Finite Group Databasebeta

← #13323119

Subgroups — #13323119

Order 1792.

592 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#11736739C211yesmaximal, characteristic
2#11736738C211yesmaximal, characteristic
3#11735892C211yesmaximal, characteristic
4C7×#1564C211yesmaximal, characteristic, complemented
5#11728502C211yesmaximal, characteristic, complemented
6#11728502C211yesmaximal, characteristic, complemented
7#11727485C211yesmaximal, characteristic, complemented
8#11726327C212yesmaximal, complemented
9#11726326C212yesmaximal, complemented
10#11726285C212yesmaximal, complemented
11#11726282C212yesmaximal, complemented
12C4×#1157C2211yescharacteristic
13C7×#431C2211yescharacteristic
14C7×#431C2211yescharacteristic
15C2×C4×C7×C8C2211yescharacteristic
16#89883C412yes—
17#89883C412yes—
18#89697C2211yescharacteristic
19#89696C2211yescharacteristic
20#89696C2211yescharacteristic
21#89609C2212yes—
22#89607C2212yes—
23#89605C2212yes—
24D7×C4⋊C8C2212yes—
25#89501—22no—
26#89500—22no—
27#89460—21no—
28#89460C411yescharacteristic
29#89460C411yescharacteristic
30C4×#1131C2212yes—
31C8×D28—21no—
32#89457C2212yes—
33#89451—21no—
34#89451C411yescharacteristic
35#89451C411yescharacteristic
36C8×Dic14—21no—
37C7×#335C2211yescharacteristic
38C7×#335C2211yescharacteristic
39C7×#334C2211yescharacteristic
40C7×#334C2211yescharacteristic
41#89275C2212yes—
42#89275C2212yes—
43#89274C2212yes—
44#89274C2212yes—
45#89253C2212yes—
46#89253C2212yes—
47#17839—71nomaximal, Sylow
48C2×#1157Q811yescharacteristic
49C22×C7×C8C2×C411yescharacteristic
50C22×C7×C8C2×C411yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C4×C22⋊D7C2×C412yes—
53C4×C22⋊D7D412yes—
54#6489Q811yescharacteristic
55C2×#1136—21no—
56C2×#1136—21no—
57C2×#1131—22no—
58C2×#1131C2×C412yes—
59C2×#1131C2×C412yes—
60C2×C8×D7—22no—
61C4×D28C2×C411yescharacteristic
62C4×D28D411yescharacteristic
63#6435Q812yes—
64C42×D7C2312yes—
65C4×Dic14C2×C411yescharacteristic
66C4×Dic14D411yescharacteristic
67C7×C4⋊C8C2311yescharacteristic
68C7×C4⋊C8C2311yescharacteristic
69C7×C4⋊C8C2311yescharacteristic
70C7×C4⋊C8C2311yescharacteristic
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C4×C7×C8—21no—
74C4×C7×C8—21no—
75C4×C7×C8C2×C411yescharacteristic
76C4×C7×C8C2×C411yescharacteristic
77C4×C7×C8C2311yescharacteristic
78C4×C7×C8C2311yescharacteristic
79C22⋊(C7⋊C8)C2×C412yes—
80(C2×D7)⋊C8—22no—
81(C2×D7)⋊C8—22no—
82(C2×D7)⋊C8C2×C412yes—
83(C2×D7)⋊C8C2×C412yes—
84#6389—22no—
85#6389C2×C412yes—
86#6389C2×C412yes—
87Dic7⋊C8—22no—
88Dic7⋊C8—22no—
89Dic7⋊C8C2×C412yes—
90Dic7⋊C8C2×C412yes—
91C8×Dic7—22no—
92C4⋊(C7⋊C8)—21no—
93C4⋊(C7⋊C8)—21no—
94C4⋊(C7⋊C8)C2×C411yescharacteristic
95C4⋊(C7⋊C8)C2×C411yescharacteristic
96C4⋊(C7⋊C8)C2312yes—
97C4⋊(C7⋊C8)C2312yes—
98C4×C7⋊C8C2312yes—
99#2991—71no—
100#2990—71no—
101#2965—71no—
102#1574—71no—
103#1574—71no—
104#1572—72no—
105#1571—72no—
106#1569—72no—
107#1567—71no—
108#1565—72no—
109#1564D711yescharacteristic, complemented
110C22×C4×C7C22×C411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×Q811yescharacteristic
113C2×C22⋊D7C4⋊C412yes—
114#1157—42no—
115C2×D28C4⋊C411yescharacteristic
116C2×C4×D7C22×C412yes—
117C2×C4×D7C2×D412yes—
118C2×C4×D7C2×Q812yes—
