Finite Group Databasebeta

← #13249337

Subgroups — #13249337

Order 1792.

586 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2322C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4C2×#89323C212yesmaximal, complemented
5#11725423C212yesmaximal, complemented
6C2×C7×#123C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#389—21no—
10C7×#388—21no—
11C7×#387—21no—
12C7×#386C411yescharacteristic
13C7×#385—21no—
14C7×#385C411yescharacteristic
15C7×#381C411yescharacteristic
16C4×C7×C22⋊C4C2211yescharacteristic
17C2×C22⋊(C7⋊C8)—21no—
18C2×C22⋊(C7⋊C8)C2212yes—
19C2×C22⋊(C7⋊C8)C2212yes—
20D4×C7⋊C8—24no—
21#89697—21no—
22#89323—21no—
23#89323—24no—
24#89320—21no—
25#9108—71nomaximal, Sylow
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C7×#123—22no—
32C7×#123C814yescomplemented
33C4×C7×D4—22no—
34C4×C7×D4—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7D411yescharacteristic
50C22×C7⋊C8—22no—
51C22×C7⋊C8—22no—
52C22×C7⋊C8—22no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C22⋊(C7⋊C8)—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)—22no—
62C22⋊(C7⋊C8)D414yes—
63C4⋊(C7⋊C8)—22no—
64C4×C7⋊C8—22no—
65#2972—71no—
66#2322D711yescharacteristic
67#1459—71no—
68C2×#324—72no—
69#1282—72no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7C22×C411yescharacteristic
72C2×C7×D4—22no—
73C2×C7×D4—22no—
74C2×C7×D4C2×C812yes—
75C2×C7×D4M1612yes—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—22no—
79C22×C4×C7—21no—
80C22×C4×C7—22no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—81no—
87C22×C4×C7—81no—
88C22×C4×C7—81no—
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C2×C812yescomplemented
93C22×C4×C7C2×D412yes—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7#4112yes—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—81no—
103C7×C22⋊C4—24no—
104C7×C22⋊C4—22no—
105C7×C22⋊C4—21no—
106C7×C22⋊C4—22no—
107C7×C22⋊C4—22no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4C2×C812yescomplemented
110C42×C7—21no—
111C42×C7—21no—
112C42×C7—81no—
113C2×C7⋊C8—24no—
114C2×C7⋊C8—42no—
115C2×C7⋊C8—82no—
116C2×C7⋊C8—82no—
117C2×C7⋊C8—82no—
118C2×#123Dic711yescharacteristic
119C2×C4×D4C2×D711yescharacteristic
120C22×C4⋊C4C2×D711yescharacteristic
121C8×D4—144no—
122#431—141no—
123C2×C22⋊C8—72no—
124C2×C22⋊C8—72no—
125C2×C22⋊C8—141no—
126#389—21no—
127#388—21no—
128#387—21no—
129#386Dic711yescharacteristic
130#385—21no—
131#385Dic711yescharacteristic
132#381Dic711yescharacteristic
133C4×C22⋊C4C2×D711yescharacteristic
134#324—144no—
135#324—141no—
136#323—141no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7C22⋊C812yes—
144C23×C7C2×C22⋊C411yescharacteristic
145C23×C7C4×D412yes—
146C23×C7C22×C811yescharacteristic
147C23×C7C2×M1611yescharacteristic
148C23×C7#13112yes—
149C7×D4—24no—
150C7×D4—24no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—24no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—81no—
177C2×C4×C7—82no—
178C2×C4×C7C22⋊C812yes—
179C2×C4×C7#12011yescharacteristic
180C2×C4×C7#12112yes—
181C2×C4×C7#12311yescharacteristic
182C7⋊C8—44no—
183C7⋊C8—82no—
184C22×D4C2×Dic711yescharacteristic
185C23×C4C2×Dic711yescharacteristic
186C23×C4C2×Dic711yescharacteristic
187C23×C4C22⋊D711yescharacteristic
188C23×C4C22×D711yescharacteristic
189C22×C8—142no—
190C22×C8—142no—
191C22×C8—142no—
192#123—22no—
193#123C7⋊C814yescomplemented
194C4×D4—22no—
