Finite Group Databasebeta

← C7×#10196 #13187412

Subgroups — C7×#10196 #13187412

Order 1792.

546 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3245C211yesmaximal, characteristic
2C7×#3206C211yesmaximal, characteristic, complemented
3C7×#3141C211yesmaximal, characteristic, complemented
4C7×#1987C211yesmaximal, characteristic, complemented
5C7×#1980C211yesmaximal, characteristic, complemented
6C7×#1502C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C7×#492—21no—
12C7×#489—22no—
13C7×#487—21no—
14C7×#487—21no—
15C7×#486—21no—
16C7×#474—21no—
17C7×#474—21no—
18C7×#466—22no—
19C7×#461—22no—
20C7×#459—21no—
21C7×#459—21no—
22C7×#448—22no—
23C7×#431C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#104—22no—
26C7×#417—22no—
27C4×C7×M16C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C7×#340—22no—
30C7×#327—22no—
31#10196C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C2×C7×#41—21no—
34C22×C7×Q8—21no—
35C2×C7×SD16—24no—
36C2×C7×D8—22no—
37C2×C7×M16D412yes—
38C22×C7×C8D412yes—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8D411yescharacteristic
43C7×C4⋊Q8D411yescharacteristic
44C7×C4⋊Q8D412yes—
45C7×#127D411yescharacteristic
46C7×#127D411yescharacteristic
47C7×#124—21no—
48C7×#124—21no—
49C7×#121—21no—
50C7×#121—21no—
51C7×#121—21no—
52C7×#121—21no—
53C4×C7×D4—21no—
54C4×C7×D4—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×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—22no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×#104—44no—
67C7×#103—24no—
68C7×#103—24no—
69C7×#102—22no—
70C7×#102—22no—
71C7×#102—24no—
72C7×C22⋊C8D412yes—
73C7×#97—21no—
74C4×C7×C8—21no—
75C4×C7×C8—21no—
76C7×#95—21no—
77#3245C2×C711yescharacteristic
78#3206C2×C711yescharacteristic, complemented
79#3141C2×C711yescharacteristic, complemented
80#1987C2×C711yescharacteristic, complemented
81#1980C2×C711yescharacteristic, complemented
82#1502C2×C711yescharacteristic, complemented
83#1283C2×C711yescharacteristic, complemented
84C7×#41—42no—
85C7×#41—42no—
86C7×#41—42no—
87C7×#41—42no—
88C2×C7×Q8—21no—
89C2×C7×Q8—22no—
90C2×C7×Q8—21no—
91C2×C7×Q8—41no—
92C2×C7×Q8—82no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7—41no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C7×SD16—84no—
109C7×D8—82no—
110C7×M16—22no—
111C7×M16—22no—
112C2×C7×C8—22no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—42no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7—42no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135#532C22×C711yescharacteristic
136#532C22×C711yescharacteristic
137C2×C4⋊Q8C22×C711yescharacteristic
138#492—21no—
139#489—22no—
140#487—21no—
141#487—21no—
142#486—21no—
143#474—21no—
144#474—21no—
145#466—22no—
146#461—22no—
147#459—21no—
148#459—21no—
149#448—22no—
150#431C22×C711yescharacteristic
151C2×C4⋊C8C22×C711yescharacteristic
152C2×#104—22no—
153#417—22no—
154C4×M16C22×C711yescharacteristic
155C4×C4⋊C4C22×C711yescharacteristic
156#340—22no—
157#327—22no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7#12711yescharacteristic
161C7×Q8—44no—
162C7×Q8—42no—
163C7×Q8—42no—
164C7×Q8—82no—
165C7×D4—42no—
166C7×D4—42no—
167C7×D4—41no—
168C7×D4—41no—
169C7×D4—41no—
170C7×D4—41no—
171C7×D4—42no—
172C7×D4—81no—
173C7×D4—81no—
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—81no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—42no—
184C2×C4×C7—81no—
185C2×C4×C7—82no—
186C2×C4×C7—81no—
187C2×C4×C7—82no—
188C2×C4×C7—82no—
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#13512yes—
196C2×C4×C7AGL(1,ℤ/8)12yes—
197C7×C8—42no—
198C7×C8—42no—
199C7×C8—42no—
200C2×#41—21no—
201C2×#41—21no—
202C22×Q8—21no—
203C2×SD16—24no—
204C2×D8—22no—
205C2×M16C7×D412yes—
206C22×C8C7×D412yes—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8C7×D412yes—
211C4⋊Q8C7×D411yescharacteristic
212C4⋊Q8C7×D411yescharacteristic
213#127C7×D411yescharacteristic
214#127C7×D411yescharacteristic
215#124—21no—
216#124—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C42—21no—
229C2×C42C23×C711yescharacteristic
230#107—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234#104—44no—
235#103—24no—
236#103—24no—
237#102—24no—
238#102—22no—
239#102—22no—
240C22⋊C8C7×D412yes—
241#97—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7#39211yescharacteristic
254C22×C7#49511yescharacteristic
255C22×C7#49611yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—82no—
260C4×C7—82no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7#44912yes—
264C4×C7#45511yescharacteristic
265C4×C7#45611yescharacteristic
266C4×C7#46312yes—
267#41—42no—
268#41—42no—
269#41—42no—
270#41—42no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C2×C7×D411yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291SD16—84no—
292D8—82no—
293M16—22no—
294M16—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—42no—
314C42C2×C7×D411yescharacteristic
315C42C2×C7×D411yescharacteristic
316C42C2×C7×D411yescharacteristic
317C42C7×#4111yescharacteristic
318C2×C7—21no—
319C2×C7—21no—
320C2×C7—161no—
321C2×C7—161no—
322C2×C7#200711yescharacteristic
323C2×C7#200811yescharacteristic
324C2×C7#201211yescharacteristic
325C23—41no—
326C23—41no—
327C23C7×#12711yescharacteristic
328Q8—44no—
329Q8—42no—
330Q8—42no—
331Q8—82no—
332D4—42no—
333D4—42no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—42no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4C7×#12011yescharacteristic
357C2×C4C7×#12011yescharacteristic
358C2×C4C7×#12011yescharacteristic
359C2×C4C7×#12111yescharacteristic
360C2×C4C7×#12111yescharacteristic
361C2×C4C7×#12111yescharacteristic
362C2×C4C7×#13512yes—
363C2×C4C7×AGL(1,ℤ/8)12yes—
364C8—42no—
365C8—42no—
366C8—42no—
367C7#1019611yescharacteristic, direct product factor, Sylow
368C22—21no—
369C22—21no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C7×#39211yescharacteristic
377C22C7×#49511yescharacteristic
378C22C7×#49611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4C7×#44912yes—
387C4C7×#45511yescharacteristic
388C4C7×#45611yescharacteristic
389C4C7×#46312yes—
390C2—21no—
391C2—21no—
392C2—161no—
393C2—161no—
394C2C7×#200711yescharacteristic
395C2C7×#200811yescharacteristic
396C2C7×#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×#3206C2semidirect
3. C7×#3141C2semidirect
4. C7×#1987C2semidirect
5. C7×#1980C2semidirect
6. C7×#1502C2semidirect
7. C7×#1283C2semidirect
31. #10196C7direct (N × H)
78. #3206C2×C7semidirect
79. #3141C2×C7semidirect
80. #1987C2×C7semidirect
81. #1980C2×C7semidirect
82. #1502C2×C7semidirect
83. #1283C2×C7semidirect
367. C7#10196direct (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×#32062
C7×#31412
C7×#19872
C7×#19802
C7×#15022
C7×#12832
#101967