Finite Group Databasebeta

← C7×#12796 #13190012

Subgroups — C7×#12796 #13190012

Order 1792.

512 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2322C211yesmaximal, characteristic
2C2×C7×#352C212yesmaximal
3C7×#1800C211yesmaximal, characteristic
4C7×#1799C211yesmaximal, characteristic
5C7×#1390C212yesmaximal
6C2×C7×#123C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C4⋊C4C411yescharacteristic, complemented
9C7×#423—21no—
10C2×C7×#100C2212yes—
11C2×C7×#99C2212yes—
12C7×#389—21no—
13C7×#388—21no—
14C7×#387—21no—
15C7×#386C2211yescharacteristic
16C7×#385—21no—
17C7×#385C411yescharacteristic, complemented
18C7×#381C411yescharacteristic, complemented
19C7×#380—21no—
20C4×C7×C22⋊C4C411yescharacteristic, complemented
21C7×#352—24no—
22C7×#343—21no—
23C7×#342—21no—
24#12796C711yesmaximal, characteristic, direct product factor, Sylow
25C22×C7×D4C2311yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C2×C7×M16—22no—
31C7×#123—21no—
32C7×#123—21no—
33C7×#123D412yes—
34C7×#123Q812yes—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—41no—
42C2×C7×C4⋊C4—41no—
43C2×C7×C4⋊C4—41no—
44C2×C7×C4⋊C4D411yescharacteristic
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—22no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4C2×C411yescharacteristic
50C2×C7×C22⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C42×C7—21no—
53C2×C42×C7D411yescharacteristic
54C7×C4⋊C8—41no—
55C7×C4⋊C8—41no—
56C7×#100—22no—
57C7×#100—22no—
58C7×#99—22no—
59C7×#99—22no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7×#95—41no—
66#2322C2×C711yescharacteristic
67C2×#352C2×C712yes—
68#1800C2×C711yescharacteristic
69#1799C2×C711yescharacteristic
70#1390C2×C712yes—
71C24×C7C22×C411yescharacteristic
72C24×C7C2×D411yescharacteristic
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4C2×D411yescharacteristic
76C2×C7×D4C2×Q811yescharacteristic
77C2×C7×D4#4111yescharacteristic
78C2×C7×D4#4111yescharacteristic
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—42no—
92C22×C4×C7—81no—
93C22×C4×C7—81no—
94C22×C4×C7—81no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C4⋊C412yescomplemented
100C22×C4×C7C2×D411yescharacteristic
101C7×M16—42no—
102C7×M16—42no—
103C2×C7×C8—42no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—81no—
111C7×C4⋊C4—82no—
112C7×C22⋊C4—21no—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—81no—
121C7×C22⋊C4C4⋊C412yescomplemented
122C42×C7—21no—
123C42×C7—21no—
124C42×C7—81no—
125C2×#123C22×C711yescharacteristic
126C2×C4×D4C22×C711yescharacteristic
127C22×C4⋊C4C4×C711yescharacteristic, complemented
128#423—21no—
129C2×#100C22×C712yes—
130C2×#99C22×C712yes—
131#389—21no—
132#388—21no—
133#387—21no—
134#386C22×C711yescharacteristic
135#385—21no—
136#385C4×C711yescharacteristic, complemented
137#381C4×C711yescharacteristic, complemented
138#380—21no—
139C4×C22⋊C4C4×C711yescharacteristic, complemented
140#352—24no—
141#343—21no—
142#342—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7C2×C22⋊C411yescharacteristic
152C23×C7C2×C4⋊C411yescharacteristic
153C23×C7#11711yescharacteristic
154C23×C7#12111yescharacteristic
155C23×C7C22⋊Q811yescharacteristic
156C7×D4—41no—
157C7×D4—41no—
158C7×D4—41no—
159C7×D4—41no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
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—81no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C2×C4×C7—82no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7—82no—
