Finite Group Databasebeta

← #13242382

Subgroups — #13242382

Order 1792.

570 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 111 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#11739522C211yesmaximal, characteristic
2C7×#1797C211yesmaximal, characteristic, complemented
3#11729281C211yesmaximal, characteristic, complemented
4#11725286C212yesmaximal, complemented
5#11725285C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×#6493C411yescharacteristic
8C2×#6487C411yescharacteristic
9C22×#1136—21no—
10C2×#6438C411yescharacteristic
11C2×C7×C4⋊C8—21no—
12C2×C7×C22⋊C8—21no—
13#89985—21no—
14#89985—21no—
15#89712—21no—
16#89712—21no—
17#89711C2211yescharacteristic
18#89708C411yescharacteristic
19#89701C2211yescharacteristic
20C7×#336C2212yes—
21#89319—21no—
22#89319C2212yes—
23#7710—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×C22⋊D7—21no—
28C22×D28C2×C411yescharacteristic
29C22×Dic14C2×C411yescharacteristic
30C22×C7×C8—22no—
31C2×C42×C7—42no—
32C2×C42×C7C2×C411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36C2×C22⋊Dic7—21no—
37#6493—42no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×C4⋊Dic7C2×C411yescharacteristic
44C2×C4⋊Dic7C2×C411yescharacteristic
45#6487—42no—
46C2×Dic7⋊C4—21no—
47C2×#1136—24no—
48C2×#1136—42no—
49C2×#1136—42no—
50C22×C7⋊C8—22no—
51#6438—42no—
52C7×C4⋊C8—42no—
53C7×C22⋊C8—42no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58#2850—71no—
59#1797D711yescharacteristic, complemented
60#1728—71no—
61#1272—72no—
62#1270—72no—
63C24×C7C22⋊C411yescharacteristic
64C23×D7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C2×C22⋊D7—42no—
83C2×C22⋊D7—42no—
84C2×C22⋊D7—42no—
85C22×Dic7—21no—
86C22×Dic7—21no—
87C22×Dic7—21no—
88C2×D28—22no—
89C2×D28—41no—
90C2×D28—41no—
91C2×Dic14—22no—
92C2×Dic14—41no—
93C2×Dic14—41no—
94C2×C7×C8—42no—
95C2×C7×C8—42no—
96C2×C7×C8—42no—
97C42×C7—22no—
98C42×C7—22no—
99C42×C7—41no—
100C42×C7—41no—
101C22⋊Dic7—42no—
102(C2×D7)⋊C4—42no—
103(C2×D7)⋊C4—42no—
104(C2×D7)⋊C4—42no—
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4—41no—
107(C2×D7)⋊C4—41no—
108(C2×D7)⋊C4—41no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111Dic7⋊C4—42no—
112#1136—48no—
113C2×C7⋊C8—24no—
114C2×C7⋊C8—42no—
115C2×C7⋊C8—42no—
116C22×M16—141no—
117C2×#124—71no—
118C2×C22⋊Q8—71no—
119C2×#121—71no—
120C22×C42C2×D711yescharacteristic
121C2×C4⋊C8—21no—
122C2×C22⋊C8—21no—
123#391—71no—
124#390—71no—
125#388—141no—
126#388—141no—
127#386—141no—
128#386—141no—
129#382—71no—
130#336—72no—
131#336—141no—
132#336C2×D712yes—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7#10412yes—
140C23×C7C2×C22⋊C411yescharacteristic
141C22×D7—41no—
142C22×D7—41no—
143C22×D7—41no—
144C22×D7—41no—
145C22×D7—41no—
146C22×D7—41no—
147C22×D7—41no—
148C2×C4×C7—24no—
149C2×C4×C7—24no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—44no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7#9912yes—
167C2×C4×C7#10012yes—
168C2×C4×C7#13112yes—
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C7#13711yescharacteristic
172C2×C4×C7#13711yescharacteristic
173C22⋊D7—84no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—41no—
183D28—82no—
184Dic14—82no—
185C7×C8—82no—
186C7⋊C8—44no—
187C22×Q8—71no—
188C22×D4—71no—
189C22×D4—141no—
190C23×C4C4×D711yescharacteristic
191C23×C4D2811yescharacteristic
192C23×C4C22⋊D711yescharacteristic
193C2×M16—144no—
194C2×M16—282no—
195C2×M16—282no—
196C22×C8—22no—
197C22×C8—142no—
198#124—282no—
199C22⋊Q8—282no—
200#121—282no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—141no—
204C2×C22⋊C4—141no—
205C2×C22⋊C4—141no—
206C2×C22⋊C4—141no—
207C2×C22⋊C4—141no—
208C2×C22⋊C4—141no—
209C2×C22⋊C4—141no—
210C2×C42—42no—
211C2×C42C4×D711yescharacteristic
