Finite Group Databasebeta

← #13264255

Subgroups — #13264255

Order 1792.

500 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#11739729C211yesmaximal, characteristic
2C7×#1740C211yesmaximal, characteristic, complemented
3#11729326C211yesmaximal, characteristic, complemented
4#11725298C212yesmaximal, complemented
5#11725295C212yesmaximal
6C2×C7×#117—21no—
7C2×C7×#117C2211yescharacteristic
8C2×C4×D28C411yescharacteristic, complemented
9#89985—21no—
10#89763C411yescharacteristic, complemented
11#89761—21no—
12#89758C411yescharacteristic, complemented
13#89732C2211yescharacteristic
14#89728—21no—
15#89721C2211yescharacteristic
16#89712—21no—
17#89709C411yescharacteristic, complemented
18#89698—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21#89323—21no—
22#89320C2212yes—
23#11754—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×C22⋊D7—21no—
26C22×D28C2×C411yescharacteristic
27C22×C4×D7—21no—
28C7×#117—24no—
29C7×#117—41no—
30C7×#117—41no—
31C7×#117—42no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C7×C22⋊C4C2311yescharacteristic
38C2×C42×C7—41no—
39C2×C42×C7C2×C411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C22⋊Dic7—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×(C2×D7)⋊C4—21no—
45C2×(C2×D7)⋊C4—21no—
46C2×C4⋊Dic7C2×C411yescharacteristic
47C2×Dic7⋊C4—21no—
48C4×D28—41no—
49C4×D28—41no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95C2×C411yescharacteristic
55C7⋊#95C2×C411yescharacteristic
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2598—71no—
60#1740D711yescharacteristic, complemented
61#1464—71no—
62#1352—72no—
63#1343—72no—
64C24×C7C2×D411yescharacteristic
65C23×D7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—22no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C2×C22⋊D7—42no—
79C2×C22⋊D7—41no—
80C2×C22⋊D7—41no—
81C2×C22⋊D7—81no—
82C22×Dic7—21no—
83C22×Dic7—21no—
84C22×Dic7—21no—
85C2×D28—41no—
86C2×D28—42no—
87C2×C4×D7—42no—
88C2×C4×D7—81no—
89C2×C4×D7—81no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—44no—
93C7×C22⋊C4—24no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C42×C7—21no—
98C42×C7—21no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—44no—
102C22⋊Dic7—81no—
103(C2×D7)⋊C4—42no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4—42no—
107(C2×D7)⋊C4—81no—
108C4⋊Dic7—41no—
109Dic7⋊C4—81no—
110C2×C7⋊C8—42no—
111C2×C4×D4—71no—
112C2×#117—21no—
113C2×#117C2×D711yescharacteristic
114#432—141no—
115#400—71no—
116#399—71no—
117#394—71no—
118#392—71no—
119#388—141no—
120#386—141no—
121#386—141no—
122#385—141no—
123#383—71no—
124#342—21no—
125#342C2×D712yes—
126#324—141no—
127#323—72no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7C2×C22⋊C411yescharacteristic
133C23×C7AGL(1,ℤ/8)11yescharacteristic
134C23×C7#13711yescharacteristic
135C22×D7—22no—
136C22×D7—41no—
137C22×D7—41no—
138C22×D7—41no—
139C22×D7—41no—
140C22×D7—41no—
141C22×D7—41no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—22no—
145C2×C4×C7—22no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—44no—
159C2×C4×C7—82no—
160C2×C4×C7—82no—
161C2×C4×C7#9911yescharacteristic
162C2×C4×C7#9911yescharacteristic
163C2×C4×C7#10412yes—
164C22⋊D7—82no—
165C22⋊D7—82no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175D28—42no—
176D28—42no—
177C4×D7—84no—
178C7⋊C8—82no—
179C22×D4—71no—
180C22×D4—141no—
181C23×C4—141no—
182C23×C4C22⋊D711yescharacteristic
183C4×D4—281no—
184C4×D4—281no—
185#117—24no—
186#117—41no—
187#117—41no—
188#117—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—141no—
192C2×C4⋊C4C4×D711yescharacteristic
