Finite Group Databasebeta

← C7×#9120 #13186336

Subgroups — C7×#9120 #13186336

Order 1792.

684 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×M16C211yesmaximal, characteristic, complemented
2C7×#2850C211yesmaximal, characteristic
3C7×#1749C211yesmaximal, characteristic, complemented
4C2×C7×#329C212yesmaximal, complemented
5C7×#1295C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C7×#124C2211yescharacteristic
8C2×C7×C22⋊Q8C411yescharacteristic
9C2×C7×#121C411yescharacteristic
10C22×C42×C7C2211yescharacteristic
11C2×C7×C22⋊C8—21no—
12C4×C7×M16—24no—
13C2×C7×#97C2212yes—
14C2×C7×#97C2212yes—
15C2×C4×C7×C8—21no—
16C7×#391C411yescharacteristic
17C7×#390C411yescharacteristic
18C7×#388—21no—
19C7×#388—21no—
20C7×#386—21no—
21C7×#386—21no—
22C7×#382C2211yescharacteristic
23C7×#336—21no—
24C7×#329—24no—
25#9120C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×M16—24no—
33C2×C7×M16—24no—
34C2×C7×M16—24no—
35C22×C7×C8—22no—
36C22×C7×C8—22no—
37C7×#124—22no—
38C7×#124—22no—
39C7×C22⋊Q8—42no—
40C7×#121—42no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4C2×C411yescharacteristic
43C2×C7×C4⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C42×C7—22no—
51C2×C42×C7—22no—
52C2×C42×C7C2×C411yescharacteristic
53C2×C42×C7D411yescharacteristic
54C2×C42×C7D411yescharacteristic
55C2×C42×C7C2311yescharacteristic
56C7×C22⋊C8—24no—
57C7×#97—22no—
58C7×#97—22no—
59C7×#97—22no—
60C7×#97D414yescomplemented
61C4×C7×C8—24no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7×#95—21no—
66C2×C4×M16C2×C711yescharacteristic, complemented
67#2850C2×C711yescharacteristic
68#1749C2×C711yescharacteristic, complemented
69C2×#329C2×C712yescomplemented
70#1295C2×C712yescomplemented
71C24×C7—21no—
72C24×C7C22⋊C411yescharacteristic
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×D4—22no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C22×C4×C7—22no—
83C22×C4×C7—22no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C7×M16—216no—
102C2×C7×C8—24no—
103C2×C7×C8—24no—
104C2×C7×C8—24no—
105C2×C7×C8—24no—
106C2×C7×C8—42no—
107C2×C7×C8—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—41no—
118C42×C7—22no—
119C42×C7—21no—
120C42×C7—22no—
121C42×C7—21no—
122C42×C7C2×D412yes—
123C42×C7#4112yes—
124C22×M16—21no—
125C2×#124C22×C711yescharacteristic
126C2×C22⋊Q8C4×C711yescharacteristic
127C2×#121C4×C711yescharacteristic
128C22×C42C22×C711yescharacteristic
129C2×C22⋊C8—21no—
130C4×M16—24no—
131C2×#97C22×C712yes—
132C2×#97C22×C712yes—
133C2×C4×C8—21no—
134#391C4×C711yescharacteristic
135#390C4×C711yescharacteristic
136#388—21no—
137#388—21no—
138#386—21no—
139#386—21no—
140#382C22×C711yescharacteristic
141#336—21no—
142#329—24no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7#10412yes—
157C23×C7C2×C22⋊C411yescharacteristic
158C7×Q8—82no—
159C7×D4—82no—
160C7×D4—84no—
161C2×C4×C7—24no—
162C2×C4×C7—24no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—44no—
176C2×C4×C7—44no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7#10012yes—
187C2×C4×C7#10412yes—
188C2×C4×C7#10412yes—
189C2×C4×C7C4×D412yes—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12112yes—
192C2×C4×C7#12311yescharacteristic
193C7×C8—28no—
194C7×C8—44no—
195C22×Q8C2×C4×C711yescharacteristic
196C22×D4—21no—
197C22×D4C2×C4×C711yescharacteristic
198C23×C4C2×C4×C711yescharacteristic
199C23×C4C7×D411yescharacteristic
