal, characteristic, complemented
7#11725546C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C22×#1143C411yescharacteristic
10C2×C4×D28C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#431C2211yescharacteristic
13C2×#6409C411yescharacteristic
14#89798—22no—
15#89798—22no—
16C2×#6383C411yescharacteristic
17C2×#6381C411yescharacteristic
18#89697C2211yescharacteristic
19#89697C2211yescharacteristic
20C2×#6377C2211yescharacteristic
21#89610—22no—
22#89498—22no—
23#89460—22no—
24C7×Q8⋊C8—22no—
25#89323—22no—
26#89254—22no—
27#9091—71nomaximal, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C22×D28C2×C411yescharacteristic
30C22×C4×D7—21no—
31C4×C7×Q8—21no—
32C4×C7×Q8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7C2311yescharacteristic
39C2×#1143—22no—
40C2×#1143—22no—
41C2×#1143—22no—
42C2×(C2×D7)⋊C4—21no—
43C2×C4⋊Dic7C2×C411yescharacteristic
44C22×C7⋊C8C2×C411yescharacteristic
45C22×C7⋊C8C2×C411yescharacteristic
46C2×#1131—24no—
47C4×D28—21no—
48C4×D28—21no—
49C4×D28D412yes—
50C4×D28D412yes—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C22⋊C8—21no—
54C7×C22⋊C8—21no—
55C4×C7×C8—21no—
56C4×C7×C8D412yes—
57#6409—22no—
58C22⋊(C7⋊C8)—21no—
59C22⋊(C7⋊C8)D412yes—
60(C2×D7)⋊C8—24no—
61#6383—22no—
62#6381—22no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65#6377—21no—
66#6377—21no—
67C4×C7⋊C8—21no—
68C4×C7⋊C8—21no—
69#2975—71no—
70C2×#440—71no—
71#1479D711yescharacteristic, complemented
72#1478—71no—
73#1461—71no—
74#1281—71no—
75#1275—71no—
76C23×D7—21no—
77C2×C7×Q8—21no—
78C2×C7×Q8—21no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8M1612yescomplemented
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×Dic7—21no—
89C2×D28—21no—
90C2×D28—21no—
91C2×D28—21no—
92C2×D28—21no—
93C2×D28—21no—
94C2×D28C22⋊C412yes—
95C2×C4×D7—22no—
96C2×C4×D7—22no—
97C2×C4×D7—82no—
98C2×C7×C8—22no—
99C2×C7×C8—41no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C42×C7—41no—
105C42×C7—41no—
106C42×C7C2×D411yescharacteristic
107C42×C7C2×D411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7#4111yescharacteristic
110#1143—28no—
111(C2×D7)⋊C4—22no—
112(C2×D7)⋊C4—22no—
113C4⋊Dic7—21no—
114C4⋊Dic7C22⋊C412yes—
115C2×C7⋊C8—21no—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—21no—
120C2×C7⋊C8—21no—
121C2×C7⋊C8—41no—
122C2×C7⋊C8M1612yescomplemented
123#1131—84no—
124C22×SD16—71no—
125C2×C4×Q8C2×D711yescharacteristic
126C2×C4×D4—71no—
127#440—142no—
128#440—142no—
129#436—142no—
130#436—142no—
131#431—71no—
132#431—71no—
133#431C2×D711yescharacteristic
134#431C2×D711yescharacteristic
135C2×#107—71no—
136C2×#103—71no—
137C2×#102—71no—
138#407—142no—
139C2×#97—71no—
140Q8⋊C8—22no—
141D4⋊C8—142no—
142#324—142no—
143C23×C7C4×D411yescharacteristic
144C22×D7—22no—
145C22×D7—41no—
146C22×D7—41no—
147C22×D7—81no—
148C22×D7—81no—
149C7×Q8—22no—
150C7×Q8—22no—
151C7×Q8—41no—
152C7×Q8—41no—
153C2×C4×C7—22no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—81no—
160C2×C4×C7C2×C22⋊C411yescharacteristic
161C2×C4×C7C4×D411yescharacteristic
162C2×C4×C7#12011yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12311yescharacteristic
166C2×C4×C7C2×M1611yescharacteristic
167C2×C4×C7C2×M1611yescharacteristic
168C2×C4×C7#13111yescharacteristic
169C2×C4×C7#13111yescharacteristic
170C2×Dic7—22no—
171C2×Dic7—81no—
172D28—22no—
173D28—22no—
174D28—41no—
175D28—41no—
176C4×D7—44no—
177C4×D7—82no—
178C7×C8—42no—
179C7×C8—81no—
180C7⋊C8—22no—
181C7⋊C8—42no—
182C7⋊C8—41no—
183C7⋊C8—81no—
184C22×Q8C4×D711yescharacteristic
185C22×D4—71no—
186C23×C4—141no—
187C2×SD16—142no—
188C2×SD16—142no—
189C2×SD16—142no—
190C2×M16—144no—
191C22×C8—71no—
192C22×C8—71no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—72no—
198C4×D4—72no—
199C4×D4—141no—