119C2×Dic14C4⋊C411yescharacteristic
120C2×C7×C8—22no—
121C2×C7×C8—21no—
122C2×C7×C8—21no—
123C2×C7×C8—21no—
124C2×C7×C8—21no—
125C2×C7×C8—41no—
126C2×C7×C8C22×C411yescharacteristic
127C2×C7×C8C22×C411yescharacteristic
128C2×C7×C8C22×C411yescharacteristic
129C2×C7×C8C22×C411yescharacteristic
130C42×C7C22×C411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7C2×Q811yescharacteristic
133C42×C7C2411yescharacteristic
134C22⋊Dic7C4⋊C412yes—
135(C2×D7)⋊C4C4⋊C412yes—
136(C2×D7)⋊C4C4⋊C412yes—
137C4⋊Dic7C4⋊C411yescharacteristic
138C4⋊Dic7C4⋊C411yescharacteristic
139Dic7⋊C4C4⋊C412yes—
140Dic7⋊C4C4⋊C412yes—
141C4×Dic7C22×C412yes—
142C4×Dic7C2×D412yes—
143C4×Dic7C2×Q812yes—
144#1136—81no—
145#1136—81no—
146C2×C7⋊C8—22no—
147C2×C7⋊C8—22no—
148C2×C7⋊C8C22×C412yes—
149C2×C7⋊C8C22×C412yes—
150#1131—24no—
151#1131—42no—
152#1131—82no—
153C8×D7—82no—
154C4×#41—71no—
155#446—71no—
156#446—71no—
157#446—141no—
158C8×Q8—141no—
159#436—71no—
160#436—71no—
161#436—141no—
162#436—142no—
163#435—72no—
164#435—72no—
165#435—142no—
166C8×D4—141no—
167#432—72no—
168#431—71no—
169#431—72no—
170#431—72no—
171#431C2×D711yescharacteristic
172#431C2×D711yescharacteristic
173#423—71no—
174#423—71no—
175#423—72no—
176C2×C4⋊C8—72no—
177C4×M16—72no—
178C2×C4×C8C2×D711yescharacteristic
179#335—72no—
180#335—72no—
181#335—72no—
182#335—72no—
183#335C2×D711yescharacteristic
184#335C2×D711yescharacteristic
185#334—72no—
186#334—72no—
187#334C2×D711yescharacteristic
188#334C2×D711yescharacteristic
189C23×C7C2×C4⋊C411yescharacteristic
190C22×D7C2×C4⋊C412yes—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—22no—
196C2×C4×C7C2×C4⋊C411yescharacteristic
197C2×C4×C7C2×C4⋊C411yescharacteristic
198C2×C4×C7C2×C4⋊C411yescharacteristic
199C2×C4×C7C23×C411yescharacteristic
200C2×C4×C7C22×D411yescharacteristic
201C2×C4×C7C22×Q811yescharacteristic
202C22⋊D7—42no—
203C2×Dic7C2×C4⋊C412yes—
204C2×Dic7C2×C4⋊C412yes—
205C2×Dic7C2×C4⋊C412yes—
206D28—22no—
207C4×D7—22no—
208C4×D7—22no—
209C4×D7—22no—
210C4×D7—22no—
211C4×D7—42no—
212Dic14—22no—
213C7×C8—21no—
214C7×C8—21no—
215C7×C8—21no—
216C7×C8—41no—
217C7×C8—41no—
218C7×C8—41no—
219C7×C8—41no—
220C7×C8#13112yes—
221C7⋊C8—22no—
222C7⋊C8—22no—
223C7⋊C8—42no—
224C7⋊C8—42no—
225C2×#41—71no—
226C2×M16—72no—
227C2×M16—72no—
228C2×M16—142no—
229C2×M16—141no—
230C2×M16—141no—
231C22×C8—142no—
232C22×C8C4×D711yescharacteristic
233C22×C8C4×D711yescharacteristic
234C4×Q8—71no—
235C4×Q8—71no—
236C4×D4—72no—
237C4×D4—72no—
238C4×D4—71no—
239C4×D4—71no—
240#117—71no—
241#117—72no—
242C2×C42—72no—
243C2×C42C22×D711yescharacteristic
244C4⋊C8—72no—
245C4⋊C8—72no—
246C4⋊C8—71no—
247C4⋊C8—71no—
248C4⋊C8—72no—
249C4⋊C8—72no—
250C4⋊C8—142no—
251C4⋊C8—142no—
252C4⋊C8—141no—
253C4⋊C8—141no—
254C4⋊C8C22×D711yescharacteristic
255C4⋊C8C22×D711yescharacteristic
256C4⋊C8C22×D711yescharacteristic
257C4⋊C8C22×D711yescharacteristic
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—72no—
261C22⋊C8—72no—
262C22⋊C8—72no—
263C22⋊C8—142no—
264C22⋊C8—142no—
265#97—72no—
266#97—72no—
267#97—142no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—72no—
271C4×C8—142no—
272C4×C8C4×D711yescharacteristic
273C4×C8C4×D711yescharacteristic