200C23×C4C7×D411yescharacteristic
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C22×C8—22no—
205C22×C8—22no—
206#124—22no—
207#124—22no—
208C22⋊Q8—42no—
209#121—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×C4×C711yescharacteristic
212C2×C4⋊C4C2×C4×C711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×C4×C711yescharacteristic
222C2×C42C7×D411yescharacteristic
223C2×C42C7×D411yescharacteristic
224C2×C42C23×C711yescharacteristic
225C22⋊C8—24no—
226#97—22no—
227#97—22no—
228#97—22no—
229#97C7×D414yescomplemented
230C4×C8—24no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235C22×C7—22no—
236C22×C7—22no—
237C22×C7—22no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7#38611yescharacteristic
250C22×C7C2×#10011yescharacteristic
251C22×C7#41311yescharacteristic
252C22×C7C2×#10411yescharacteristic
253C22×C7C2×#10411yescharacteristic
254C22×C7C2×#10411yescharacteristic
255C22×C7#42111yescharacteristic
256C4×C7—22no—
257C4×C7—22no—
258C4×C7—21no—
259C4×C7—22no—
260C4×C7—21no—
261C4×C7—22no—
262C4×C7—22no—
263C4×C7—81no—
264C4×C7—81no—
265C4×C7—81no—
266C24—21no—
267C24C7×C22⋊C411yescharacteristic
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C7×C22⋊C411yescharacteristic
291C22×C4C7×C22⋊C411yescharacteristic
292C22×C4C7×C22⋊C411yescharacteristic
293C22×C4C22×C4×C711yescharacteristic
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296M16—216no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—21no—
317C42C2×C7×D412yes—
318C42C7×#4112yes—
319C2×C7—42no—
320C2×C7—161no—
321C2×C7#152411yescharacteristic
322C2×C7#152511yescharacteristic
323C2×C7#152711yescharacteristic
324C2×C7#152911yescharacteristic
325C2×C7#187111yescharacteristic
326C2×C7#189611yescharacteristic
327C2×C7#190011yescharacteristic
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C7×#10412yes—
342C23C2×C7×C22⋊C411yescharacteristic
343Q8—82no—
344D4—82no—
345D4—84no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4C7×#10012yes—
372C2×C4C7×#10412yes—
373C2×C4C7×#10412yes—
374C2×C4C4×C7×D412yes—
375C2×C4C7×#12011yescharacteristic
376C2×C4C7×#12112yes—
377C2×C4C7×#12311yescharacteristic
378C8—28no—
379C8—44no—
380C7#912011yescharacteristic, direct product factor, Sylow
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C7×#38611yescharacteristic
396C22C2×C7×#10011yescharacteristic
397C22C7×#41311yescharacteristic
398C22C2×C7×#10411yescharacteristic
399C22C2×C7×#10411yescharacteristic
400C22C2×C7×#10411yescharacteristic
401C22C7×#42111yescharacteristic
402C4—22no—
403C4—22no—
404C4—21no—
405C4—22no—
406C4—21no—
407C4—22no—
408C4—22no—
409C4—81no—
410C4—81no—
411C4—81no—
412C2—42no—
413C2—161no—
414C2C7×#152411yescharacteristic
415C2C7×#152511yescharacteristic
416C2C7×#152711yescharacteristic
417C2C7×#152911yescharacteristic
418C2C7×#187111yescharacteristic
419C2C7×#189611yescharacteristic
420C2C7×#190011yescharacteristic
subgroupG/H
center, socleC23×C7C2×C22⋊C4
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×M16C2semidirect
3. C7×#1749C2semidirect
4. C2×C7×#329C2semidirect
5. C7×#1295C2semidirect
25. #9120C7direct (N × H)
60. C7×#97D4semidirect
66. C2×C4×M16C2×C7semidirect
68. #1749C2×C7semidirect
69. C2×#329C2×C7semidirect
70. #1295C2×C7semidirect
229. #97C7×D4semidirect
380. C7#9120direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×M162
C7×#28502
C7×#17492
C2×C7×#3292
C7×#12952
#91207