Finite Group Databasebeta

← #13247357

Subgroups — #13247357

Order 1792.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2353C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725431C212yesmaximal, complemented
5#11725418C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C7×#388—21no—
8C7×#387—21no—
9C7×#386—21no—
10C7×#383C411yescharacteristic, complemented
11C7×#381C411yescharacteristic, complemented
12C7×#380C2211yescharacteristic
13C4×C7×C22⋊C4—21no—
14C4×C7×C22⋊C4C411yescharacteristic, complemented
15C7×#376C411yescharacteristic, complemented
16C2×C7×#95C2211yescharacteristic
17C2×#6408—21no—
18#89984C2212yes—
19C2×C22⋊(C7⋊C8)C2212yes—
20#89323—21no—
21#89321—21no—
22#89320—21no—
23#8763—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C22⋊C4—21no—
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7—21no—
39C2×C42×C7C2×C411yescharacteristic
40C2×C22⋊Dic7—22no—
41C2×C4⋊Dic7—22no—
42C22×C7⋊C8—22no—
43C7×#95—21no—
44C7×#95—21no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95C2×C411yescharacteristic
49C7×#95C2×C411yescharacteristic
50C7×#95C2×C411yescharacteristic
51C7×#95C2×C411yescharacteristic
52#6408—42no—
53C7⋊#95—22no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56#2353D711yescharacteristic
57#1799—71no—
58#1459—71no—
59#1294—72no—
60#1289—72no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—81no—
76C22×C4×C7—81no—
77C22×C4×C7—81no—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7#4112yes—
84C22×C4×C7#4112yes—
85C22×Dic7—42no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—81no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—81no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—42no—
97C42×C7—81no—
98C22⋊Dic7—42no—
99C22⋊Dic7—42no—
100C4⋊Dic7—82no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—82no—
103C2×C4×D4C2×D711yescharacteristic
104C2×#99—141no—
105C2×C22⋊C8—72no—
106#388—21no—
107#387—21no—
108#386—21no—
109#383Dic711yescharacteristic, complemented
110#381Dic711yescharacteristic, complemented
111#380—72no—
112#380C2×D711yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4Dic711yescharacteristic, complemented
115#376Dic711yescharacteristic, complemented
116C2×#95C2×D711yescharacteristic
117#342—141no—
118#324—141no—
119#323—141no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7C2×C22⋊C411yescharacteristic
128C23×C7#12312yes—
129C23×C7#13512yes—
130C23×C7AGL(1,ℤ/8)11yescharacteristic
131C23×C7#13711yescharacteristic
132C7×D4—42no—
133C7×D4—42no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
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—81no—
158C2×C4×C7—81no—
159C2×C4×C7—84no—
160C2×C4×C7#11711yescharacteristic
161C2×C4×C7#11711yescharacteristic
162C2×C4×C7#12312yes—
163C2×Dic7—82no—
164C2×Dic7—82no—
165C2×Dic7—82no—
166C7⋊C8—82no—
167C22×D4C22⋊D711yescharacteristic
168C23×C4—21no—
169C23×C4C2×Dic711yescharacteristic
170C23×C4C22×D711yescharacteristic
171C22×C8—142no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—142no—
176C2×C4⋊C4C22⋊D711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—142no—
181C2×C22⋊C4C22⋊D711yescharacteristic
182C2×C22⋊C4C22⋊D711yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C2×Dic711yescharacteristic
186#99—282no—
187C22⋊C8—142no—
188C22⋊C8—142no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—142no—
195#95C2×Dic711yescharacteristic
196#95C2×Dic711yescharacteristic
197#95C2×Dic711yescharacteristic
198#95C2×Dic711yescharacteristic
199C22×C7—22no—
200C22×C7—21no—
201C22×C7—22no—
202C22×C7—21no—
203C22×C7—42no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7#38111yescharacteristic
212C22×C7#41611yescharacteristic
213C22×C7#41711yescharacteristic
214C22×C7#48212yes—
215C22×C7#48512yes—
216C4×C7—22no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—42no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—81no—
224C4×C7—81no—
225C4×C7—81no—
226Dic7—162no—
227C24C22⋊Dic711yescharacteristic
228C24C2×C22⋊D711yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—22no—
232C2×D4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—282no—
245C22×C4C22⋊Dic711yescharacteristic
246C22×C4C22⋊Dic711yescharacteristic
247C22×C4C22⋊Dic711yescharacteristic
248C22×C4#115712yes—
249C22×C4#115712yes—
250C22×C4C22×Dic711yescharacteristic
251C22×C4C2×C22⋊D711yescharacteristic
252C2×C8—282no—
253C2×C8—562no—
254C4⋊C4—22no—
255C4⋊C4—81no—
256C4⋊C4—562no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C22⋊C4—282no—
264C22⋊C4—282no—
265C42—21no—
266C42—21no—
267C42—42no—
268C42—81no—
269C2×C7—22no—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7—22no—
273C2×C7—81no—
274C2×C7#182511yescharacteristic
275C2×C7#182711yescharacteristic
276C2×C7#182811yescharacteristic
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#649212yes—
285C23#651211yescharacteristic
286C23#651312yes—
287C23#651411yescharacteristic
288C23C2×C22⋊Dic711yescharacteristic
289D4—42no—
290D4—42no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—84no—
317C2×C4—562no—
318C2×C4—562no—
319C2×C4—562no—
320C2×C4#648911yescharacteristic
321C2×C4#648911yescharacteristic
322C2×C4#649212yes—
323C8—562no—
324C7#876311yescharacteristic, complemented, Sylow
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#8976812yes—
338C22#8977512yes—
339C22#8998311yescharacteristic
340C22#9000411yescharacteristic
341C22#9000511yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—1122no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2—81no—
358C2#1172932911yescharacteristic
359C2#1172939811yescharacteristic
360C2#1172940711yescharacteristic
subgroupG/H
centerC22#89983
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2353C2
socleC22×C7#381
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2353D7
hypocenterC7#8763
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
4. #11725431C2semidirect
5. #11725418C2semidirect
10. C7×#383C4semidirect
11. C7×#381C4semidirect
14. C4×C7×C22⋊C4C4semidirect
15. C7×#376C4semidirect
109. #383Dic7semidirect
110. #381Dic7semidirect
114. C4×C22⋊C4Dic7semidirect
115. #376Dic7semidirect
324. C7#8763semidirect

Maximal subgroups

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

subgroupindex
C7×#23532
#117293732
#117293542
#117254312
#117254182
#87637