193C2×C4⋊C4D2811yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—141no—
196C2×C22⋊C4—141no—
197C2×C22⋊C4—141no—
198C2×C22⋊C4—141no—
199C2×C22⋊C4—141no—
200C2×C22⋊C4C22⋊D711yescharacteristic
201C2×C22⋊C4C22×D711yescharacteristic
202C2×C42—41no—
203C2×C42C4×D711yescharacteristic
204C2×C42D2811yescharacteristic
205C4⋊C8—281no—
206C22⋊C8—142no—
207#97—281no—
208#95—71no—
209#95—71no—
210#95—141no—
211#95—141no—
212#95—141no—
213#95—141no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—81no—
223C22×C7C2×#9911yescharacteristic
224C22×C7#41711yescharacteristic
225C22×C7C2×#10411yescharacteristic
226C2×D7—42no—
227C2×D7—42no—
228C2×D7—42no—
229C2×D7—81no—
230C2×D7—81no—
231C2×D7—81no—
232C2×D7—81no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—22no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—84no—
241Dic7—81no—
242Dic7—81no—
243Dic7—81no—
244C24—141no—
245C24C2×C22⋊D711yescharacteristic
246C2×D4—281no—
247C2×D4—282no—
248C2×D4—282no—
249C2×D4—281no—
250C2×D4—281no—
251C2×D4—561no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4—282no—
262C22×C4—561no—
263C22×C4—561no—
264C22×C4(C2×D7)⋊C411yescharacteristic
265C22×C4(C2×D7)⋊C411yescharacteristic
266C22×C4(C2×D7)⋊C411yescharacteristic
267C22×C4(C2×D7)⋊C411yescharacteristic
268C22×C4C2×C4×D711yescharacteristic
269C22×C4C2×D2811yescharacteristic
270C2×C8—282no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—44no—
274C4⋊C4—281no—
275C4⋊C4—561no—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—282no—
281C22⋊C4—282no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C22⋊C4—561no—
285C22⋊C4—561no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—44no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—41no—
294C2×C7—41no—
295C2×C7#150411yescharacteristic
296C2×C7#212011yescharacteristic
297C2×C7#212811yescharacteristic
298D7—82no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—142no—
304C23—281no—
305C23—281no—
306C23—281no—
307C23—281no—
308C23—281no—
309C23—281no—
310C23C2×(C2×D7)⋊C411yescharacteristic
311C23#649511yescharacteristic
312C23#650511yescharacteristic
313D4—282no—
314D4—282no—
315D4—562no—
316D4—562no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—281no—
337C2×C4—281no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—281no—
345C2×C4—564no—
346C2×C4#637912yes—
347C2×C4#638011yescharacteristic
348C2×C4#638011yescharacteristic
349C8—562no—
350C7#1175411yescharacteristic, complemented, Sylow
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—282no—
361C22—282no—
362C22—282no—
363C22—561no—
364C22—561no—
365C22—561no—
366C22—561no—
367C22C2×#637911yescharacteristic
368C22C2×#638011yescharacteristic
369C22#8973811yescharacteristic
370C4—22no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—84no—
378C4—561no—
379C4—561no—
380C4—561no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—562no—
386C2#1172715211yescharacteristic
387C2#1172781011yescharacteristic
388C2#1172781511yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1740C2
socleC22×C7C2×#99
prime radical#89732C22
cosocleC2×C7×C22⋊C4C23
hypercenter#1740D7
hypocenterC7#11754
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C7×#1740C2semidirect
3. #11729326C2semidirect
4. #11725298C2semidirect
8. C2×C4×D28C4semidirect
10. #89763C4semidirect
12. #89758C4semidirect
17. #89709C4semidirect
60. #1740D7semidirect
350. C7#11754semidirect

Maximal subgroups

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

subgroupindex
#117397292
C7×#17402
#117293262
#117252982
#117252952
#117547