Finite Group Databasebeta

← #13264125

Subgroups — #13264125

Order 1792.

606 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2397C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic, complemented
3#11729354C211yesmaximal, characteristic, complemented
4#11725457C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C411yescharacteristic, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C7×#399—21no—
11C7×#399C411yescharacteristic, complemented
12C7×#399C2211yescharacteristic
13C7×#392—21no—
14C7×#392C411yescharacteristic, complemented
15C7×#392C2211yescharacteristic
16C7×#376C411yescharacteristic, complemented
17C2×#6408—21no—
18C4×C22⋊Dic7—21no—
19C2×C22⋊(C7⋊C8)—21no—
20#89323—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11726—71nomaximal, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C4×C7×D4—44no—
34C4×C7×D4—44no—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7C2×C411yescharacteristic
50C2×C22⋊Dic7—22no—
51C2×C4×Dic7—41no—
52C22×C7⋊C8—41no—
53C7×#95—21no—
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56#6408—24no—
57C7⋊#95—41no—
58C22⋊(C7⋊C8)—22no—
59C22⋊(C7⋊C8)—41no—
60#2397D711yescharacteristic
61C4×#99—71no—
62#1459—71no—
63#1352—72no—
64#1343—72no—
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—41no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—42no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7—42no—
89C22×C4×C7—42no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22×C411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×Dic7—42no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C42×C7—24no—
110C42×C7—21no—
111C42×C7—21no—
112C22⋊Dic7—24no—
113C22⋊Dic7—42no—
114C4×Dic7—82no—
115C2×C7⋊C8—42no—
116C2×C7⋊C8—82no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4Dic711yescharacteristic, complemented
120C2×C4×D4C2×D711yescharacteristic
121C2×#99—141no—
122C2×C22⋊C8—141no—
123#399—21no—
124#399Dic711yescharacteristic, complemented
125#399C2×D711yescharacteristic
126#392—21no—
127#392Dic711yescharacteristic, complemented
128#392C2×D711yescharacteristic
129C4×C22⋊C4—141no—
130#376Dic711yescharacteristic, complemented
131#342—72no—
132#324—141no—
133#323—72no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—22no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7#10412yes—
150C23×C7C2×C22⋊C411yescharacteristic
151C23×C7#13512yes—
152C7×D4—44no—
153C7×D4—42no—
154C7×D4—48no—
155C7×D4—48no—
156C7×D4—42no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7—82no—
181C2×C4×C7—82no—
182C2×C4×C7#9911yescharacteristic
183C2×C4×C7#9911yescharacteristic
184C2×Dic7—44no—
185C2×Dic7—82no—
186C2×Dic7—82no—
187C7⋊C8—82no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C2×Dic711yescharacteristic
191C22×D4C22⋊D711yescharacteristic
192C23×C4—21no—
193C23×C4C22⋊D711yescharacteristic
194C23×C4C22⋊D711yescharacteristic
195C22×C8—281no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic711yescharacteristic
205C2×C4⋊C4C22⋊D711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—142no—
212C2×C22⋊C4C2×Dic711yescharacteristic
213C2×C22⋊C4C22×D711yescharacteristic
214C2×C42—21no—
215C2×C42—281no—
216C2×C42C2×Dic711yescharacteristic
217#99—144no—
218C22⋊C8—142no—
219C22⋊C8—281no—
220#95—21no—
221#95—281no—
222#95C2×Dic711yescharacteristic
223#95C2×Dic711yescharacteristic
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—42no—
231C22×C7—42no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—42no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7C2×#9911yescharacteristic
244C22×C7#41611yescharacteristic
245C22×C7C2×#10411yescharacteristic
246C4×C7—21no—
247C4×C7—42no—
248C4×C7—42no—
249C4×C7—41no—
250C4×C7—41no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7—81no—
254C4×C7#35312yes—
255Dic7—84no—
256C24—21no—
257C24C22⋊Dic711yescharacteristic
258C24C2×C22⋊D711yescharacteristic
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—282no—
284C22×C4C22⋊Dic711yescharacteristic
285C22×C4C22⋊Dic711yescharacteristic
286C22×C4C22⋊Dic711yescharacteristic
287C22×C4C22×Dic711yescharacteristic
288C22×C4C2×C22⋊D711yescharacteristic
289C2×C8—282no—
290C2×C8—562no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—144no—
303C22⋊C4—282no—
304C42—24no—
305C42—21no—
306C42—21no—
307C42—562no—
308C2×C7—21no—
309C2×C7—21no—
310C2×C7—42no—
311C2×C7—42no—
312C2×C7—82no—
313C2×C7#150111yescharacteristic
314C2×C7#212011yescharacteristic
315C2×C7C2×#35311yescharacteristic
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23#641112yes—
332C23#651312yes—
333C23C2×C22⋊Dic711yescharacteristic
334D4—44no—
335D4—42no—
336D4—48no—
337D4—48no—
338D4—42no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—284no—
365C2×C4—562no—
366C2×C4—562no—
367C2×C4#640811yescharacteristic
368C2×C4#640811yescharacteristic
369C8—562no—
370C7#1172611yescharacteristic, complemented, Sylow
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×#640811yescharacteristic
391C22#9000511yescharacteristic
392C22C2×#641111yescharacteristic
393C4—21no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—564no—
402C4#8934012yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—82no—
408C2#1172946111yescharacteristic
409C2#1172947011yescharacteristic
410C2C2×#8934011yescharacteristic
subgroupG/H
centerC2×C4#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2397C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2397D7
hypocenterC7#11726
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2397

Complements

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

NH ≅ G/Ntype
2. C4×#6408C2semidirect
3. #11729354C2semidirect
4. #11725457C2semidirect
5. #11725451C2semidirect
8. C2×C4×C7×D4C4semidirect
11. C7×#399C4semidirect
14. C7×#392C4semidirect
16. C7×#376C4semidirect
119. C2×C4×D4Dic7semidirect
124. #399Dic7semidirect
127. #392Dic7semidirect
130. #376Dic7semidirect
370. C7#11726semidirect

Maximal subgroups

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

subgroupindex
C7×#23972
C4×#64082
#117293542
#117254572
#117254512
#117267