Finite Group Databasebeta

← #13264567

Subgroups — #13264567

Order 1792.

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

normal subgroups only

This group has 434 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×#2395C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729357C211yesmaximal, characteristic
4#11725457C212yesmaximal, complemented
5#11725455C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#117C411yescharacteristic, complemented
8C7×#400C411yescharacteristic, complemented
9C7×#399C2211yescharacteristic
10C7×#398C411yescharacteristic, complemented
11C7×#397—21no—
12C7×#397C2211yescharacteristic
13C7×#394—21no—
14C4×C7×C4⋊C4—21no—
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19#89698—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11822—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4C2311yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7—21no—
44C2×C42×C7D411yescharacteristic
45C2×C22⋊Dic7—22no—
46C2×C4⋊Dic7—41no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95C2×C411yescharacteristic
52C7×#95C2×C411yescharacteristic
53#6408—42no—
54C7⋊#95—41no—
55C22⋊(C7⋊C8)—22no—
56C4⋊(C7⋊C8)—41no—
57#6377—41no—
58#2395D711yescharacteristic
59#1799—71no—
60#1466—71no—
61#1352—72no—
62#1344—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
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—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×Dic7—42no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C42×C7—21no—
98C42×C7—21no—
99C42×C7—42no—
100C42×C7—41no—
101C42×C7—41no—
102C22⋊Dic7—42no—
103C22⋊Dic7—42no—
104C4⋊Dic7—82no—
105C2×C7⋊C8—42no—
106C2×C4×D4C2×D711yescharacteristic
107C2×#117Dic711yescharacteristic, complemented
108#432—141no—
109C2×#99—141no—
110#400Dic711yescharacteristic, complemented
111#399C2×D711yescharacteristic
112#398Dic711yescharacteristic, complemented
113#397—21no—
114#397C2×D711yescharacteristic
115#394—21no—
116#380—141no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4Dic711yescharacteristic, complemented
120#342—72no—
121#323—72no—
122#323—141no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7C2×C22⋊C411yescharacteristic
133C23×C7AGL(1,ℤ/8)11yescharacteristic
134C23×C7#13711yescharacteristic
135C7×D4—42no—
136C7×D4—42no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
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—81no—
163C2×C4×C7#9911yescharacteristic
164C2×C4×C7#9911yescharacteristic
165C2×Dic7—82no—
166C2×Dic7—82no—
167C2×Dic7—82no—
168C7⋊C8—82no—
169C22×D4C22⋊D711yescharacteristic
170C23×C4C2×Dic711yescharacteristic
171C23×C4C22⋊D711yescharacteristic
172C4×D4—41no—
173C4×D4—41no—
174#117—41no—
175#117—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—281no—
179C2×C4⋊C4C2×Dic711yescharacteristic
180C2×C4⋊C4C2×Dic711yescharacteristic
181C2×C4⋊C4C2×Dic711yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—142no—
187C2×C22⋊C4C22⋊D711yescharacteristic
188C2×C22⋊C4C22×D711yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C22⋊D711yescharacteristic
192C4⋊C8—281no—
193#99—282no—
194C22⋊C8—142no—
195#97—281no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—281no—
201#95C2×Dic711yescharacteristic
202#95C2×Dic711yescharacteristic
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7C2×#9911yescharacteristic
218C22×C7#41711yescharacteristic
219C22×C7#42111yescharacteristic
220C4×C7—21no—
221C4×C7—42no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7#35212yes—
230Dic7—162no—
231C24C22⋊Dic711yescharacteristic
232C24C2×C22⋊D711yescharacteristic
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—282no—
248C22×C4C22⋊Dic711yescharacteristic
249C22×C4C22⋊Dic711yescharacteristic
250C22×C4C22⋊Dic711yescharacteristic
251C22×C4C22×Dic711yescharacteristic
252C22×C4C2×C22⋊D711yescharacteristic
253C2×C8—282no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—562no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—282no—
268C22⋊C4—282no—
269C42—21no—
270C42—21no—
271C42—42no—
272C42—41no—
273C42—41no—
274C2×C7—21no—
275C2×C7—21no—
276C2×C7—41no—
277C2×C7—41no—
278C2×C7—81no—
279C2×C7#151311yescharacteristic
280C2×C7C2×#35211yescharacteristic
281C2×C7#212611yescharacteristic
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#651211yescharacteristic
292C23#651411yescharacteristic
293C23C2×C22⋊Dic711yescharacteristic
294D4—42no—
295D4—42no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
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—81no—
322C2×C4—562no—
323C2×C4—562no—
324C2×C4—562no—
325C2×C4#640811yescharacteristic
326C2×C4#640811yescharacteristic
327C8—562no—
328C7#1182211yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#640811yescharacteristic
344C22#9000411yescharacteristic
345C22#9000811yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—1122no—
356C4#8933412yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2C2×#8933411yescharacteristic
363C2#1172946311yescharacteristic
364C2#1172957311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2395C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2395D7
hypocenterC7#11822
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
4. #11725457C2semidirect
5. #11725455C2semidirect
7. C2×C7×#117C4semidirect
8. C7×#400C4semidirect
10. C7×#398C4semidirect
16. C4×C7×C22⋊C4C4semidirect
107. C2×#117Dic7semidirect
110. #400Dic7semidirect
112. #398Dic7semidirect
119. C4×C22⋊C4Dic7semidirect
328. C7#11822semidirect

Maximal subgroups

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

subgroupindex
C7×#23952
#117293732
#117293572
#117254572
#117254552
#118227