Finite Group Databasebeta

← C7×#9223 #13186439

Subgroups — C7×#9223 #13186439

Order 1792.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#476C211yesmaximal, characteristic
2C2×C7×#436C211yesmaximal, characteristic
3C7×#1990C211yesmaximal, characteristic, complemented
4C7×#1802C211yesmaximal, characteristic
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C7×#1296C211yesmaximal, characteristic, complemented
7C7×#1277C211yesmaximal, characteristic, complemented
8C22×C7×M16—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#476—22no—
12C7×#476C414yescomplemented
13C7×#436—22no—
14C7×#436—22no—
15C2×C7×#106—21no—
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#102—21no—
20C2×C7×#102—21no—
21C2×C7×C22⋊C8—21no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#384C2211yescharacteristic
24C7×#340—21no—
25C7×#340—21no—
26C7×#340—21no—
27C7×#333—22no—
28#9223C711yesmaximal, characteristic, direct product factor, Sylow
29C22×C7×Q8—21no—
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C2×C7×M16—24no—
33C2×C7×M16—42no—
34C2×C7×M16—42no—
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C22×C7×C8—21no—
38C22×C7×C8D411yescharacteristic
39C22×C7×C8D411yescharacteristic
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8C2×C412yes—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C4×C7×D4C2×C412yes—
47C4×C7×D4D412yes—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C22⋊C4—21no—
55C2×C42×C7C2311yescharacteristic
56C7×#106—24no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8C2×C412yes—
62C7×C4⋊C8D412yes—
63C7×#102—24no—
64C7×#102—42no—
65C7×C22⋊C8—24no—
66C4×C7×C8—21no—
67C4×C7×C8—21no—
68C2×#476C2×C711yescharacteristic
69C2×#436C2×C711yescharacteristic
70#1990C2×C711yescharacteristic, complemented
71#1802C2×C711yescharacteristic
72C2×#333C2×C711yescharacteristic, complemented
73#1296C2×C711yescharacteristic, complemented
74#1277C2×C711yescharacteristic, complemented
75C24×C7—21no—
76C2×C7×Q8—22no—
77C2×C7×Q8—41no—
78C2×C7×Q8—81no—
79C2×C7×Q8—81no—
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4C22⋊C412yes—
86C22×C4×C7—21no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7#4111yescharacteristic
98C7×M16—48no—
99C2×C7×C8—21no—
100C2×C7×C8—21no—
101C2×C7×C8—22no—
102C2×C7×C8—21no—
103C2×C7×C8—22no—
104C2×C7×C8—21no—
105C2×C7×C8—21no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8SD1612yes—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—81no—
124C7×C4⋊C4C22⋊C412yes—
125C7×C22⋊C4—22no—
126C7×C22⋊C4—22no—
127C42×C7C22×C411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C22×M16—21no—
132C2×C4⋊Q8C22×C711yescharacteristic
133C2×C4×D4C22×C711yescharacteristic
134#476—22no—
135#476C4×C714yescomplemented
136#436—22no—
137#436—22no—
138C2×#106—21no—
139C2×C4⋊C8C22×C711yescharacteristic
140C2×C4⋊C8C22×C711yescharacteristic
141C2×C4⋊C8C22×C711yescharacteristic
142C2×#102—21no—
143C2×#102—21no—
144C2×C22⋊C8—21no—
145C2×C4×C8C22×C711yescharacteristic
146#384C22×C711yescharacteristic
147#340—21no—
148#340—21no—
149#340—21no—
150#333—22no—
151C23×C7—22no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7#12111yescharacteristic
159C7×Q8—82no—
160C7×Q8—82no—
161C7×D4—22no—
162C7×D4—22no—
163C7×D4—41no—
164C7×D4—41no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—22no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7—81no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7C2×C22⋊C411yescharacteristic
181C2×C4×C7C4×D411yescharacteristic
182C2×C4×C7C4×D411yescharacteristic
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12311yescharacteristic
186C2×C4×C7C2×SD1611yescharacteristic
187C2×C4×C7#13511yescharacteristic
188C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C7#13711yescharacteristic
191C2×C4×C7#13711yescharacteristic
192C2×C4×C7#13711yescharacteristic
193C2×C4×C7#13711yescharacteristic
194C7×C8—22no—
195C7×C8—42no—
196C7×C8—42no—
197C7×C8—41no—
198C7×C8—81no—
199C22×Q8—21no—
200C22×D4C7×D411yescharacteristic
201C23×C4—21no—
202C2×M16—24no—
203C2×M16—42no—
