Finite Group Databasebeta

← C7×#10013 #13187229

Subgroups — C7×#10013 #13187229

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3103C211yesmaximal, characteristic, complemented
2C7×#3100C211yesmaximal, characteristic
3C7×#2398C211yesmaximal, characteristic, complemented
4C7×#1963C211yesmaximal, characteristic, complemented
5C7×#1487C211yesmaximal, characteristic, complemented
6C7×#1486C211yesmaximal, characteristic, complemented
7C7×#1274C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#481—21no—
12C7×#480—21no—
13C7×#475—21no—
14C7×#474—21no—
15C7×#460—21no—
16C7×#459—21no—
17C7×#448—21no—
18C7×#447—21no—
19C4×C7×SD16—22no—
20C4×C7×D8—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C2×C7×#104—22no—
25C7×#400—21no—
26C7×#397—21no—
27C7×#394—21no—
28C7×#393—21no—
29C4×C7×C22⋊C4—22no—
30C7×#376C2211yescharacteristic
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10013C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×Q8D411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C23×C4×C7—21no—
38C2×C7×SD16—22no—
39C2×C7×D8—22no—
40C2×C7×M16—21no—
41C7×C4⋊Q8D411yescharacteristic
42C7×#127D411yescharacteristic
43C7×#124—21no—
44C7×#121—21no—
45C7×#121—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7—21no—
63C2×C42×C7D412yes—
64C2×C42×C7C2311yescharacteristic
65C7×#107—22no—
66C7×#106—22no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×#104—44no—
72C7×#103—22no—
73C7×#102—22no—
74C7×#102—22no—
75C7×#102—22no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C4×C7×C8D412yes—
79C4×C7×C8D412yes—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—21no—
83C7×#95—21no—
84#3103C2×C711yescharacteristic, complemented
85#3100C2×C711yescharacteristic
86#2398C2×C711yescharacteristic, complemented
87#1963C2×C711yescharacteristic, complemented
88#1487C2×C711yescharacteristic, complemented
89#1486C2×C711yescharacteristic, complemented
90#1274C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C7×#41—42no—
93C7×#41—42no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—42no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—42no—
104C2×C7×D4—41no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—42no—
113C22×C4×C7—42no—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C7×SD16—44no—
118C7×D8—44no—
119C7×M16—42no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—41no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—44no—
132C7×C22⋊C4—44no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C42×C7—22no—
138C42×C7—22no—
139C42×C7—41no—
140C42×C7—41no—
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7C2×D411yescharacteristic
144C42×C7#4111yescharacteristic
145#532C22×C711yescharacteristic
146C2×C4×Q8C22×C711yescharacteristic
147C2×C4×D4C22×C711yescharacteristic
148#481—21no—
149#480—21no—
150#475—21no—
151#474—21no—
152#460—21no—
153#459—21no—
154#448—21no—
155#447—21no—
156C4×SD16—22no—
157C4×D8—22no—
158#431C22×C711yescharacteristic
159#431C22×C711yescharacteristic
160#423C22×C711yescharacteristic
161C2×#104—22no—
162#400—21no—
163#397—21no—
164#394—21no—
165#393—21no—
166C4×C22⋊C4—22no—
167#376C22×C711yescharacteristic
168Q8⋊C8—22no—
169D4⋊C8—22no—
170C23×C7—22no—
171C23×C7—42no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—42no—
175C23×C7—41no—
176C23×C7#12011yescharacteristic
177C7×Q8—22no—
178C7×Q8—22no—
179C7×Q8—41no—
180C7×Q8—41no—
181C7×Q8—42no—
182C7×D4—22no—
183C7×D4—22no—
184C7×D4—41no—
185C7×D4—41no—
186C7×D4—42no—
187C7×D4—42no—
188C7×D4—42no—
189C7×D4—82no—
190C2×C4×C7—22no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—82no—
204C2×C4×C7—81no—
205C2×C4×C7—81no—
