Finite Group Databasebeta

← #13272476

Subgroups — #13272476

Order 1792.

492 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2566C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3C4×#6408C211yesmaximal, characteristic
4#11725447C212yesmaximal
5#11725447C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#396—21no—
9C7×#396—21no—
10C7×#390C2211yescharacteristic
11C7×#390C2211yescharacteristic
12C7×#385—21no—
13C7×#385—21no—
14C7×#384C411yescharacteristic, complemented
15C7×#382C411yescharacteristic, complemented
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C2×#6408C2212yes—
18C2×#6408C2212yes—
19#89984—21no—
20C4×C22⋊Dic7—21no—
21#89321—21no—
22#89321—21no—
23#13300—71nomaximal, Sylow
24C22×C7×D4C2311yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C7×#125—41no—
28C7×#125—41no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C2×C22⋊Dic7—22no—
49C2×C22⋊Dic7—22no—
50C2×C4⋊Dic7—41no—
51C2×C4×Dic7—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55#6408—22no—
56#6408—24no—
57#6408Q814yes—
58C7⋊#95—41no—
59C7⋊#95—41no—
60#2566D711yescharacteristic
61#1799—71no—
62C4×#99—71no—
63#1393—72no—
64#1393—72no—
65C24×C7C2×D411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4C2×Q812yes—
70C2×C7×D4#4112yes—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—81no—
80C22×C4×C7—81no—
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22×C411yescharacteristic
86C22×Dic7—42no—
87C22×Dic7—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—81no—
96C7×C4⋊C4—81no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—81no—
102C7×C22⋊C4—81no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—41no—
106C42×C7—41no—
107C22⋊Dic7—24no—
108C22⋊Dic7—42no—
109C22⋊Dic7—42no—
110C22⋊Dic7—42no—
111C4⋊Dic7—82no—
112C4×Dic7—82no—
113C2×#125Dic711yescharacteristic, complemented
114C2×C4×D4C2×D711yescharacteristic
115C2×#99—72no—
116C2×#99—72no—
117#396—21no—
118#396—21no—
119#390C2×D711yescharacteristic
120#390C2×D711yescharacteristic
121#385—21no—
122#385—21no—
123#384Dic711yescharacteristic, complemented
124#382Dic711yescharacteristic, complemented
125#380—141no—
126C4×C4⋊C4Dic711yescharacteristic, complemented
127C4×C22⋊C4—141no—
128#342—141no—
129#342—141no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7C2×C22⋊C411yescharacteristic
139C23×C7C22⋊Q812yes—
140C23×C7#12411yescharacteristic
141C23×C7C4⋊Q811yescharacteristic
142C7×D4—42no—
143C7×D4—42no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—22no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—41no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
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—81no—
168C2×C4×C7—81no—
169C2×C4×C7#9911yescharacteristic
170C2×C4×C7#9911yescharacteristic
171C2×C4×C7C4×Q812yes—
172C2×Dic7—44no—
173C2×Dic7—82no—
174C2×Dic7—82no—
175C2×Dic7—82no—
176C2×Dic7—82no—
177C2×Dic7—82no—
178C22×D4C22×D711yescharacteristic
179C23×C4C22⋊D711yescharacteristic
180C23×C4C22⋊D711yescharacteristic
181#125—41no—
182#125—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—281no—
191C2×C4⋊C4C2×Dic711yescharacteristic
192C2×C4⋊C4C2×Dic711yescharacteristic
193C2×C4⋊C4C2×Dic711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—142no—
199C2×C22⋊C4—142no—
200C2×C22⋊C4C22⋊D711yescharacteristic
201C2×C22⋊C4C22⋊D711yescharacteristic
202C2×C42—281no—
203C2×C42C2×Dic711yescharacteristic
204C2×C42C2×Dic711yescharacteristic
205C2×C42C2×Dic711yescharacteristic
206#99—74no—
207#99—142no—
208#99—144no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—281no—
213#95—281no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—42no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7#39111yescharacteristic
229C22×C7C2×#9911yescharacteristic
230C22×C7#41011yescharacteristic
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—22no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—81no—
239C4×C7—81no—
240C4×C7—81no—
241Dic7—84no—
242Dic7—162no—
243C24C2×C22⋊D711yescharacteristic
244C24C2×C22⋊D711yescharacteristic
245C2×D4—41no—
246C2×D4—41no—
247C2×D4Q8×D712yes—
248C2×D4#116112yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—282no—
260C22×C4—282no—
261C22×C4C22⋊Dic711yescharacteristic
262C22×C4C22⋊Dic711yescharacteristic
263C22×C4C22⋊Dic711yescharacteristic
264C22×C4C22⋊Dic711yescharacteristic
265C22×C4C22×Dic711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—562no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—144no—
283C22⋊C4—282no—
284C22⋊C4—282no—
285C22⋊C4—282no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42—562no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—42no—
294C2×C7—41no—
295C2×C7—41no—
296C2×C7#199611yescharacteristic
297C2×C7#212611yescharacteristic
298C2×C7#212711yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23#650711yescharacteristic
308C23#650912yes—
309C23#651011yescharacteristic
310C23C2×C22⋊Dic711yescharacteristic
311D4—42no—
312D4—42no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—284no—
339C2×C4—562no—
340C2×C4—562no—
341C2×C4—562no—
342C2×C4—562no—
343C2×C4—562no—
344C2×C4#640811yescharacteristic
345C2×C4#640811yescharacteristic
346C2×C4Q8×Dic712yes—
347C7#1330011yescharacteristic, complemented, Sylow
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22C2×#640811yescharacteristic
363C22#9000211yescharacteristic
364C22#9001211yescharacteristic
365C4—22no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—564no—
376C4—1122no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2#1172945511yescharacteristic
383C2#1172957311yescharacteristic
384C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2566C2
socleC22×C7C2×#99
prime radical, cosocleC22×C7×D4C23
hypercenter#2566D7
hypocenterC7#13300
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2566

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
14. C7×#384C4semidirect
15. C7×#382C4semidirect
16. C4×C7×C4⋊C4C4semidirect
113. C2×#125Dic7semidirect
123. #384Dic7semidirect
124. #382Dic7semidirect
126. C4×C4⋊C4Dic7semidirect
347. C7#13300semidirect

Maximal subgroups

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

subgroupindex
C7×#25662
#117293732
C4×#64082
#117254472
#133007