200C4×D4—141no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4C4×D711yescharacteristic
204C2×C22⋊C4—141no—
205C2×C42—21no—
206C2×C42C22×D711yescharacteristic
207#107—142no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—141no—
211C4⋊C8—141no—
212#103—142no—
213#102—142no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—72no—
217C22⋊C8—144no—
218C22⋊C8—141no—
219#97—141no—
220#97—141no—
221C4×C8—21no—
222C4×C8—141no—
223C4×C8—141no—
224C4×C8C22⋊D712yes—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7#38611yescharacteristic
228C22×C7#43511yescharacteristic
229C22×C7#43611yescharacteristic
230C2×D7—42no—
231C2×D7—42no—
232C2×D7—42no—
233C2×D7—82no—
234C2×D7—82no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—42no—
240C4×C7—81no—
241C4×C7#40711yescharacteristic
242C4×C7#40811yescharacteristic
243C4×C7#43511yescharacteristic
244C4×C7#43611yescharacteristic
245Dic7—42no—
246C24—141no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8#113112yescomplemented
253C2×D4—72no—
254C2×D4—141no—
255C2×D4—141no—
256C2×D4—141no—
257C2×D4—141no—
258C2×D4—141no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—141no—
262C22×C4—142no—
263C22×C4—142no—
264C22×C4—562no—
265C22×C4C2×C4×D711yescharacteristic
266C22×C4D4×D711yescharacteristic
267C22×C4#115911yescharacteristic
268SD16—148no—
269M16—564no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—72no—
273C2×C8—141no—
274C2×C8—141no—
275C2×C8—141no—
276C2×C8—141no—
277C2×C8—141no—
278C2×C8—142no—
279C2×C8—281no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—72no—
285C4⋊C4—141no—
286C22⋊C4—142no—
287C22⋊C4—142no—
288C42—41no—
289C42—41no—
290C42#115711yescharacteristic
291C42D4×D711yescharacteristic
292C42D4×D711yescharacteristic
293C42C2×C22⋊D711yescharacteristic
294C2×C7—21no—
295C2×C7—21no—
296C2×C7#187011yescharacteristic
297C2×C7#190211yescharacteristic
298C2×C7#190411yescharacteristic
299D7—82no—
300C23—142no—
301C23—281no—
302C23—281no—
303C23—561no—
304C23—561no—
305C23(C22⋊D7)⋊C411yescharacteristic
306Q8—22no—
307Q8—22no—
308Q8—41no—
309Q8—41no—
310D4—142no—
311D4—142no—
312D4—281no—
313D4—281no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—142no—
322C2×C4—284no—
323C2×C4—562no—
324C2×C4—561no—
325C2×C4D7×C22⋊C411yescharacteristic
326C2×C4#644611yescharacteristic
327C2×C4#644711yescharacteristic
328C2×C4C2×#113111yescharacteristic
329C2×C4#646411yescharacteristic
330C2×C4D7×M1611yescharacteristic
331C2×C4#647011yescharacteristic
332C2×C4C4×C22⋊D711yescharacteristic
333C2×C4#650011yescharacteristic
334C2×C4#650111yescharacteristic
335C8—42no—
336C8—81no—
337C8—142no—
338C8—282no—
339C8—281no—
340C8—561no—
341C7#909111yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—282no—
345C22—282no—
346C22—282no—
347C22—562no—
348C22—562no—
349C22#8950111yescharacteristic
350C22#8950211yescharacteristic
351C22#8972811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—81no—
358C4—282no—
359C4#8949811yescharacteristic
360C4#8949911yescharacteristic
361C4#8988311yescharacteristic
362C4#8990011yescharacteristic
363C2—21no—
364C2—21no—
365C2—562no—
366C2#1172771211yescharacteristic
367C2#1172810311yescharacteristic
368C2#1172816111yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1479C2
socleC22×C7#386
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1479D7
hypocenterC7#9091
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. C7×#1479C2semidirect
4. #11729378C2semidirect
6. #11725547C2semidirect
7. #11725546C2semidirect
71. #1479D7semidirect
82. C2×C7×Q8M16semidirect
122. C2×C7⋊C8M16semidirect
252. C2×Q8#1131semidirect
341. C7#9091semidirect

Maximal subgroups

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

subgroupindex
C2×#897982
#117367332
C7×#14792
#117293782
#117285112
#117255472
#117255462
#90917