Finite Group Databasebeta

← #13243249

Subgroups — #13243249

Order 1792.

684 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2920C211yesmaximal, characteristic
2C4×#6489C211yesmaximal, characteristic
3C2×#89697C211yesmaximal, characteristic
4#11725542C214yesmaximal, complemented
5C22×C42×C7C2211yescharacteristic
6C2×#6489—21no—
7C7×#432—21no—
8C7×#432C412yescomplemented
9C7×#431—21no—
10C7×#431C2212yes—
11C7×#423C412yescomplemented
12C7×#407—21no—
13C4×C7×M16—21no—
14C4×C22⋊Dic7—21no—
15C4×C22⋊Dic7C412yes—
16C2×C22⋊(C7⋊C8)—21no—
17C4×C4⋊Dic7—21no—
18C4×C4⋊Dic7C412yes—
19#89704—21no—
20C42×Dic7C2212yes—
21#89697—21no—
22#89697—21no—
23#89697C412yescomplemented
24#89697C2212yes—
25C2×C4⋊(C7⋊C8)C412yescomplemented
26C2×C4×C7⋊C8—21no—
27#7870—71nomaximal, Sylow
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—22no—
32C2×C42×C7C2×C411yescharacteristic
33C2×C42×C7C2×C411yescharacteristic
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7C2×C411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C42×C7Q811yescharacteristic
42C2×C42×C7Q811yescharacteristic
43C2×C42×C7C2311yescharacteristic
44C2×C22⋊Dic7—22no—
45#6489—28no—
46C2×C4⋊Dic7—22no—
47C2×C4×Dic7—22no—
48C2×C4×Dic7—22no—
49C2×C4×Dic7C2×C414yes—
50C22×C7⋊C8—22no—
51C7×C4⋊C8—22no—
52C7×C4⋊C8C2×C414yes—
53C7×C22⋊C8—22no—
54C7×C22⋊C8—22no—
55C7×#97—22no—
56C4×C7×C8—22no—
57C7⋊#95—22no—
58C7⋊#95—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C4⋊(C7⋊C8)—22no—
62C4⋊(C7⋊C8)C2×C414yes—
63C4×C7⋊C8—22no—
64C4×C7⋊C8—22no—
65#2920D711yescharacteristic
66C2×#431—71no—
67C4×#117—71no—
68#1276—74no—
69C24×C7C22⋊C411yescharacteristic
70C22×C4×C7—24no—
71C22×C4×C7—24no—
72C22×C4×C7—42no—
73C22×C4×C7C4211yescharacteristic
74C22×C4×C7C4211yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C4⋊C411yescharacteristic
83C22×C4×C7C4⋊C411yescharacteristic
84C22×C4×C7C4⋊C411yescharacteristic
85C22×C4×C7C4⋊C411yescharacteristic
86C22×C4×C7C4⋊C411yescharacteristic
87C22×C4×C7C4⋊C411yescharacteristic
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×Dic7—24no—
92C7×M16—44no—
93C2×C7×C8—24no—
94C42×C7C4211yescharacteristic
95C42×C7C4211yescharacteristic
96C42×C7C22⋊C411yescharacteristic
97C42×C7C22⋊C411yescharacteristic
98C42×C7C22⋊C411yescharacteristic
99C42×C7C22⋊C411yescharacteristic
100C42×C7C4⋊C411yescharacteristic
101C42×C7C4⋊C411yescharacteristic
102C42×C7C4⋊C411yescharacteristic
103C42×C7C4⋊C411yescharacteristic
104C42×C7C4⋊C411yescharacteristic
105C42×C7C4⋊C411yescharacteristic
106C42×C7C22×C411yescharacteristic
107C42×C7C22×C411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7C2×Q811yescharacteristic
110C22⋊Dic7—28no—
111C4⋊Dic7—28no—
112C4×Dic7—28no—
113C4×Dic7—28no—
114C4×Dic7—44no—
115C2×C7⋊C8—24no—
116C2×C7⋊C8—44no—
117C2×#117—141no—
118C22×C42C2×D711yescharacteristic
119#432—21no—
120#432Dic712yescomplemented
121#431—21no—
122#431—72no—