206C2×C4×C7—81no—
207C2×C4×C7—81no—
208C2×C4×C7#12011yescharacteristic
209C2×C4×C7#12011yescharacteristic
210C2×C4×C7#12111yescharacteristic
211C2×C4×C7#12111yescharacteristic
212C2×C4×C7#12111yescharacteristic
213C2×C4×C7#12711yescharacteristic
214C2×C4×C7#13512yes—
215C2×C4×C7AGL(1,ℤ/8)12yes—
216C7×C8—42no—
217C7×C8—42no—
218C7×C8—81no—
219C2×#41—21no—
220C22×Q8C7×D411yescharacteristic
221C22×D4C7×D411yescharacteristic
222C23×C4—21no—
223C2×SD16—22no—
224C2×D8—22no—
225C2×M16—21no—
226C4⋊Q8C7×D411yescharacteristic
227#127C7×D411yescharacteristic
228#124—21no—
229#121—21no—
230#121—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C7×D411yescharacteristic
243C2×C4⋊C4C7×D411yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42C7×D412yes—
249C2×C42C23×C711yescharacteristic
250#107—22no—
251#106—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256#104—44no—
257#103—22no—
258#102—22no—
259#102—22no—
260#102—22no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C4×C8C7×D412yes—
264C4×C8C7×D412yes—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—44no—
272C22×C7—42no—
273C22×C7—82no—
274C22×C7—81no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7#39211yescharacteristic
280C22×C7#44912yes—
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—42no—
286C4×C7—82no—
287C4×C7—81no—
288C4×C7—81no—
289C4×C7—81no—
290C4×C7#46312yes—
291C4×C7#49511yescharacteristic
292C4×C7#49611yescharacteristic
293C24—21no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C7×D411yescharacteristic
317C22×C4C2×C7×D411yescharacteristic
318C22×C4C2×C7×D411yescharacteristic
319SD16—44no—
320D8—44no—
321M16—42no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—22no—
340C42—22no—
341C42—41no—
342C42—41no—
343C42C2×C7×D411yescharacteristic
344C42C2×C7×D411yescharacteristic
345C42C2×C7×D411yescharacteristic
346C42C7×#4111yescharacteristic
347C2×C7—22no—
348C2×C7—82no—
349C2×C7—161no—
350C2×C7#201011yescharacteristic
351C2×C7#201011yescharacteristic
352C2×C7#201211yescharacteristic
353C23—22no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23C7×#12011yescharacteristic
360Q8—22no—
361Q8—22no—
362Q8—41no—
363Q8—41no—
364Q8—42no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—82no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4C7×#12011yescharacteristic
392C2×C4C7×#12011yescharacteristic
393C2×C4C7×#12111yescharacteristic
394C2×C4C7×#12111yescharacteristic
395C2×C4C7×#12111yescharacteristic
396C2×C4C7×#12711yescharacteristic
397C2×C4C7×#13512yes—
398C2×C4C7×AGL(1,ℤ/8)12yes—
399C8—42no—
400C8—42no—
401C8—81no—
402C7#1001311yescharacteristic, direct product factor, Sylow
403C22—21no—
404C22—21no—
405C22—44no—
406C22—42no—
407C22—82no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C7×#39211yescharacteristic
414C22C7×#44912yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4C7×#46312yes—
425C4C7×#49511yescharacteristic
426C4C7×#49611yescharacteristic
427C2—22no—
428C2—82no—
429C2—161no—
430C2C7×#201011yescharacteristic
431C2C7×#201011yescharacteristic
432C2C7×#201211yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC7×#3100C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3103C2semidirect
3. C7×#2398C2semidirect
4. C7×#1963C2semidirect
5. C7×#1487C2semidirect
6. C7×#1486C2semidirect
7. C7×#1274C2semidirect
33. #10013C7direct (N × H)
84. #3103C2×C7semidirect
86. #2398C2×C7semidirect
87. #1963C2×C7semidirect
88. #1487C2×C7semidirect
89. #1486C2×C7semidirect
90. #1274C2×C7semidirect
402. C7#10013direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31032
C7×#31002
C7×#23982
C7×#19632
C7×#14872
C7×#14862
C7×#12742
#100137