274C4×C8C22×D711yescharacteristic
275C4×C8C22×D711yescharacteristic
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7C22×C4⋊C411yescharacteristic
280C2×D7—22no—
281C2×D7—22no—
282C2×D7—22no—
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7#43012yes—
287C4×C7C2×#13111yescharacteristic
288C4×C7#56811yescharacteristic
289C4×C7#57611yescharacteristic
290C4×C7#57811yescharacteristic
291Dic7—22no—
292Dic7—22no—
293Dic7—22no—
294#41—282no—
295C2×Q8—71no—
296C2×D4—72no—
297C2×D4—71no—
298C22×C4—72no—
299C22×C4—72no—
300C22×C4—72no—
301C22×C4C2×C4×D711yescharacteristic
302C22×C4D4×D711yescharacteristic
303C22×C4Q8×D711yescharacteristic
304M16—144no—
305M16—282no—
306M16—562no—
307M16—561no—
308M16—561no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—72no—
316C2×C8—72no—
317C2×C8—142no—
318C2×C8—142no—
319C2×C8—562no—
320C2×C8C2×C4×D711yescharacteristic
321C2×C8C2×C4×D711yescharacteristic
322C2×C8C2×C4×D711yescharacteristic
323C2×C8C2×C4×D711yescharacteristic
324C4⋊C4—72no—
325C4⋊C4—72no—
326C4⋊C4—71no—
327C4⋊C4—71no—
328C22⋊C4—72no—
329C22⋊C4—72no—
330C22⋊C4—72no—
331C42—72no—
332C42—72no—
333C42—72no—
334C42C2×C4×D711yescharacteristic
335C42D4×D711yescharacteristic
336C42Q8×D711yescharacteristic
337C42C23×D711yescharacteristic
338C2×C7—41no—
339C2×C7#290611yescharacteristic
340C2×C7#291311yescharacteristic
341C2×C7C2×#43011yescharacteristic
342D7—42no—
343C23—72no—
344C23D7×C4⋊C411yescharacteristic
345Q8—142no—
346D4—142no—
347D4—282no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—72no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—142no—
357C2×C4—142no—
358C2×C4—142no—
359C2×C4—142no—
360C2×C4—282no—
361C2×C4D7×C4⋊C411yescharacteristic
362C2×C4D7×C4⋊C411yescharacteristic
363C2×C4D7×C4⋊C411yescharacteristic
364C2×C4C22×C4×D711yescharacteristic
365C2×C4C2×D4×D711yescharacteristic
366C2×C4C2×Q8×D711yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—41no—
371C8—41no—
372C8—41no—
373C8—41no—
374C8—142no—
375C8—142no—
376C8—282no—
377C8—282no—
378C8#646412yes—
379C7#1783911yescharacteristic, complemented, Sylow
380C22—21no—
381C22—21no—
382C22—21no—
383C22—142no—
384C22—142no—
385C22—142no—
386C22C2×D7×C4⋊C411yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—142no—
391C4—142no—
392C4—142no—
393C4#8966512yes—
394C4C2×#646411yescharacteristic
395C4#9043611yescharacteristic
396C4#9046011yescharacteristic
397C4#9046211yescharacteristic
398C2—41no—
399C2—282no—
400C2#1173645711yescharacteristic
401C2#1173662411yescharacteristic
402C2C2×#8966511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1564C2
socleC22×C7C22×C4⋊C4
prime radicalC2×#1157Q8
cosocleC42×C7C24
hypercenter#1564D7
hypocenterC7#17839
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
4. C7×#1564C2semidirect
5. #11728502C2semidirect
6. #11728502C2semidirect
7. #11727485C2semidirect
8. #11726327C2semidirect
9. #11726326C2semidirect
10. #11726285C2semidirect
11. #11726282C2semidirect
109. #1564D7semidirect
379. C7#17839semidirect

Maximal subgroups

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

subgroupindex
#117367392
#117367382
#117358922
C7×#15642
#117285022
#117274852
#117263272
#117263262
#117262852
#117262822
#178397