123#431—72no—
124#431—141no—
125#431—141no—
126#431C2×D712yes—
127#423Dic712yescomplemented
128C2×C4⋊C8—72no—
129#407—21no—
130C2×C22⋊C8—141no—
131C4×M16—21no—
132C2×C4×C8—141no—
133C4×C4⋊C4—72no—
134C4×C4⋊C4—141no—
135C4×C22⋊C4—72no—
136C4×C22⋊C4—141no—
137#376—141no—
138C43—72no—
139C23×C7—22no—
140C23×C7—22no—
141C23×C7—41no—
142C23×C7#9511yescharacteristic
143C23×C7#9511yescharacteristic
144C23×C7C2×C22⋊C411yescharacteristic
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—22no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—22no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7#9511yescharacteristic
162C2×C4×C7#9511yescharacteristic
163C2×C4×C7#9511yescharacteristic
164C2×C4×C7#9511yescharacteristic
165C2×C4×C7#9511yescharacteristic
166C2×C4×C7#9511yescharacteristic
167C2×C4×C7#10414yes—
168C2×C4×C7C2×C4211yescharacteristic
169C2×C4×C7C2×C22⋊C411yescharacteristic
170C2×C4×C7C2×C22⋊C411yescharacteristic
171C2×C4×C7C2×C4⋊C411yescharacteristic
172C2×C4×C7C2×C4⋊C411yescharacteristic
173C2×C4×C7C2×C4⋊C411yescharacteristic
174C2×C4×C7#13114yes—
175C2×Dic7—28no—
176C2×Dic7—44no—
177C2×Dic7—44no—
178C7×C8—44no—
179C7⋊C8—44no—
180C23×C4C4×D711yescharacteristic
181C23×C4D2811yescharacteristic
182C23×C4C22⋊D711yescharacteristic
183C2×M16—22no—
184C22×C8—142no—
185#117—148no—
186C2×C4⋊C4—142no—
187C2×C22⋊C4—142no—
188C2×C42—74no—
189C2×C42—142no—
190C2×C42—142no—
191C2×C42Dic1411yescharacteristic
192C2×C42Dic1411yescharacteristic
193C2×C42C4×D711yescharacteristic
194C2×C42C4×D711yescharacteristic
195C2×C42C4×D711yescharacteristic
196C2×C42D2811yescharacteristic
197C2×C42C2×Dic711yescharacteristic
198C2×C42C2×Dic711yescharacteristic
199C2×C42C22⋊D711yescharacteristic
200C2×C42C22⋊D711yescharacteristic
201C2×C42C22⋊D711yescharacteristic
202C2×C42C22×D711yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—74no—
205C4⋊C8—142no—
206C4⋊C8C2×Dic714yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—142no—
210C22⋊C8—142no—
211#97—22no—
212C4×C8—22no—
213C4×C8—142no—
214C4×C8—142no—
215#95—142no—
216#95—142no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—44no—
225C22×C7#33912yes—
226C22×C7C2×#9511yescharacteristic
227C22×C7#40812yes—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7#33912yes—
236C4×C7#40512yes—
237C4×C7C2×#10412yes—
238C4×C7#42112yes—
239C4×C7#42412yes—
240Dic7—48no—
241C24(C2×D7)⋊C411yescharacteristic
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—144no—
246C22×C4C4×Dic711yescharacteristic
247C22×C4C4×Dic711yescharacteristic
248C22×C4Dic7⋊C411yescharacteristic
249C22×C4Dic7⋊C411yescharacteristic
250C22×C4Dic7⋊C411yescharacteristic
251C22×C4Dic7⋊C411yescharacteristic
252C22×C4C4⋊Dic711yescharacteristic
253C22×C4C4⋊Dic711yescharacteristic
254C22×C4(C2×D7)⋊C411yescharacteristic
255C22×C4(C2×D7)⋊C411yescharacteristic
256C22×C4(C2×D7)⋊C411yescharacteristic
257C22×C4(C2×D7)⋊C411yescharacteristic
