Finite Group Databasebeta

← C7×#10186 #13187402

Subgroups — C7×#10186 #13187402

Order 1792.

528 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×#3229C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic
3C2×C7×#461C211yesmaximal, characteristic
4C7×#1546C211yesmaximal, characteristic, complemented
5C7×#1481C211yesmaximal, characteristic, complemented
6C7×#1460C211yesmaximal, characteristic
7C7×#1301C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C7×#533C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#498—21no—
13C7×#487—21no—
14C7×#483—22no—
15C7×#478—22no—
16C7×#476—21no—
17C7×#475—22no—
18C7×#474—21no—
19C7×#461—22no—
20C7×#461—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#428—21no—
24C7×#423C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C7×#418—21no—
27C2×C7×#103—21no—
28C7×#332—22no—
29C7×Q8⋊C8—22no—
30C7×#323—22no—
31#10186C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8—21no—
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C2×C7×SD16—24no—
36C2×C7×SD16—42no—
37C2×C7×SD16—42no—
38C2×C7×M16—21no—
39C22×C7×C8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8D411yescharacteristic
44C7×C4⋊Q8D412yes—
45C7×C4⋊Q8D411yescharacteristic
46C7×#125—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C7×#117—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—21no—
60C2×C42×C7C2311yescharacteristic
61C7×#107—22no—
62C7×#106—24no—
63C7×#106—22no—
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×#102—22no—
73C7×#102—22no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78#3229C2×C711yescharacteristic, complemented
79#3077C2×C711yescharacteristic
80C2×#461C2×C711yescharacteristic
81#1546C2×C711yescharacteristic, complemented
82#1481C2×C711yescharacteristic, complemented
83#1460C2×C711yescharacteristic
84#1301C2×C711yescharacteristic, complemented
85C24×C7—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—22no—
88C2×C7×Q8—21no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4SD1612yes—
100C22×C4×C7—21no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7—82no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7#4111yescharacteristic
108C7×SD16—48no—
109C7×M16—41no—
110C7×M16—41no—
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—22no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C42×C7—41no—
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C22×SD16—21no—
138#533C22×C711yescharacteristic
139C2×C4⋊Q8C22×C711yescharacteristic
140C2×C4×D4C22×C711yescharacteristic
141#498—21no—
142#487—21no—
143#483—22no—
144#478—22no—
145#476—21no—
146#475—22no—
147#474—21no—
148#461—22no—
149#461—22no—
150#431C22×C711yescharacteristic
151#431C22×C711yescharacteristic
152#428—21no—
153#423C22×C711yescharacteristic
154C2×C4⋊C8C22×C711yescharacteristic
155#418—21no—
156C2×#103—21no—
157#332—22no—
158Q8⋊C8—22no—
159#323—22no—
160C23×C7—22no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—81no—
164C23×C7—81no—
165C23×C7#12111yescharacteristic
166C7×Q8—42no—
167C7×Q8—42no—
168C7×Q8—42no—
169C7×Q8—42no—
170C7×Q8—81no—
171C7×Q8—81no—
172C7×D4—22no—
173C7×D4—22no—
174C7×D4—41no—
175C7×D4—41no—
176C2×C4×C7—22no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—82no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—82no—
186C2×C4×C7—81no—
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12011yescharacteristic
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12711yescharacteristic
193C2×C4×C7C2×SD1611yescharacteristic
194C2×C4×C7#13511yescharacteristic
195C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C7#13711yescharacteristic
197C7×C8—42no—
198C7×C8—42no—
199C7×C8—42no—
200C7×C8—81no—
201C22×Q8—21no—
202C22×D4C7×D411yescharacteristic
203C23×C4—21no—
204C2×SD16—24no—
205C2×SD16—42no—
206C2×SD16—42no—
207C2×M16—21no—
208C22×C8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8C7×D412yes—
213C4⋊Q8C7×D411yescharacteristic
214C4⋊Q8C7×D411yescharacteristic
215#125—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224#117—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C7×D411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C42C23×C711yescharacteristic
230#107—22no—
231#106—22no—
232#106—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C7×D412yes—
238C4⋊C8C7×D412yes—
239#103—24no—
240#103—24no—
241#102—22no—
242#102—22no—
243C22⋊C8—21no—
244C22⋊C8C7×D412yes—
245C4×C8—21no—
246C4×C8—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—82no—
251C22×C7—82no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7#39211yescharacteristic
255C22×C7#46511yescharacteristic
256C22×C7#46811yescharacteristic
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—82no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7—81no—
264C4×C7#44811yescharacteristic
265C4×C7#44911yescharacteristic
266C4×C7#45011yescharacteristic
267C4×C7#45211yescharacteristic
268C4×C7#47112yes—
269C24—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C7×SD1612yes—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C22×C4C7×#4111yescharacteristic
292SD16—48no—
293M16—41no—
294M16—41no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—41no—
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C2×C7×D411yescharacteristic
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—82no—
324C2×C7#200011yescharacteristic
325C2×C7#200911yescharacteristic
326C2×C7#201911yescharacteristic
327C23—22no—
328C23—41no—
329C23—41no—
330C23—81no—
331C23—81no—
332C23C7×#12111yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—42no—
336Q8—42no—
337Q8—81no—
338Q8—81no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4C7×#12011yescharacteristic
355C2×C4C7×#12011yescharacteristic
356C2×C4C7×#12011yescharacteristic
357C2×C4C7×#12111yescharacteristic
358C2×C4C7×#12111yescharacteristic
359C2×C4C7×#12711yescharacteristic
360C2×C4C2×C7×SD1611yescharacteristic
361C2×C4C7×#13511yescharacteristic
362C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
363C2×C4C7×#13711yescharacteristic
364C8—42no—
365C8—42no—
366C8—42no—
367C8—81no—
368C7#1018611yescharacteristic, direct product factor, Sylow
369C22—21no—
370C22—21no—
371C22—42no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22C7×#39211yescharacteristic
377C22C7×#46511yescharacteristic
378C22C7×#46811yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4C7×#44811yescharacteristic
387C4C7×#44911yescharacteristic
388C4C7×#45011yescharacteristic
389C4C7×#45211yescharacteristic
390C4C7×#47112yes—
391C2—21no—
392C2—21no—
393C2—82no—
394C2C7×#200011yescharacteristic
395C2C7×#200911yescharacteristic
396C2C7×#201911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
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
1. C7×#3229C2semidirect
4. C7×#1546C2semidirect
5. C7×#1481C2semidirect
7. C7×#1301C2semidirect
31. #10186C7direct (N × H)
78. #3229C2×C7semidirect
81. #1546C2×C7semidirect
82. #1481C2×C7semidirect
84. #1301C2×C7semidirect
368. C7#10186direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32292
C7×#30772
C2×C7×#4612
C7×#15462
C7×#14812
C7×#14602
C7×#13012
#101867