Finite Group Databasebeta

← C7×#13300 #13190516

Subgroups — C7×#13300 #13190516

Order 1792.

492 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2566C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C4×C7×#99C211yesmaximal, characteristic
4C7×#1393C212yesmaximal
5C7×#1393C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99C2212yes—
9C2×C7×#99C2212yes—
10C7×#396—21no—
11C7×#396—21no—
12C7×#390C2211yescharacteristic
13C7×#390C2211yescharacteristic
14C7×#385—21no—
15C7×#385—21no—
16C7×#384C411yescharacteristic, complemented
17C7×#382C411yescharacteristic, complemented
18C7×#380—21no—
19C4×C7×C4⋊C4C411yescharacteristic, complemented
20C4×C7×C22⋊C4—21no—
21C7×#342—21no—
22C7×#342—21no—
23#13300C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4C2311yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C7×#125—41no—
28C7×#125—41no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—41no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C42×C7—41no—
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C2×C42×C7C2×C411yescharacteristic
52C7×#99—22no—
53C7×#99—24no—
54C7×#99Q814yes—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—41no—
59C7×#95—41no—
60#2566C2×C711yescharacteristic
61#1799C2×C711yescharacteristic
62C4×#99C2×C711yescharacteristic
63#1393C2×C712yes—
64#1393C2×C712yes—
65C24×C7C2×D411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4C2×Q812yes—
70C2×C7×D4#4112yes—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—42no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—81no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—24no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—81no—
108C42×C7—21no—
109C42×C7—21no—
110C42×C7—41no—
111C42×C7—41no—
112C42×C7—82no—
113C2×#125C4×C711yescharacteristic, complemented
114C2×C4×D4C22×C711yescharacteristic
115C2×#99C22×C712yes—
116C2×#99C22×C712yes—
117#396—21no—
118#396—21no—
119#390C22×C711yescharacteristic
120#390C22×C711yescharacteristic
121#385—21no—
122#385—21no—
123#384C4×C711yescharacteristic, complemented
124#382C4×C711yescharacteristic, complemented
125#380—21no—
126C4×C4⋊C4C4×C711yescharacteristic, complemented
127C4×C22⋊C4—21no—
128#342—21no—
129#342—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7C2×C22⋊C411yescharacteristic
139C23×C7C22⋊Q812yes—
140C23×C7#12411yescharacteristic
141C23×C7C4⋊Q811yescharacteristic
142C7×D4—42no—
143C7×D4—42no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—22no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—41no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—44no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7—82no—
171C2×C4×C7—82no—
172C2×C4×C7—82no—
173C2×C4×C7—82no—
174C2×C4×C7—82no—
175C2×C4×C7#9911yescharacteristic
176C2×C4×C7#9911yescharacteristic
177C2×C4×C7C4×Q812yes—
178C22×D4C23×C711yescharacteristic
179C23×C4C7×D411yescharacteristic
180C23×C4C7×D411yescharacteristic
181#125—41no—
182#125—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4C2×C4×C711yescharacteristic
192C2×C4⋊C4C2×C4×C711yescharacteristic
193C2×C4⋊C4C2×C4×C711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C7×D411yescharacteristic
201C2×C22⋊C4C7×D411yescharacteristic
202C2×C42—41no—
203C2×C42C2×C4×C711yescharacteristic
204C2×C42C2×C4×C711yescharacteristic
205C2×C42C2×C4×C711yescharacteristic
206#99—22no—
207#99—24no—
208#99C7×Q814yes—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—41no—
213#95—41no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—42no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7#39111yescharacteristic
229C22×C7C2×#9911yescharacteristic
230C22×C7#41011yescharacteristic
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—22no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—81no—
239C4×C7—84no—
240C4×C7—81no—
241C4×C7—81no—
242C4×C7—162no—
243C24C2×C7×D411yescharacteristic
244C24C2×C7×D411yescharacteristic
245C2×D4—41no—
246C2×D4—41no—
247C2×D4C2×C7×Q812yes—
248C2×D4C7×#4112yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4C7×C22⋊C411yescharacteristic
262C22×C4C7×C22⋊C411yescharacteristic
263C22×C4C7×C22⋊C411yescharacteristic
264C22×C4C7×C22⋊C411yescharacteristic
265C22×C4C22×C4×C711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—82no—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42—82no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—42no—
294C2×C7—41no—
295C2×C7—41no—
296C2×C7#199611yescharacteristic
297C2×C7#212611yescharacteristic
298C2×C7#212711yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C2×C7×C22⋊C411yescharacteristic
308C23C7×C22⋊Q812yes—
309C23C7×#12411yescharacteristic
310C23C7×C4⋊Q811yescharacteristic
311D4—42no—
312D4—42no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4C7×#9911yescharacteristic
345C2×C4C7×#9911yescharacteristic
346C2×C4C4×C7×Q812yes—
347C7#1330011yescharacteristic, direct product factor, Sylow
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22C7×#39111yescharacteristic
363C22C2×C7×#9911yescharacteristic
364C22C7×#41011yescharacteristic
365C4—22no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—84no—
374C4—81no—
375C4—81no—
376C4—162no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2C7×#199611yescharacteristic
383C2C7×#212611yescharacteristic
384C2C7×#212711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
16. C7×#384C4semidirect
17. C7×#382C4semidirect
19. C4×C7×C4⋊C4C4semidirect
23. #13300C7direct (N × H)
113. C2×#125C4×C7semidirect
123. #384C4×C7semidirect
124. #382C4×C7semidirect
126. C4×C4⋊C4C4×C7semidirect
347. C7#13300direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25662
C7×#17992
C4×C7×#992
C7×#13932
#133007