212C2×C42D2811yescharacteristic
213C2×C42C22⋊D711yescharacteristic
214C2×C42C22×D711yescharacteristic
215C4⋊C8—42no—
216C22⋊C8—42no—
217#95—141no—
218#95—141no—
219#95—141no—
220#95—141no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7#40811yescharacteristic
229C22×C7C2×#9911yescharacteristic
230C22×C7C2×#10011yescharacteristic
231C22×C7#41711yescharacteristic
232C22×C7#41711yescharacteristic
233C22×C7C2×#10411yescharacteristic
234C22×C7#42111yescharacteristic
235C2×D7—81no—
236C2×D7—81no—
237C2×D7—81no—
238C2×D7—81no—
239C2×D7—81no—
240C2×D7—81no—
241C2×D7—81no—
242C4×C7—22no—
243C4×C7—22no—
244C4×C7—21no—
245C4×C7—22no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—41no—
250Dic7—81no—
251Dic7—81no—
252Dic7—81no—
253C24—141no—
254C24(C2×D7)⋊C411yescharacteristic
255C2×Q8—142no—
256C2×Q8—281no—
257C2×Q8—281no—
258C2×D4—142no—
259C2×D4—281no—
260C2×D4—281no—
261C2×D4—282no—
262C2×D4—282no—
263C2×D4—282no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4—141no—
278C22×C4(C2×D7)⋊C411yescharacteristic
279C22×C4(C2×D7)⋊C411yescharacteristic
280C22×C4(C2×D7)⋊C411yescharacteristic
281C22×C4C2×C4×D711yescharacteristic
282C22×C4C2×D2811yescharacteristic
283C22×C4C2×C22⋊D711yescharacteristic
284M16—288no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—144no—
289C2×C8—282no—
290C2×C8—282no—
291C4⋊C4—281no—
292C4⋊C4—281no—
293C4⋊C4—282no—
294C22⋊C4—282no—
295C22⋊C4—282no—
296C22⋊C4—281no—
297C22⋊C4—281no—
298C22⋊C4—281no—
299C22⋊C4—281no—
300C22⋊C4—282no—
301C22⋊C4—282no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C2×C7—42no—
307C2×C7#147111yescharacteristic
308C2×C7#148311yescharacteristic
309C2×C7#149511yescharacteristic
310C2×C7#150411yescharacteristic
311C2×C7#151311yescharacteristic
312C2×C7#152711yescharacteristic
313C2×C7#156011yescharacteristic
314D7—161no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—281no—
322C23—281no—
323C23—281no—
324C23—281no—
325C23—281no—
326C23—281no—
327C23—281no—
328C23#639912yes—
329C23C2×(C2×D7)⋊C411yescharacteristic
330Q8—562no—
331D4—562no—
332D4—564no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—281no—
352C2×C4—281no—
353C2×C4—281no—
354C2×C4—281no—
355C2×C4—281no—
356C2×C4—281no—
357C2×C4—281no—
358C2×C4—281no—
359C2×C4—281no—
360C2×C4#638012yes—
361C2×C4#639712yes—
362C2×C4#646412yes—
363C2×C4#647111yescharacteristic
364C2×C4#647211yescharacteristic
365C2×C4#649511yescharacteristic
366C2×C4#650511yescharacteristic
367C8—82no—
368C8—284no—
369C7#771011yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—561no—
378C22—561no—
379C22—561no—
380C22—561no—
381C22—561no—
382C22—561no—
383C22—561no—
384C22C2×#638011yescharacteristic
385C22#8973811yescharacteristic
386C22#8977711yescharacteristic
387C22#8988211yescharacteristic
388C22C2×#639711yescharacteristic
389C22#8990511yescharacteristic
390C22C2×#639911yescharacteristic
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—561no—
400C4—561no—
401C4—561no—
402C2—42no—
403C2—1121no—
404C2#1172726211yescharacteristic
405C2#1172767511yescharacteristic
406C2#1172773011yescharacteristic
407C2#1172818111yescharacteristic
408C2#1172825911yescharacteristic
409C2#1172851811yescharacteristic
410C2#1172855811yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1797C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C4
cosocleC2×C42×C7C23
hypercenter#1797D7
hypocenterC7#7710
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C7×#1797C2semidirect
3. #11729281C2semidirect
4. #11725286C2semidirect
5. #11725285C2semidirect
59. #1797D7semidirect
369. C7#7710semidirect

Maximal subgroups

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

subgroupindex
#117395222
C7×#17972
#117292812
#117252862
#117252852
#77107