Finite Group Databasebeta

← C7×#10200 #13187416

Subgroups — C7×#10200 #13187416

Order 1792.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3245C211yesmaximal, characteristic
2C7×#3227C211yesmaximal, characteristic, complemented
3C7×#3141C211yesmaximal, characteristic, complemented
4C7×#1981C211yesmaximal, characteristic, complemented
5C7×#1980C211yesmaximal, characteristic, complemented
6C7×#1502C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C7×#498—21no—
12C7×#489—22no—
13C7×#487—21no—
14C7×#486—21no—
15C7×#486—21no—
16C7×#481—22no—
17C7×#476—22no—
18C7×#474—21no—
19C7×#474—21no—
20C7×#466—22no—
21C7×#459—21no—
22C7×#459—21no—
23C7×#431C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#104—22no—
26C7×#417—22no—
27C2×C7×#103—22no—
28C4×C7×M16C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C7×#327—22no—
31#10200C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C2×C7×#41—21no—
34C22×C7×Q8—21no—
35C2×C7×D8—22no—
36C2×C7×M16D412yes—
37C22×C7×C8D412yes—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×C4⋊Q8D412yes—
44C7×#127D411yescharacteristic
45C7×#127D411yescharacteristic
46C7×#124—21no—
47C7×#124—21no—
48C7×#121—21no—
49C7×#121—21no—
50C7×#121—21no—
51C7×#121—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⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C42×C7—21no—
60C2×C42×C7C2311yescharacteristic
61C7×#107—22no—
62C7×#106—24no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×#104—44no—
67C7×#103—24no—
68C7×#103—44no—
69C7×#102—22no—
70C7×#102—22no—
71C7×#102—24no—
72C7×#102—24no—
73C7×C22⋊C8D412yes—
74C7×#97—21no—
75C4×C7×C8—21no—
76C4×C7×C8—21no—
77C7×#95—21no—
78#3245C2×C711yescharacteristic
79#3227C2×C711yescharacteristic, complemented
80#3141C2×C711yescharacteristic, complemented
81#1981C2×C711yescharacteristic, complemented
82#1980C2×C711yescharacteristic, complemented
83#1502C2×C711yescharacteristic, complemented
84#1283C2×C711yescharacteristic, complemented
85C7×#41—42no—
86C7×#41—42no—
87C7×#41—42no—
88C7×#41—42no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—22no—
92C2×C7×Q8—42no—
93C2×C7×Q8—42no—
94C2×C7×Q8—41no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C22×C4×C7—41no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C7×D8—82no—
111C7×M16—22no—
112C7×M16—22no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—42no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C42×C7—42no—
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7#4111yescharacteristic
136#532C22×C711yescharacteristic
137#532C22×C711yescharacteristic
138C2×C4⋊Q8C22×C711yescharacteristic
139#498—21no—
140#489—22no—
141#487—21no—
142#486—21no—
143#486—21no—
144#481—22no—
145#476—22no—
146#474—21no—
147#474—21no—
148#466—22no—
149#459—21no—
150#459—21no—
151#431C22×C711yescharacteristic
152C2×C4⋊C8C22×C711yescharacteristic
153C2×#104—22no—
154#417—22no—
155C2×#103—22no—
156C4×M16C22×C711yescharacteristic
157C4×C4⋊C4C22×C711yescharacteristic
158#327—22no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7#12711yescharacteristic
162C7×Q8—44no—
163C7×Q8—42no—
164C7×Q8—42no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×D4—42no—
168C7×D4—42no—
169C7×D4—41no—
170C7×D4—41no—
171C7×D4—41no—
172C7×D4—41no—
173C7×D4—42no—
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—81no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—81no—
187C2×C4×C7—82no—
188C2×C4×C7—81no—
189C2×C4×C7—82no—
190C2×C4×C7—82no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#13512yes—
198C2×C4×C7AGL(1,ℤ/8)12yes—
199C7×C8—42no—
200C7×C8—42no—
201C7×C8—42no—
202C2×#41—21no—
203C2×#41—21no—
204C22×Q8—21no—
205C2×D8—22no—
206C2×M16C7×D412yes—
207C22×C8C7×D412yes—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C7×D412yes—
212C4⋊Q8C7×D411yescharacteristic
213C4⋊Q8C7×D411yescharacteristic
214#127C7×D411yescharacteristic
215#127C7×D411yescharacteristic
216#124—21no—
217#124—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C42—21no—
230C2×C42C23×C711yescharacteristic
231#107—22no—
232#106—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#104—44no—
237#103—24no—
238#103—44no—
239#102—24no—
240#102—22no—
241#102—22no—
242#102—24no—
243C22⋊C8C7×D412yes—
244#97—21no—
245C4×C8—21no—
246C4×C8—21no—
247#95—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7#39211yescharacteristic
257C22×C7#49511yescharacteristic
258C22×C7#49611yescharacteristic
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—82no—
263C4×C7—82no—
264C4×C7—81no—
265C4×C7—81no—
266C4×C7#44912yes—
267C4×C7#45311yescharacteristic
268C4×C7#45411yescharacteristic
269C4×C7#46312yes—
270#41—42no—
271#41—42no—
272#41—42no—
273#41—42no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C7×D411yescharacteristic
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C2×C7×D411yescharacteristic
295D8—82no—
296M16—22no—
297M16—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—42no—
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C7×#4111yescharacteristic
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—161no—
324C2×C7—161no—
325C2×C7#200611yescharacteristic
326C2×C7#200911yescharacteristic
327C2×C7#201211yescharacteristic
328C23—41no—
329C23—41no—
330C23C7×#12711yescharacteristic
331Q8—44no—
332Q8—42no—
333Q8—42no—
334Q8—81no—
335Q8—81no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—42no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4C7×#12011yescharacteristic
361C2×C4C7×#12011yescharacteristic
362C2×C4C7×#12011yescharacteristic
363C2×C4C7×#12111yescharacteristic
364C2×C4C7×#12111yescharacteristic
365C2×C4C7×#12111yescharacteristic
366C2×C4C7×#13512yes—
367C2×C4C7×AGL(1,ℤ/8)12yes—
368C8—42no—
369C8—42no—
370C8—42no—
371C7#1020011yescharacteristic, direct product factor, Sylow
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C7×#39211yescharacteristic
381C22C7×#49511yescharacteristic
382C22C7×#49611yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4C7×#44912yes—
391C4C7×#45311yescharacteristic
392C4C7×#45411yescharacteristic
393C4C7×#46312yes—
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2C7×#200611yescharacteristic
399C2C7×#200911yescharacteristic
400C2C7×#201211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3245C2
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. C7×#3227C2semidirect
3. C7×#3141C2semidirect
4. C7×#1981C2semidirect
5. C7×#1980C2semidirect
6. C7×#1502C2semidirect
7. C7×#1283C2semidirect
31. #10200C7direct (N × H)
79. #3227C2×C7semidirect
80. #3141C2×C7semidirect
81. #1981C2×C7semidirect
82. #1980C2×C7semidirect
83. #1502C2×C7semidirect
84. #1283C2×C7semidirect
371. C7#10200direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32452
C7×#32272
C7×#31412
C7×#19812
C7×#19802
C7×#15022
C7×#12832
#102007