Finite Group Databasebeta

← C7×#10181 #13187397

Subgroups — C7×#10181 #13187397

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3206C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C7×#3037C211yesmaximal, characteristic
4C7×#1549C211yesmaximal, characteristic, complemented
5C7×#1480C211yesmaximal, characteristic
6C7×#1461C211yesmaximal, characteristic, complemented
7C7×#1301C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#492—21no—
13C7×#487—21no—
14C7×#477—22no—
15C7×#477—22no—
16C7×#464—22no—
17C7×#462—21no—
18C7×#461—22no—
19C7×#460—21no—
20C7×#448—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C2×C7×#107—21no—
24C7×#423C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C7×#417—21no—
27C2×C7×#103—21no—
28C7×#332—22no—
29C7×D4⋊C8—22no—
30C7×#324—22no—
31#10181C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×Q8—21no—
34C22×C7×Q8D411yescharacteristic
35C7×#137—42no—
36C7×#137—42no—
37C2×C7×Q16—22no—
38C2×C7×SD16—22no—
39C2×C7×SD16—24no—
40C2×C7×M16—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8D412yes—
46C7×C4⋊Q8D411yescharacteristic
47C7×#127D411yescharacteristic
48C7×#124—21no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63C7×#107—24no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8D412yes—
70C7×#103—24no—
71C7×#103—24no—
72C7×#103—22no—
73C7×#102—22no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78#3206C2×C711yescharacteristic
79C2×#477C2×C711yescharacteristic, complemented
80#3037C2×C711yescharacteristic
81#1549C2×C711yescharacteristic, complemented
82#1480C2×C711yescharacteristic
83#1461C2×C711yescharacteristic, complemented
84#1301C2×C711yescharacteristic, complemented
85C7×#41—41no—
86C7×#41—41no—
87C7×#41—41no—
88C7×#41—41no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—22no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—41no—
97C2×C7×Q8—82no—
98C2×C7×Q8SD1612yescomplemented
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—41no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C7×Q16—44no—
111C7×SD16—44no—
112C7×SD16—84no—
113C7×M16—41no—
114C7×M16—41no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C2×#137—21no—
140#532C22×C711yescharacteristic
141C2×C4⋊Q8C22×C711yescharacteristic
142C2×C4×Q8C22×C711yescharacteristic
143#492—21no—
144#487—21no—
145#477—22no—
146#477—22no—
147#464—22no—
148#462—21no—
149#461—22no—
150#460—21no—
151#448—22no—
152#431C22×C711yescharacteristic
153#431C22×C711yescharacteristic
154C2×#107—21no—
155#423C22×C711yescharacteristic
156C2×C4⋊C8C22×C711yescharacteristic
157#417—21no—
158C2×#103—21no—
159#332—22no—
160D4⋊C8—22no—
161#324—22no—
162C23×C7—41no—
163C23×C7#12111yescharacteristic
164C7×Q8—22no—
165C7×Q8—22no—
166C7×Q8—44no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—41no—
170C7×Q8—41no—
171C7×Q8—82no—
172C7×D4—42no—
173C7×D4—41no—
174C7×D4—41no—
175C7×D4—42no—
176C7×D4—81no—
177C7×D4—81no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—82no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12711yescharacteristic
196C2×C4×C7C2×SD1611yescharacteristic
197C2×C4×C7#13511yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C7#13711yescharacteristic
200C7×C8—42no—
201C7×C8—42no—
202C7×C8—42no—
203C7×C8—81no—
204C2×#41—21no—
205C22×Q8—21no—
206C22×Q8C7×D411yescharacteristic
207#137—42no—
208#137—42no—
209C2×Q16—22no—
210C2×SD16—22no—
211C2×SD16—24no—
212C2×M16—21no—
213C22×C8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C7×D412yes—
218C4⋊Q8C7×D411yescharacteristic
219#127C7×D411yescharacteristic
220#124—21no—
221#121—21no—
222#121—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C7×D411yescharacteristic
233C2×C42—21no—
234C2×C42C23×C711yescharacteristic
235#107—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8C7×D412yes—
241C4⋊C8C7×D412yes—
242#103—24no—
243#103—22no—
244#103—24no—
245#102—22no—
246C22⋊C8—21no—
247C22⋊C8C7×D412yes—
248C4×C8—21no—
249C4×C8—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7#39211yescharacteristic
256C22×C7#46511yescharacteristic
257C22×C7#46811yescharacteristic
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—42no—
262C4×C7—81no—
263C4×C7—81no—
264C4×C7—82no—
265C4×C7#44811yescharacteristic
266C4×C7#44911yescharacteristic
267C4×C7#45011yescharacteristic
268C4×C7#45211yescharacteristic
269C4×C7#47312yes—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8C7×SD1612yescomplemented
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—41no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C7×D411yescharacteristic
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C7×#4111yescharacteristic
295Q16—44no—
296SD16—44no—
297SD16—84no—
298M16—41no—
299M16—41no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—41no—
319C42—41no—
320C42C2×C7×D411yescharacteristic
321C42C2×C7×D411yescharacteristic
322C42C2×C7×D411yescharacteristic
323C42C2×C7×D411yescharacteristic
324C2×C7—21no—
325C2×C7—21no—
326C2×C7—161no—
327C2×C7#200011yescharacteristic
328C2×C7#200811yescharacteristic
329C2×C7#202011yescharacteristic
330C23—41no—
331C23C7×#12111yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—44no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—82no—
340D4—42no—
341D4—42no—
342D4—41no—
343D4—41no—
344D4—81no—
345D4—81no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C7×#12011yescharacteristic
359C2×C4C7×#12011yescharacteristic
360C2×C4C7×#12011yescharacteristic
361C2×C4C7×#12111yescharacteristic
362C2×C4C7×#12111yescharacteristic
363C2×C4C7×#12711yescharacteristic
364C2×C4C2×C7×SD1611yescharacteristic
365C2×C4C7×#13511yescharacteristic
366C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C7×#13711yescharacteristic
368C8—42no—
369C8—42no—
370C8—42no—
371C8—81no—
372C7#1018111yescharacteristic, direct product factor, Sylow
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C7×#39211yescharacteristic
379C22C7×#46511yescharacteristic
380C22C7×#46811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4C7×#44811yescharacteristic
389C4C7×#44911yescharacteristic
390C4C7×#45011yescharacteristic
391C4C7×#45211yescharacteristic
392C4C7×#47312yes—
393C2—21no—
394C2—21no—
395C2—161no—
396C2C7×#200011yescharacteristic
397C2C7×#200811yescharacteristic
398C2C7×#202011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
4. C7×#1549C2semidirect
6. C7×#1461C2semidirect
7. C7×#1301C2semidirect
31. #10181C7direct (N × H)
79. C2×#477C2×C7semidirect
81. #1549C2×C7semidirect
83. #1461C2×C7semidirect
84. #1301C2×C7semidirect
98. C2×C7×Q8SD16semidirect
283. C2×Q8C7×SD16semidirect
372. C7#10181direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32062
C2×C7×#4772
C7×#30372
C7×#15492
C7×#14802
C7×#14612
C7×#13012
#101817