204C2×M16—42no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C7×D411yescharacteristic
209C22×C8C7×D411yescharacteristic
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C2×C4×C712yes—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4C2×C4×C712yes—
217C4×D4C7×D412yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C7×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C42C23×C711yescharacteristic
226#106—24no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8C2×C4×C712yes—
232C4⋊C8C7×D412yes—
233#102—24no—
234#102—42no—
235C22⋊C8—24no—
236C4×C8—21no—
237C4×C8—21no—
238C22×C7—42no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7#38611yescharacteristic
246C22×C7#46111yescharacteristic
247C22×C7#46311yescharacteristic
248C22×C7#46811yescharacteristic
249C22×C7#47011yescharacteristic
250C22×C7#47111yescharacteristic
251C22×C7#47211yescharacteristic
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—42no—
255C4×C7—82no—
256C4×C7—161no—
257C4×C7#41711yescharacteristic
258C4×C7#41911yescharacteristic
259C4×C7C4×SD1611yescharacteristic
260C4×C7#44011yescharacteristic
261C4×C7#45011yescharacteristic
262C4×C7#45211yescharacteristic
263C4×C7#46811yescharacteristic
264C4×C7#47011yescharacteristic
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4C7×C22⋊C412yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C2×C7×D411yescharacteristic
286C22×C4C2×C7×D411yescharacteristic
287C22×C4C7×#4111yescharacteristic
288M16—48no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8C7×SD1612yes—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4C7×C22⋊C412yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C42C22×C4×C711yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C7×#4111yescharacteristic
321C2×C7—82no—
322C2×C7#169111yescharacteristic
323C2×C7#169511yescharacteristic
324C2×C7#169711yescharacteristic
325C2×C7#170311yescharacteristic
326C2×C7#187711yescharacteristic
327C2×C7#189411yescharacteristic
328C2×C7#190711yescharacteristic
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23C7×#12111yescharacteristic
337Q8—82no—
338Q8—82no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C2×C7×C22⋊C411yescharacteristic
359C2×C4C4×C7×D411yescharacteristic
360C2×C4C4×C7×D411yescharacteristic
361C2×C4C7×#12011yescharacteristic
362C2×C4C7×#12111yescharacteristic
363C2×C4C7×#12311yescharacteristic
364C2×C4C2×C7×SD1611yescharacteristic
365C2×C4C7×#13511yescharacteristic
366C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C7×#13711yescharacteristic
369C2×C4C7×#13711yescharacteristic
370C2×C4C7×#13711yescharacteristic
371C2×C4C7×#13711yescharacteristic
372C8—22no—
373C8—42no—
374C8—42no—
375C8—41no—
376C8—81no—
377C7#922311yescharacteristic, direct product factor, Sylow
378C22—42no—
379C22—42no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C7×#38611yescharacteristic
386C22C7×#46111yescharacteristic
387C22C7×#46311yescharacteristic
388C22C7×#46811yescharacteristic
389C22C7×#47011yescharacteristic
390C22C7×#47111yescharacteristic
391C22C7×#47211yescharacteristic
392C4—21no—
393C4—21no—
394C4—42no—
395C4—82no—
396C4—161no—
397C4C7×#41711yescharacteristic
398C4C7×#41911yescharacteristic
399C4C4×C7×SD1611yescharacteristic
400C4C7×#44011yescharacteristic
401C4C7×#45011yescharacteristic
402C4C7×#45211yescharacteristic
403C4C7×#46811yescharacteristic
404C4C7×#47011yescharacteristic
405C2—82no—
406C2C7×#169111yescharacteristic
407C2C7×#169511yescharacteristic
408C2C7×#169711yescharacteristic
409C2C7×#170311yescharacteristic
410C2C7×#187711yescharacteristic
411C2C7×#189411yescharacteristic
412C2C7×#190711yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1990C2semidirect
5. C2×C7×#333C2semidirect
6. C7×#1296C2semidirect
7. C7×#1277C2semidirect
12. C7×#476C4semidirect
28. #9223C7direct (N × H)
70. #1990C2×C7semidirect
72. C2×#333C2×C7semidirect
73. #1296C2×C7semidirect
74. #1277C2×C7semidirect
135. #476C4×C7semidirect
377. C7#9223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4762
C2×C7×#4362
C7×#19902
C7×#18022
C2×C7×#3332
C7×#12962
C7×#12772
#92237