Finite Group Databasebeta

← C7×#9961 #13187177

Subgroups — C7×#9961 #13187177

Order 1792.

552 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3189C211yesmaximal, characteristic
2C2×C7×#483C211yesmaximal, characteristic
3C7×#2503C211yesmaximal, characteristic
4C7×#1996C211yesmaximal, characteristic
5C7×#1823C211yesmaximal, characteristic
6C7×#1520C211yesmaximal, characteristic
7C7×#1282C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#484—21no—
12C7×#484—21no—
13C7×#483—22no—
14C7×#483—22no—
15C7×#467—22no—
16C7×#451—21no—
17C7×#451—21no—
18C2×C7×#106—21no—
19C2×C7×#103—21no—
20C2×C7×C22⋊C8C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C7×#400—22no—
23C7×#391—21no—
24C7×#382—21no—
25C7×#381—22no—
26C7×#380C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#342—22no—
29C7×#340—22no—
30C7×#328—22no—
31#9961C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C23×C4×C7—21no—
35C23×C4×C7C2311yescharacteristic
36C2×C7×Q16—22no—
37C22×C7×C8—21no—
38C22×C7×C8D412yes—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C2×C7×C4⋊C4—21no—
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⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D412yes—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4D412yes—
62C2×C42×C7—21no—
63C7×#107—22no—
64C7×#106—24no—
65C7×#103—22no—
66C7×#103—22no—
67C7×#103—24no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8D412yes—
72C7×#95—24no—
73C7×#95—24no—
74C7×#95—21no—
75C7×#95—21no—
76#3189C2×C711yescharacteristic
77C2×#483C2×C711yescharacteristic
78#2503C2×C711yescharacteristic
79#1996C2×C711yescharacteristic
80#1823C2×C711yescharacteristic
81#1520C2×C711yescharacteristic
82#1282C2×C711yescharacteristic
83C24×C7C2×D411yescharacteristic
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—42no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7—41no—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7#4111yescharacteristic
108C7×Q16—82no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—42no—
112C2×C7×C8—42no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—42no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—22no—
127C7×C22⋊C4—22no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—42no—
133C42×C7—42no—
134C2×C22⋊Q8C22×C711yescharacteristic
135C2×C22⋊Q8C22×C711yescharacteristic
136C22×C4⋊C4C22×C711yescharacteristic
137#484—21no—
138#484—21no—
139#483—22no—
140#483—22no—
141#467—22no—
142#451—21no—
143#451—21no—
144C2×#106—21no—
145C2×#103—21no—
146C2×C22⋊C8C22×C711yescharacteristic
147C2×C22⋊C8C22×C711yescharacteristic
148#400—22no—
149#391—21no—
150#382—21no—
151#381—22no—
152#380C22×C711yescharacteristic
153C4×C22⋊C4C22×C711yescharacteristic
154#342—22no—
155#340—22no—
156#328—22no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7#12011yescharacteristic
160C23×C7#12111yescharacteristic
161C23×C7#12711yescharacteristic
162C23×C7#13512yes—
163C23×C7#13712yes—
164C7×Q8—42no—
165C7×Q8—42no—
166C7×Q8—42no—
167C7×Q8—81no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—22no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—44no—
185C2×C4×C7—82no—
186C2×C4×C7—81no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C7×C8—42no—
198C7×C8—42no—
199C22×Q8C7×D411yescharacteristic
200C22×Q8C7×D411yescharacteristic
201C23×C4—21no—
202C23×C4C23×C711yescharacteristic
203C2×Q16—22no—
204C22×C8—21no—
205C22×C8C7×D412yes—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C7×D412yes—
223C2×C4⋊C4C7×D411yescharacteristic
224C2×C4⋊C4C7×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4C7×D412yes—
229C2×C42—21no—
230#107—22no—
231#106—24no—
232#103—24no—
233#103—22no—
234#103—22no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8C7×D412yes—
239#95—24no—
240#95—24no—
241#95—21no—
242#95—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7#39211yescharacteristic
252C22×C7#45212yes—
253C22×C7#46412yes—
254C22×C7#49511yescharacteristic
255C22×C7#49711yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—82no—
260C4×C7—81no—
261C4×C7—81no—
262C4×C7—82no—
263C4×C7—81no—
264C4×C7—81no—
265C4×C7—161no—
266C4×C7#45812yes—
267C24C2×C7×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—42no—
273C2×Q8—42no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4C2×C7×D411yescharacteristic
287C22×C4C2×C7×D411yescharacteristic
288C22×C4C2×C7×D411yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C22×C4C7×#4111yescharacteristic
292Q16—82no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C42—42no—
318C2×C7—22no—
319C2×C7—21no—
320C2×C7—22no—
321C2×C7—21no—
322C2×C7#200911yescharacteristic
323C2×C7#201511yescharacteristic
324C2×C7#202211yescharacteristic
325C23—41no—
326C23—41no—
327C23C7×#12011yescharacteristic
328C23C7×#12111yescharacteristic
329C23C7×#12711yescharacteristic
330C23C7×#13512yes—
331C23C7×#13712yes—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4C7×#12011yescharacteristic
362C2×C4C7×#12011yescharacteristic
363C2×C4C7×#12111yescharacteristic
364C2×C4C7×#12111yescharacteristic
365C8—42no—
366C8—42no—
367C7#996111yescharacteristic, direct product factor, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22C7×#39211yescharacteristic
377C22C7×#45212yes—
378C22C7×#46412yes—
379C22C7×#49511yescharacteristic
380C22C7×#49711yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—161no—
391C4C7×#45812yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2C7×#200911yescharacteristic
397C2C7×#201511yescharacteristic
398C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
31. #9961C7direct (N × H)
367. C7#9961direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31892
C2×C7×#4832
C7×#25032
C7×#19962
C7×#18232
C7×#15202
C7×#12822
#99617