Finite Group Databasebeta

← C7×#11774 #13188990

Subgroups — C7×#11774 #13188990

Order 1792.

544 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2535C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic, complemented
3C7×#1465C211yesmaximal, characteristic, complemented
4C7×#1344C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C4×C7×D4C411yescharacteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#99—21no—
10C7×#407—21no—
11C7×#397C411yescharacteristic, complemented
12C7×#397C2211yescharacteristic
13C7×#392C411yescharacteristic, complemented
14C7×#392C2211yescharacteristic
15C7×#386—21no—
16C7×#386—21no—
17C7×#382C411yescharacteristic, complemented
18C7×#380—21no—
19C4×C7×C22⋊C4—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11774C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—41no—
32C7×#121—44no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—41no—
49C2×C42×C7C2×C411yescharacteristic
50C7×#99—24no—
51C7×C22⋊C8—22no—
52C7×C22⋊C8—41no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—41no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2535C2×C711yescharacteristic
60C4×#99C2×C711yescharacteristic, complemented
61#1465C2×C711yescharacteristic, complemented
62#1344C2×C712yescomplemented
63#1343C2×C712yescomplemented
64C24×C7—21no—
65C24×C7C22⋊C411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—22no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—81no—
77C2×C7×D4—81no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7—42no—
89C22×C4×C7—81no—
90C22×C4×C7—81no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C7×M16—82no—
97C2×C7×C8—42no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—42no—
101C7×C22⋊C4—24no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—81no—
110C7×C22⋊C4—81no—
111C42×C7—21no—
112C42×C7—21no—
113C42×C7—82no—
114C2×#121—21no—
115C2×C4×D4C4×C711yescharacteristic, complemented
116C2×C4×D4C22×C711yescharacteristic
117C2×#99—21no—
118#407—21no—
119#397C4×C711yescharacteristic, complemented
120#397C22×C711yescharacteristic
121#392C4×C711yescharacteristic, complemented
122#392C22×C711yescharacteristic
123#386—21no—
124#386—21no—
125#382C4×C711yescharacteristic, complemented
126#380—21no—
127C4×C22⋊C4—21no—
128#342C22×C712yes—
129#324—21no—
130#323C22×C712yes—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—22no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7C2×C22⋊C411yescharacteristic
148C23×C7AGL(1,ℤ/8)12yes—
149C7×D4—44no—
150C7×D4—42no—
151C7×D4—42no—
152C7×D4—84no—
153C7×D4—84no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—44no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7—84no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C2×C4×C7#9911yescharacteristic
181C2×C4×C7#9911yescharacteristic
182C2×C4×C7#10412yes—
183C7×C8—82no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4C2×C4×C711yescharacteristic
187C22×D4C7×D411yescharacteristic
188C23×C4—21no—
189C23×C4C7×D411yescharacteristic
190C23×C4C7×D411yescharacteristic
191C2×M16—41no—
192#121—44no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×C4×C711yescharacteristic
199C2×C4⋊C4C7×D411yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4C2×C4×C711yescharacteristic
207C2×C22⋊C4C23×C711yescharacteristic
208C2×C42—41no—
209C2×C42C2×C4×C711yescharacteristic
210#99—24no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—41no—
217#95C2×C4×C711yescharacteristic
218#95C2×C4×C711yescharacteristic
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—22no—
222C22×C7—42no—
223C22×C7—42no—
224C22×C7—42no—
225C22×C7—42no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—42no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7C2×#9911yescharacteristic
239C22×C7#41811yescharacteristic
240C22×C7C2×#10411yescharacteristic
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—22no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—84no—
250C4×C7—81no—
251C24—21no—
252C24C7×C22⋊C411yescharacteristic
253C24C2×C7×D411yescharacteristic
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—81no—
264C2×D4—81no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4C7×C22⋊C411yescharacteristic
279C22×C4C7×C22⋊C411yescharacteristic
280C22×C4C7×C22⋊C411yescharacteristic
281C22×C4C22×C4×C711yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283M16—82no—
284C2×C8—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C22⋊C4—24no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C42—21no—
299C42—21no—
300C42—82no—
301C2×C7—21no—
302C2×C7—21no—
303C2×C7—42no—
304C2×C7—81no—
305C2×C7—82no—
306C2×C7#150711yescharacteristic
307C2×C7#212011yescharacteristic
308C2×C7#212711yescharacteristic
309C23—21no—
310C23—21no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23C2×C7×C22⋊C411yescharacteristic
326C23C7×AGL(1,ℤ/8)12yes—
327D4—44no—
328D4—42no—
329D4—42no—
330D4—84no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4C7×#9911yescharacteristic
359C2×C4C7×#9911yescharacteristic
360C2×C4C7×#10412yes—
361C8—82no—
362C7#1177411yescharacteristic, direct product factor, Sylow
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×C7×#9911yescharacteristic
383C22C7×#41811yescharacteristic
384C22C2×C7×#10411yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—84no—
394C4—81no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—82no—
400C2C7×#150711yescharacteristic
401C2C7×#212011yescharacteristic
402C2C7×#212711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ 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
2. C4×C7×#99C2semidirect
3. C7×#1465C2semidirect
4. C7×#1344C2semidirect
5. C7×#1343C2semidirect
7. C2×C4×C7×D4C4semidirect
11. C7×#397C4semidirect
13. C7×#392C4semidirect
17. C7×#382C4semidirect
23. #11774C7direct (N × H)
60. C4×#99C2×C7semidirect
61. #1465C2×C7semidirect
62. #1344C2×C7semidirect
63. #1343C2×C7semidirect
115. C2×C4×D4C4×C7semidirect
119. #397C4×C7semidirect
121. #392C4×C7semidirect
125. #382C4×C7semidirect
362. C7#11774direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25352
C4×C7×#992
C7×#14652
C7×#13442
C7×#13432
#117747