Finite Group Databasebeta

← #13264916

Subgroups — #13264916

Order 1792.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2580C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725457C212yesmaximal, complemented
5#11725452C212yesmaximal, complemented
6C2×C4×C7×Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#399C411yescharacteristic, complemented
9C7×#399C2211yescharacteristic
10C7×#393C411yescharacteristic, complemented
11C7×#393C2211yescharacteristic
12C7×#390—21no—
13C7×#389—21no—
14C7×#385—21no—
15C7×#385—21no—
16C7×#382C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19C2×C22⋊(C7⋊C8)—21no—
20#89323—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11897—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C4×C7×Q8—41no—
29C4×C7×Q8—41no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
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⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7C2×C411yescharacteristic
46C2×C22⋊Dic7—22no—
47C2×C4⋊Dic7—41no—
48C22×C7⋊C8—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95C2×C411yescharacteristic
53C7×#95C2×C411yescharacteristic
54#6408—42no—
55C7⋊#95—41no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—41no—
58#2580D711yescharacteristic
59#1799—71no—
60#1459—71no—
61#1352—72no—
62#1351—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—42no—
66C2×C7×Q8—41no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4SD1612yes—
70C2×C7×D4Q1612yes—
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×C7C2×D411yescharacteristic
86C22×Dic7—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—81no—
93C7×C4⋊C4—81no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—81no—
99C7×C22⋊C4—81no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—42no—
103C22⋊Dic7—42no—
104C22⋊Dic7—42no—
105C4⋊Dic7—82no—
106C2×C7⋊C8—42no—
107C2×C7⋊C8—82no—
108C2×C4×Q8Dic711yescharacteristic, complemented
109C2×C4×D4C2×D711yescharacteristic
110C2×#99—141no—
111C2×C22⋊C8—141no—
112#399Dic711yescharacteristic, complemented
113#399C2×D711yescharacteristic
114#393Dic711yescharacteristic, complemented
115#393C2×D711yescharacteristic
116#390—21no—
117#389—21no—
118#385—21no—
119#385—21no—
120#382Dic711yescharacteristic, complemented
121#380—141no—
122#342—72no—
123#324—141no—
124#323—72no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7#10312yes—
134C23×C7C2×C22⋊C411yescharacteristic
135C23×C7C2×SD1611yescharacteristic
136C23×C7C2×Q1611yescharacteristic
137C7×Q8—42no—
138C7×Q8—42no—
139C7×D4—42no—
140C7×D4—42no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—41no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
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—81no—
165C2×C4×C7—81no—
166C2×C4×C7#9911yescharacteristic
167C2×C4×C7#9911yescharacteristic
168C2×C4×C7#10312yes—
169C2×Dic7—82no—
170C2×Dic7—82no—
171C2×Dic7—82no—
172C7⋊C8—82no—
173C22×Q8C2×Dic711yescharacteristic
174C22×D4C22⋊D711yescharacteristic
175C23×C4C22⋊D711yescharacteristic
176C23×C4C22⋊D711yescharacteristic
177C22×C8—281no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—281no—
187C2×C4⋊C4C2×Dic711yescharacteristic
188C2×C4⋊C4C22⋊D711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—142no—
194C2×C22⋊C4C2×Dic711yescharacteristic
195C2×C22⋊C4C22×D711yescharacteristic
196C2×C42—21no—
197C2×C42C2×Dic711yescharacteristic
198#99—282no—
199C22⋊C8—142no—
200C22⋊C8—281no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—281no—
205#95C2×Dic711yescharacteristic
206#95C2×Dic711yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—22no—
211C22×C7—21no—
212C22×C7—42no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7C2×#9911yescharacteristic
222C22×C7C2×#10311yescharacteristic
223C22×C7#42111yescharacteristic
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—22no—
227C4×C7—42no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—81no—
234Dic7—162no—
235C24C22⋊Dic711yescharacteristic
236C24C2×C22⋊D711yescharacteristic
237C2×Q8—42no—
238C2×Q8—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4#114312yes—
242C2×D4#114412yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—282no—
254C22×C4C22⋊Dic711yescharacteristic
255C22×C4C22⋊Dic711yescharacteristic
256C22×C4C22⋊Dic711yescharacteristic
257C22×C4C22×Dic711yescharacteristic
258C22×C4C2×C22⋊D711yescharacteristic
259C2×C8—282no—
260C2×C8—562no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—562no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—282no—
276C22⋊C4—282no—
277C42—21no—
278C42—21no—
279C42—42no—
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—41no—
283C2×C7—42no—
284C2×C7—41no—
285C2×C7#152011yescharacteristic
286C2×C7#212011yescharacteristic
287C2×C7#212711yescharacteristic
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#640912yes—
297C23C2×#114311yescharacteristic
298C23C2×#114411yescharacteristic
299C23C2×C22⋊Dic711yescharacteristic
300Q8—42no—
301Q8—42no—
302D4—42no—
303D4—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—562no—
330C2×C4—562no—
331C2×C4—562no—
332C2×C4#640811yescharacteristic
333C2×C4#640811yescharacteristic
334C2×C4#640912yes—
335C8—562no—
336C7#1189711yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22C2×#640811yescharacteristic
352C22C2×#640911yescharacteristic
353C22#9000811yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—1122no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—42no—
369C2—41no—
370C2#1172944411yescharacteristic
371C2#1172947011yescharacteristic
372C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2580C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2580D7
hypocenterC7#11897
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
4. #11725457C2semidirect
5. #11725452C2semidirect
6. C2×C4×C7×Q8C4semidirect
8. C7×#399C4semidirect
10. C7×#393C4semidirect
16. C7×#382C4semidirect
108. C2×C4×Q8Dic7semidirect
112. #399Dic7semidirect
114. #393Dic7semidirect
120. #382Dic7semidirect
336. C7#11897semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117293732
#117293542
#117254572
#117254522
#118977