Finite Group Databasebeta

← #13264257

Subgroups — #13264257

Order 1792.

500 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2598C211yesmaximal, characteristic
2#11729346C211yesmaximal, characteristic, complemented
3#11729326C211yesmaximal, characteristic, complemented
4#11725457C212yesmaximal, complemented
5#11725451C212yesmaximal
6C2×C4×C7×D4C411yescharacteristic, complemented
7C2×C7×#117C2211yescharacteristic
8C2×#6489—21no—
9C7×#400C411yescharacteristic, complemented
10C7×#399C2211yescharacteristic
11C7×#394C411yescharacteristic, complemented
12C7×#392C2211yescharacteristic
13C7×#388—21no—
14C7×#386—21no—
15C7×#386—21no—
16C7×#385—21no—
17C7×#383C411yescharacteristic, complemented
18#89698—21no—
19#89323—21no—
20#89321—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11754—71nomaximal, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7D411yescharacteristic
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C7×#117—41no—
31C7×#117—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4C2311yescharacteristic
43C2×C42×C7C2×C411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×C22⋊Dic7—22no—
46#6489—24no—
47#6489—42no—
48C2×C4⋊Dic7—41no—
49C2×C4×Dic7—41no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2598D711yescharacteristic
60#1740—71no—
61#1464—71no—
62#1352—72no—
63#1343—72no—
64C24×C7—21no—
65C24×C7C2×D411yescharacteristic
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—81no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×Dic7—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—81no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—81no—
101C7×C22⋊C4—81no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C22⋊Dic7—24no—
107C22⋊Dic7—42no—
108C4⋊Dic7—44no—
109C4×Dic7—44no—
110C2×C7⋊C8—42no—
111C2×C4×D4Dic711yescharacteristic, complemented
112C2×#117—141no—
113C2×#117C2×D711yescharacteristic
114#432—141no—
115#400Dic711yescharacteristic, complemented
116#399C2×D711yescharacteristic
117#394Dic711yescharacteristic, complemented
118#392C2×D711yescharacteristic
119#388—21no—
120#386—21no—
121#386—21no—
122#385—21no—
123#383Dic711yescharacteristic, complemented
124#342—72no—
125#342—141no—
126#324—141no—
127#323—72no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—22no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7C2×C22⋊C411yescharacteristic
140C23×C7AGL(1,ℤ/8)11yescharacteristic
141C23×C7#13711yescharacteristic
142C7×D4—42no—
143C7×D4—42no—
144C7×D4—82no—
145C7×D4—82no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—84no—
172C2×C4×C7#9911yescharacteristic
173C2×C4×C7#9911yescharacteristic
174C2×C4×C7#10412yes—
175C2×Dic7—44no—
176C2×Dic7—82no—
177C2×Dic7—82no—
178C7⋊C8—82no—
179C22×D4—21no—
180C22×D4C2×Dic711yescharacteristic
181C23×C4—21no—
182C23×C4C22⋊D711yescharacteristic
183C4×D4—41no—
184C4×D4—41no—
185#117—41no—
186#117—41no—
187#117—144no—
188#117—282no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—281no—
191C2×C4⋊C4C2×Dic711yescharacteristic
192C2×C4⋊C4C2×Dic711yescharacteristic
193C2×C4⋊C4C22⋊D711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—142no—
200C2×C22⋊C4C22⋊D711yescharacteristic
201C2×C22⋊C4C22×D711yescharacteristic
202C2×C42—281no—
203C2×C42C2×Dic711yescharacteristic
204C2×C42C22⋊D711yescharacteristic
205C4⋊C8—281no—
206C22⋊C8—142no—
207#97—281no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95C2×Dic711yescharacteristic
213#95C2×Dic711yescharacteristic
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—42no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7C2×#9911yescharacteristic
231C22×C7#41711yescharacteristic
232C22×C7C2×#10411yescharacteristic
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—22no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—81no—
241C4×C7—81no—
242C4×C7—81no—
243Dic7—84no—
244C24—21no—
245C24C2×C22⋊D711yescharacteristic
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—81no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—282no—
264C22×C4C22⋊Dic711yescharacteristic
265C22×C4C22⋊Dic711yescharacteristic
266C22×C4C22⋊Dic711yescharacteristic
267C22×C4C22⋊Dic711yescharacteristic
268C22×C4C22×Dic711yescharacteristic
269C22×C4C2×C22⋊D711yescharacteristic
270C2×C8—282no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C4⋊C4—284no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—144no—
285C22⋊C4—282no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—284no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—41no—
294C2×C7—41no—
295C2×C7—82no—
296C2×C7#150411yescharacteristic
297C2×C7#212011yescharacteristic
298C2×C7#212811yescharacteristic
299C23—21no—
300C23—21no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#651211yescharacteristic
311C23#651411yescharacteristic
312C23C2×C22⋊Dic711yescharacteristic
313D4—42no—
314D4—42no—
315D4—82no—
316D4—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—284no—
344C2×C4—562no—
345C2×C4—562no—
346C2×C4#640811yescharacteristic
347C2×C4#640811yescharacteristic
348C2×C4#641112yes—
349C8—562no—
350C7#1175411yescharacteristic, complemented, Sylow
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#640811yescharacteristic
368C22#9000411yescharacteristic
369C22C2×#641111yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—564no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2#1172946211yescharacteristic
387C2#1172947011yescharacteristic
388C2#1172961511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2598C2
socleC22×C7C2×#99
prime radicalC7×#392C22
cosocleC2×C7×C22⋊C4C23
hypercenter#2598D7
hypocenterC7#11754
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
2. #11729346C2semidirect
3. #11729326C2semidirect
4. #11725457C2semidirect
6. C2×C4×C7×D4C4semidirect
9. C7×#400C4semidirect
11. C7×#394C4semidirect
17. C7×#383C4semidirect
111. C2×C4×D4Dic7semidirect
115. #400Dic7semidirect
117. #394Dic7semidirect
123. #383Dic7semidirect
350. C7#11754semidirect

Maximal subgroups

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

subgroupindex
C7×#25982
#117293462
#117293262
#117254572
#117254512
#117547