191C2×C4×C7#9911yescharacteristic
192C2×C4×C7#9911yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7C22⋊Q811yescharacteristic
195C7×C8—82no—
196C22×D4C23×C711yescharacteristic
197C23×C4C2×C4×C711yescharacteristic
198C23×C4C2×C4×C711yescharacteristic
199C23×C4C2×C4×C711yescharacteristic
200C23×C4C7×D411yescharacteristic
201C2×M16—22no—
202#123—21no—
203#123—21no—
204#123C7×D412yes—
205#123C7×Q812yes—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C7×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C711yescharacteristic
220C2×C22⋊C4C2×C4×C711yescharacteristic
221C2×C22⋊C4C2×C4×C711yescharacteristic
222C2×C22⋊C4C7×D411yescharacteristic
223C2×C42—21no—
224C2×C42C7×D411yescharacteristic
225C4⋊C8—41no—
226C4⋊C8—41no—
227#100—22no—
228#100—22no—
229#99—22no—
230#99—22no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—41no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—42no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—81no—
253C22×C7#38011yescharacteristic
254C22×C7C2×#9911yescharacteristic
255C22×C7#41011yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—42no—
261C4×C7—42no—
262C4×C7—41no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—81no—
266C4×C7—81no—
267C4×C7—162no—
268C24C22×C4×C711yescharacteristic
269C24C2×C7×D411yescharacteristic
270C2×D4—41no—
271C2×D4—41no—
272C2×D4C2×C7×D411yescharacteristic
273C2×D4C2×C7×Q811yescharacteristic
274C2×D4C7×#4111yescharacteristic
275C2×D4C7×#4111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4C7×C22⋊C411yescharacteristic
293C22×C4C7×C22⋊C411yescharacteristic
294C22×C4C7×C22⋊C411yescharacteristic
295C22×C4C7×C22⋊C411yescharacteristic
296C22×C4C7×C4⋊C412yescomplemented
297C22×C4C2×C7×D411yescharacteristic
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C22⋊C4—21no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—81no—
318C22⋊C4C7×C4⋊C412yescomplemented
319C42—21no—
320C42—21no—
321C42—81no—
322C2×C7—21no—
323C2×C7—21no—
324C2×C7—41no—
325C2×C7—41no—
326C2×C7—41no—
327C2×C7—41no—
328C2×C7#179911yescharacteristic
329C2×C7#212111yescharacteristic
330C2×C7#212311yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C2×C7×C22⋊C411yescharacteristic
340C23C2×C7×C4⋊C411yescharacteristic
341C23C7×#11711yescharacteristic
342C23C7×#12111yescharacteristic
343C23C7×C22⋊Q811yescharacteristic
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4C7×#9911yescharacteristic
380C2×C4C7×#9911yescharacteristic
381C2×C4C7×#12111yescharacteristic
382C2×C4C7×C22⋊Q811yescharacteristic
383C8—82no—
384C7#1279611yescharacteristic, direct product factor, Sylow
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C7×#38011yescharacteristic
402C22C2×C7×#9911yescharacteristic
403C22C7×#41011yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4—162no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2—41no—
421C2—41no—
422C2C7×#179911yescharacteristic
423C2C7×#212111yescharacteristic
424C2C7×#212311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
8. C22×C7×C4⋊C4C4semidirect
17. C7×#385C4semidirect
18. C7×#381C4semidirect
20. C4×C7×C22⋊C4C4semidirect
24. #12796C7direct (N × H)
99. C22×C4×C7C4⋊C4semidirect
121. C7×C22⋊C4C4⋊C4semidirect
127. C22×C4⋊C4C4×C7semidirect
136. #385C4×C7semidirect
137. #381C4×C7semidirect
139. C4×C22⋊C4C4×C7semidirect
296. C22×C4C7×C4⋊C4semidirect
318. C22⋊C4C7×C4⋊C4semidirect
384. C7#12796direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23222
C2×C7×#3522
C7×#18002
C7×#17992
C7×#13902
#127967