258C22×C4(C2×D7)⋊C411yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4C22⋊Dic711yescharacteristic
261C22×C4C2×C4×D711yescharacteristic
262C22×C4C2×D2811yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264M16—44no—
265C2×C8—24no—
266C2×C8—144no—
267C2×C8—284no—
268C4⋊C4—148no—
269C22⋊C4—148no—
270C42—148no—
271C42—148no—
272C42—284no—
273C42C4×Dic711yescharacteristic
274C42C4×Dic711yescharacteristic
275C42Dic7⋊C411yescharacteristic
276C42Dic7⋊C411yescharacteristic
277C42Dic7⋊C411yescharacteristic
278C42Dic7⋊C411yescharacteristic
279C42C4⋊Dic711yescharacteristic
280C42C4⋊Dic711yescharacteristic
281C42(C2×D7)⋊C411yescharacteristic
282C42(C2×D7)⋊C411yescharacteristic
283C42C22⋊Dic711yescharacteristic
284C42C22⋊Dic711yescharacteristic
285C42C2×Dic1411yescharacteristic
286C42C2×C4×D711yescharacteristic
287C42C22×Dic711yescharacteristic
288C42C2×C22⋊D711yescharacteristic
289C2×C7—22no—
290C2×C7—22no—
291C2×C7—41no—
292C2×C7#172911yescharacteristic
293C2×C7C2×#33911yescharacteristic
294C2×C7#173311yescharacteristic
295C23—22no—
296C23—22no—
297C23—41no—
298C23C7⋊#9511yescharacteristic
299C23C7⋊#9511yescharacteristic
300C23C2×(C2×D7)⋊C411yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—148no—
318C2×C4—284no—
319C2×C4—284no—
320C2×C4C7⋊#9511yescharacteristic
321C2×C4C7⋊#9511yescharacteristic
322C2×C4C7⋊#9511yescharacteristic
323C2×C4C7⋊#9511yescharacteristic
324C2×C4C7⋊#9511yescharacteristic
325C2×C4C7⋊#9511yescharacteristic
326C2×C4#641114yes—
327C2×C4#647014yes—
328C2×C4C2×C4×Dic711yescharacteristic
329C2×C4C2×Dic7⋊C411yescharacteristic
330C2×C4C2×Dic7⋊C411yescharacteristic
331C2×C4C2×C4⋊Dic711yescharacteristic
332C2×C4C2×(C2×D7)⋊C411yescharacteristic
333C2×C4C2×C22⋊Dic711yescharacteristic
334C8—44no—
335C8—284no—
336C7#787011yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22#8932712yes—
346C22#8990112yes—
347C22C2×C7⋊#9511yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—288no—
356C4#8932712yes—
357C4#8989112yes—
358C4#8989412yes—
359C4#8991112yes—
360C4C2×#641112yes—
361C2—22no—
362C2—22no—
363C2—41no—
364C2C2×#8932711yescharacteristic
365C2#1172973411yescharacteristic
366C2#1172973611yescharacteristic
subgroupG/H
centerC42C2×C22⋊D7
commutatorC2×C4×C7C2×C42
FrattiniC2×C42C22×D7
FittingC7×#2920C2
socleC22×C7C2×#95
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2920D7
hypocenterC7#7870
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C42 ◁ C22×C42 ◁ #2920

Complements

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

NH ≅ G/Ntype
4. #11725542C2semidirect
8. C7×#432C4semidirect
11. C7×#423C4semidirect
23. #89697C4semidirect
25. C2×C4⋊(C7⋊C8)C4semidirect
120. #432Dic7semidirect
127. #423Dic7semidirect
336. C7#7870semidirect

Maximal subgroups

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

subgroupindex
C7×#29202
C4×#64892
C2×#896972
#117255422
#78707