Finite Group Databasebeta

← C7×#9244 #13186460

Subgroups — C7×#9244 #13186460

Order 1792.

464 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#464C211yesmaximal, characteristic
2C7×#2984C211yesmaximal, characteristic, complemented
3C7×#1527C211yesmaximal, characteristic
4C7×#1491C211yesmaximal, characteristic, complemented
5C7×#1469C211yesmaximal, characteristic
6C7×#1290C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C22×C7×Q16C411yescharacteristic
9C22×C7×SD16C411yescharacteristic
10C2×C7×#124C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#464—21no—
13C7×#464—21no—
14C7×#464—21no—
15C7×#464—21no—
16C7×#446—21no—
17C7×#446—22no—
18C7×C8×Q8—21no—
19C7×#432—21no—
20C7×#432C2211yescharacteristic
21C7×#432C2211yescharacteristic
22C7×#431—21no—
23C7×#431C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#103C411yescharacteristic
26C2×C7×#102C411yescharacteristic
27C4×C7×M16C2211yescharacteristic
28C7×#329—22no—
29C7×Q8⋊C8—21no—
30C7×Q8⋊C8—21no—
31C7×#323—21no—
32C7×#323—21no—
33#9244C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8C2×C411yescharacteristic
35C22×C7×Q8C2×C411yescharacteristic
36C22×C7×D4C2×C411yescharacteristic
37C2×C7×Q16—21no—
38C2×C7×Q16—21no—
39C2×C7×Q16—42no—
40C2×C7×SD16—21no—
41C2×C7×SD16—21no—
42C2×C7×SD16—42no—
43C2×C7×M16—21no—
44C22×C7×C8C2×C411yescharacteristic
45C22×C7×C8C2×C411yescharacteristic
46C7×#124—21no—
47C7×#124—21no—
48C7×#124—21no—
49C7×#124—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8D411yescharacteristic
53C4×C7×Q8D411yescharacteristic
54C4×C7×Q8D411yescharacteristic
55C4×C7×Q8D411yescharacteristic
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4C2×C411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×C4⋊C8—22no—
63C7×C4⋊C8—22no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×#103—21no—
70C7×#103—21no—
71C7×#102—21no—
72C7×#102—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D411yescharacteristic
76C7×C22⋊C8D411yescharacteristic
77C7×#97—22no—
78C7×#97—21no—
79C7×#97—21no—
80C7×#97D412yes—
81C4×C7×C8—22no—
82C4×C7×C8—21no—
83C4×C7×C8—21no—
84C2×#464C2×C711yescharacteristic
85#2984C2×C711yescharacteristic, complemented
86#1527C2×C711yescharacteristic
87#1491C2×C711yescharacteristic, complemented
88#1469C2×C711yescharacteristic
89#1290C2×C711yescharacteristic, complemented
90#1283C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—22no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—41no—
97C2×C7×Q8—41no—
98C2×C7×Q8—41no—
99C2×C7×Q8C22⋊C411yescharacteristic
100C2×C7×Q8C22⋊C411yescharacteristic
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—41no—
104C2×C7×D4—41no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7C22×C411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C7×Q16—42no—
112C7×Q16—42no—
113C7×SD16—42no—
114C7×SD16—42no—
115C7×M16—22no—
116C7×M16—41no—
117C2×C7×C8—21no—
118C2×C7×C8—21no—
119C2×C7×C8—21no—
120C2×C7×C8—21no—
121C2×C7×C8—22no—
122C2×C7×C8—41no—
123C2×C7×C8—41no—
124C2×C7×C8—41no—
125C2×C7×C8—41no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4C22⋊C411yescharacteristic
132C7×C4⋊C4C22⋊C411yescharacteristic
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C42×C7—21no—
138C42×C7—21no—
139C42×C7—21no—
140C42×C7—21no—
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7C2×D411yescharacteristic
144C42×C7#4111yescharacteristic
145C22×Q16C4×C711yescharacteristic
146C22×SD16C4×C711yescharacteristic
147C2×#124C22×C711yescharacteristic
148C2×C4×Q8C22×C711yescharacteristic
149#464—21no—
150#464—21no—
151#464—21no—
152#464—21no—
153#446—21no—
154#446—22no—
155C8×Q8—21no—
156#432—21no—
157#432C22×C711yescharacteristic
158#432C22×C711yescharacteristic
159#431—21no—
160#431C22×C711yescharacteristic
161C2×C4⋊C8C22×C711yescharacteristic
162C2×#103C4×C711yescharacteristic
163C2×#102C4×C711yescharacteristic
164C4×M16C22×C711yescharacteristic
165#329—22no—
166Q8⋊C8—21no—
167Q8⋊C8—21no—
168#323—21no—
169#323—21no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7—81no—
174C23×C7—81no—
175C23×C7C4×D411yescharacteristic
176C7×Q8—21no—
177C7×Q8—21no—
178C7×Q8—21no—
179C7×Q8—42no—
180C7×Q8—41no—
