Finite Group Databasebeta

← C7×#9953 #13187169

Subgroups — C7×#9953 #13187169

Order 1792.

594 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3187C211yesmaximal, characteristic
2C2×C7×#480C211yesmaximal, characteristic
3C7×#2503C211yesmaximal, characteristic, complemented
4C7×#1964C211yesmaximal, characteristic
5C7×#1823C211yesmaximal, characteristic, complemented
6C7×#1520C211yesmaximal, characteristic, complemented
7C7×#1282C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#483—21no—
12C7×#481—21no—
13C7×#480—22no—
14C7×#480—22no—
15C7×#465—22no—
16C7×#450—21no—
17C7×#448—21no—
18C2×C7×#107—21no—
19C2×C7×#102—21no—
20C2×C7×#99—22no—
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C7×#400—22no—
24C7×#391—21no—
25C7×#382—21no—
26C7×#381—22no—
27C7×#380C2211yescharacteristic
28C4×C7×C22⋊C4C2211yescharacteristic
29C7×#340—22no—
30C7×#328—22no—
31#9953C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C23×C4×C7C2311yescharacteristic
37C2×C7×SD16—22no—
38C22×C7×C8—21no—
39C22×C7×C8D412yes—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D412yes—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4D412yes—
62C2×C42×C7—21no—
63C7×#107—24no—
64C7×#106—22no—
65C7×#103—22no—
66C7×#102—24no—
67C7×#102—22no—
68C7×#99—44no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8D412yes—
73C7×#95—24no—
74C7×#95—24no—
75C7×#95—21no—
76C7×#95—21no—
77#3187C2×C711yescharacteristic
78C2×#480C2×C711yescharacteristic
79#2503C2×C711yescharacteristic, complemented
80#1964C2×C711yescharacteristic
81#1823C2×C711yescharacteristic, complemented
82#1520C2×C711yescharacteristic, complemented
83#1282C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C24×C7C2×D411yescharacteristic
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—42no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—42no—
95C2×C7×D4—42no—
96C22×C4×C7—22no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—42no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7—41no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7#4111yescharacteristic
113C7×SD16—82no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—42no—
117C2×C7×C8—42no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—42no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—42no—
136C42×C7—42no—
137C2×C22⋊Q8C22×C711yescharacteristic
138C2×#121C22×C711yescharacteristic
139C22×C4⋊C4C22×C711yescharacteristic
140#483—21no—
141#481—21no—
142#480—22no—
143#480—22no—
144#465—22no—
145#450—21no—
146#448—21no—
147C2×#107—21no—
148C2×#102—21no—
149C2×#99—22no—
150C2×C22⋊C8C22×C711yescharacteristic
151C2×C22⋊C8C22×C711yescharacteristic
152#400—22no—
153#391—21no—
154#382—21no—
155#381—22no—
156#380C22×C711yescharacteristic
157C4×C22⋊C4C22×C711yescharacteristic
158#340—22no—
159#328—22no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7—81no—
165C23×C7—81no—
166C23×C7#12011yescharacteristic
167C23×C7#12111yescharacteristic
168C23×C7#12711yescharacteristic
169C23×C7#13512yes—
170C23×C7AGL(1,ℤ/8)12yes—
171C7×Q8—42no—
172C7×Q8—81no—
173C7×D4—42no—
174C7×D4—42no—
175C7×D4—82no—
176C7×D4—81no—
177C7×D4—81no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—44no—
194C2×C4×C7—42no—
195C2×C4×C7—82no—
196C2×C4×C7—81no—
197C2×C4×C7—81no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C2×C4×C7—81no—
201C2×C4×C7#12011yescharacteristic
202C2×C4×C7#12011yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C7×C8—42no—
206C7×C8—42no—
207C22×Q8C7×D411yescharacteristic
208C22×D4—21no—
209C22×D4C7×D411yescharacteristic
210C23×C4—21no—
211C23×C4C23×C711yescharacteristic
212C2×SD16—22no—
213C22×C8—21no—
214C22×C8C7×D412yes—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219#121—21no—
220#121—21no—
221#121—21no—
222#121—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C7×D412yes—
231C2×C4⋊C4C7×D411yescharacteristic
232C2×C4⋊C4C7×D411yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C7×D412yes—
237C2×C42—21no—
238#107—24no—
239#106—22no—
240#103—22no—
241#102—24no—
242#102—22no—
243#99—44no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8C7×D412yes—
248#95—24no—
249#95—24no—
250#95—21no—
251#95—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—42no—
259C22×C7—42no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—162no—
264C22×C7#39211yescharacteristic
265C22×C7#44912yes—
266C22×C7#46312yes—
267C22×C7#49511yescharacteristic
268C22×C7#49611yescharacteristic
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—82no—
273C4×C7—81no—
274C4×C7—82no—
275C4×C7—81no—
276C4×C7—81no—
277C4×C7—161no—
278C4×C7#45812yes—
279C24—21no—
280C24C2×C7×D411yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—42no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304C22×C4C2×C7×D411yescharacteristic
305C22×C4C2×C7×D411yescharacteristic
306C22×C4C2×C7×D411yescharacteristic
307C22×C4C7×#4111yescharacteristic
308SD16—82no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C42—42no—
332C2×C7—22no—
333C2×C7—21no—
334C2×C7—22no—
335C2×C7—21no—
336C2×C7—161no—
337C2×C7#200911yescharacteristic
338C2×C7#201211yescharacteristic
339C2×C7#202211yescharacteristic
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23C7×#12011yescharacteristic
347C23C7×#12111yescharacteristic
348C23C7×#12711yescharacteristic
349C23C7×#13512yes—
350C23C7×AGL(1,ℤ/8)12yes—
351Q8—42no—
352Q8—81no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4C7×#12011yescharacteristic
382C2×C4C7×#12011yescharacteristic
383C2×C4C7×#12111yescharacteristic
384C2×C4C7×#12111yescharacteristic
385C8—42no—
386C8—42no—
387C7#995311yescharacteristic, direct product factor, Sylow
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—162no—
400C22C7×#39211yescharacteristic
401C22C7×#44912yes—
402C22C7×#46312yes—
403C22C7×#49511yescharacteristic
404C22C7×#49611yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—161no—
414C4C7×#45812yes—
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2C7×#200911yescharacteristic
421C2C7×#201211yescharacteristic
422C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#2503C2semidirect
5. C7×#1823C2semidirect
6. C7×#1520C2semidirect
7. C7×#1282C2semidirect
31. #9953C7direct (N × H)
79. #2503C2×C7semidirect
81. #1823C2×C7semidirect
82. #1520C2×C7semidirect
83. #1282C2×C7semidirect
387. C7#9953direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31872
C2×C7×#4802
C7×#25032
C7×#19642
C7×#18232
C7×#15202
C7×#12822
#99537