195C4×D4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22⋊D711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4C2×Dic711yescharacteristic
206C2×C22⋊C4C2×Dic711yescharacteristic
207C2×C22⋊C4C2×Dic711yescharacteristic
208C2×C22⋊C4C22⋊D711yescharacteristic
209C2×C42—21no—
210C2×C42C22⋊D711yescharacteristic
211C4⋊C8—142no—
212C22⋊C8—74no—
213C22⋊C8—142no—
214C22⋊C8—142no—
215C22⋊C8—142no—
216C22⋊C8—142no—
217C4×C8—142no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223C22×C7—21no—
224C22×C7—22no—
225C22×C7—21no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—42no—
231C22×C7—42no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7—81no—
235C22×C7#38611yescharacteristic
236C22×C7C2×C22⋊C811yescharacteristic
237C22×C7#40811yescharacteristic
238C22×C7C8×D412yes—
239C22×C7#43512yes—
240C4×C7—22no—
241C4×C7—21no—
242C4×C7—22no—
243C4×C7—21no—
244C4×C7—42no—
245C4×C7—42no—
246C4×C7—41no—
247C4×C7—41no—
248C4×C7—81no—
249C4×C7—81no—
250C24C22⋊Dic711yescharacteristic
251C24C22×Dic711yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4C2×C7⋊C812yes—
255C2×D4#113612yes—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C2×C7⋊C812yescomplemented
270C22×C4C22⋊Dic711yescharacteristic
271C22×C4C22⋊Dic711yescharacteristic
272C22×C4C22⋊Dic711yescharacteristic
273C22×C4D4×D712yes—
274C22×C4#115912yes—
275C22×C4C2×C22⋊D711yescharacteristic
276C22×C4C2×C22⋊D711yescharacteristic
277C2×C8—144no—
278C2×C8—282no—
279C2×C8—562no—
280C2×C8—562no—
281C2×C8—562no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—81no—
288C22⋊C4—24no—
289C22⋊C4—21no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—81no—
294C22⋊C4C2×C7⋊C812yescomplemented
295C42—21no—
296C42—21no—
297C42—81no—
298C2×C7—21no—
299C2×C7—21no—
300C2×C7—22no—
301C2×C7—22no—
302C2×C7—42no—
303C2×C7#186911yescharacteristic
304C2×C7#191311yescharacteristic
305C2×C7#191511yescharacteristic
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23C22⋊(C7⋊C8)12yes—
313C23C22×C7⋊C811yescharacteristic
314C23C2×#113611yescharacteristic
315C23D4×Dic712yes—
316C23#651112yes—
317C23C2×C22⋊Dic711yescharacteristic
318D4—24no—
319D4—24no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4C22⋊(C7⋊C8)12yes—
348C2×C4#649811yescharacteristic
349C2×C4#650011yescharacteristic
350C2×C4#650212yes—
351C8—284no—
352C8—562no—
353C7#910811yescharacteristic, complemented, Sylow
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22D4×C7⋊C812yes—
367C22#8978312yes—
368C22C2×C22⋊(C7⋊C8)11yescharacteristic
369C22#8999211yescharacteristic
370C22#9000611yescharacteristic
371C4—21no—
372C4—22no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—42no—
386C2#1172936411yescharacteristic
387C2#1172960211yescharacteristic
388C2#1172962011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC23×C7C22×C8
FrattiniC23×C4C22×D7
FittingC7×#2322C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC23×C4×C7C23
hypercenter#2322D7
hypocenterC7#9108
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2322

Complements

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

NH ≅ G/Ntype
4. C2×#89323C2semidirect
5. #11725423C2semidirect
32. C7×#123C8semidirect
92. C22×C4×C7C2×C8semidirect
109. C7×C22⋊C4C2×C8semidirect
193. #123C7⋊C8semidirect
269. C22×C4C2×C7⋊C8semidirect
294. C22⋊C4C2×C7⋊C8semidirect
353. C7#9108semidirect

Maximal subgroups

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

subgroupindex
C7×#23222
#117397462
#117293542
C2×#893232
#117254232
#91087