181C7×Q8—41no—
182C7×Q8—81no—
183C7×Q8#10012yes—
184C7×D4—41no—
185C7×D4—41no—
186C7×D4—81no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—81no—
201C2×C4×C7C2×C22⋊C411yescharacteristic
202C2×C4×C7C4×D411yescharacteristic
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C2×C4×C7#12111yescharacteristic
206C2×C4×C7#12311yescharacteristic
207C7×C8—41no—
208C7×C8—41no—
209C7×C8—42no—
210C7×C8—42no—
211C7×C8—41no—
212C7×C8—41no—
213C7×C8—81no—
214C22×Q8C2×C4×C711yescharacteristic
215C22×Q8C2×C4×C711yescharacteristic
216C22×D4C2×C4×C711yescharacteristic
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—42no—
220C2×SD16—21no—
221C2×SD16—21no—
222C2×SD16—42no—
223C2×M16—21no—
224C22×C8C2×C4×C711yescharacteristic
225C22×C8C2×C4×C711yescharacteristic
226#124—21no—
227#124—21no—
228#124—21no—
229#124—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8C7×D411yescharacteristic
233C4×Q8C7×D411yescharacteristic
234C4×Q8C7×D411yescharacteristic
235C4×Q8C7×D411yescharacteristic
236C2×C4⋊C4—21no—
237C2×C4⋊C4C2×C4×C711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42C23×C711yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249#103—21no—
250#103—21no—
251#102—21no—
252#102—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8C7×D411yescharacteristic
256C22⋊C8C7×D411yescharacteristic
257#97—22no—
258#97—21no—
259#97—21no—
260#97C7×D412yes—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—161no—
270C22×C7—161no—
271C22×C7#38611yescharacteristic
272C22×C7#44311yescharacteristic
273C22×C7#44411yescharacteristic
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—22no—
278C4×C7—21no—
279C4×C7—21no—
280C4×C7—41no—
281C4×C7—41no—
282C4×C7—41no—
283C4×C7—81no—
284C4×C7C2×#10011yescharacteristic
285C4×C7#41311yescharacteristic
286C24—21no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8C7×C22⋊C411yescharacteristic
295C2×Q8C7×C22⋊C411yescharacteristic
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4C22×C4×C711yescharacteristic
304C22×C4C2×C7×D411yescharacteristic
305C22×C4C7×#4111yescharacteristic
306Q16—42no—
307Q16—42no—
308SD16—42no—
309SD16—42no—
310M16—22no—
311M16—41no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4C7×C22⋊C411yescharacteristic
327C4⋊C4C7×C22⋊C411yescharacteristic
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42C2×C7×D411yescharacteristic
337C42C2×C7×D411yescharacteristic
338C42C2×C7×D411yescharacteristic
339C42C7×#4111yescharacteristic
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—161no—
343C2×C7#187611yescharacteristic
344C2×C7#190011yescharacteristic
345C2×C7#190511yescharacteristic
346C23—41no—
347C23—41no—
348C23—41no—
349C23—81no—
350C23—81no—
351C23C4×C7×D411yescharacteristic
352Q8—21no—
353Q8—21no—
354Q8—21no—
355Q8—42no—
356Q8—41no—
357Q8—41no—
358Q8—81no—
359Q8C7×#10012yes—
360D4—41no—
361D4—41no—
362D4—81no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×C7×C22⋊C411yescharacteristic
378C2×C4C4×C7×D411yescharacteristic
379C2×C4C7×#12011yescharacteristic
380C2×C4C7×#12111yescharacteristic
381C2×C4C7×#12111yescharacteristic
382C2×C4C7×#12311yescharacteristic
383C8—41no—
384C8—41no—
385C8—42no—
386C8—42no—
387C8—41no—
388C8—41no—
389C8—81no—
390C7#924411yescharacteristic, direct product factor, Sylow
391C22—21no—
392C22—21no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—161no—
397C22—161no—
398C22C7×#38611yescharacteristic
399C22C7×#44311yescharacteristic
400C22C7×#44411yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4C2×C7×#10011yescharacteristic
412C4C7×#41311yescharacteristic
413C2—21no—
414C2—21no—
415C2—161no—
416C2C7×#187611yescharacteristic
417C2C7×#190011yescharacteristic
418C2C7×#190511yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ 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×#2984C2semidirect
4. C7×#1491C2semidirect
6. C7×#1290C2semidirect
7. C7×#1283C2semidirect
33. #9244C7direct (N × H)
85. #2984C2×C7semidirect
87. #1491C2×C7semidirect
89. #1290C2×C7semidirect
90. #1283C2×C7semidirect
390. C7#9244direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4642
C7×#29842
C7×#15272
C7×#14912
C7×#14692
C7×#12902
C7×#12832
#92447