Finite Group Databasebeta

← C7×#11820 #13189036

Subgroups — C7×#11820 #13189036

Order 1792.

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

normal subgroups only

This group has 626 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×#2388C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic, complemented
3C7×#1466C211yesmaximal, characteristic, complemented
4C7×#1351C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#432—21no—
10C2×C7×#99—21no—
11C7×#395C411yescharacteristic, complemented
12C7×#394—21no—
13C7×#394C411yescharacteristic, complemented
14C7×#393C2211yescharacteristic
15C7×#392—21no—
16C7×#392C2211yescharacteristic
17C7×#380—21no—
18C4×C7×C22⋊C4—21no—
19C4×C7×C22⋊C4C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11820C711yesmaximal, characteristic, direct product factor, Sylow
24C2×C7×#41—44no—
25C2×C7×#41—41no—
26C2×C7×#41—41no—
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4C2×C411yescharacteristic
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7D411yescharacteristic
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C4×C7×D4—44no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
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⋊C4—22no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C2×C42×C7—21no—
50C2×C42×C7D411yescharacteristic
51C7×C4⋊C8—41no—
52C7×#99—42no—
53C7×C22⋊C8—22no—
54C7×#97—41no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—41no—
58C7×#95C2×C411yescharacteristic
59C7×#95C2×C411yescharacteristic
60#2388C2×C711yescharacteristic
61#1799C2×C711yescharacteristic, complemented
62#1466C2×C711yescharacteristic, complemented
63#1351C2×C712yescomplemented
64#1343C2×C712yescomplemented
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C7×#41—44no—
69C7×#41—44no—
70C7×#41—44no—
71C7×#41—44no—
72C2×C7×Q8—42no—
73C2×C7×Q8—41no—
74C2×C7×D4—21no—
75C2×C7×D4—21no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—42no—
87C22×C4×C7—24no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—42no—
96C22×C4×C7—42no—
97C22×C4×C7—42no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—42no—
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22⋊C411yescharacteristic
104C22×C4×C7C22×C411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C2×C7×C8—42no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—42no—
109C7×C4⋊C4—82no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C42×C7—21no—
120C42×C7—21no—
121C42×C7—42no—
122C22×#41C4×C711yescharacteristic, complemented
123C2×C4×D4—21no—
124C2×C4×D4C22×C711yescharacteristic
125#432—21no—
126C2×#99—21no—
127#395C4×C711yescharacteristic, complemented
128#394—21no—
129#394C4×C711yescharacteristic, complemented
130#393C22×C711yescharacteristic
131#392—21no—
132#392C22×C711yescharacteristic
133#380—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4C4×C711yescharacteristic, complemented
136#342C22×C712yes—
137#323—21no—
138#323C22×C712yes—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—22no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7C2×C22⋊C411yescharacteristic
156C23×C7AGL(1,ℤ/8)11yescharacteristic
157C23×C7#13711yescharacteristic
158C7×Q8—42no—
159C7×Q8—42no—
160C7×D4—42no—
161C7×D4—42no—
162C7×D4—42no—
163C7×D4—44no—
164C7×D4—44no—
165C7×D4—48no—
166C7×D4—42no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—44no—
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—44no—
184C2×C4×C7—42no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7—82no—
194C2×C4×C7—82no—
195C2×C4×C7#9911yescharacteristic
196C2×C4×C7#9911yescharacteristic
197C7×C8—82no—
198C2×#41—44no—
199C2×#41—41no—
200C2×#41—41no—
201C22×Q8C2×C4×C711yescharacteristic
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×C4×C711yescharacteristic
205C22×D4C7×D411yescharacteristic
206C23×C4—21no—
207C23×C4C2×C4×C711yescharacteristic
208C23×C4C7×D411yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C2×C4×C711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C7×D411yescharacteristic
222C2×C22⋊C4C23×C711yescharacteristic
223C2×C42—21no—
224C2×C42C7×D411yescharacteristic
225C4⋊C8—41no—
226#99—42no—
227C22⋊C8—22no—
228#97—41no—
229#95—21no—
230#95—21no—
231#95—41no—
232#95C2×C4×C711yescharacteristic
233#95C2×C4×C711yescharacteristic
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—42no—
241C22×C7—42no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—42no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7C2×#9911yescharacteristic
256C22×C7#41711yescharacteristic
257C22×C7#42111yescharacteristic
258C4×C7—21no—
259C4×C7—42no—
260C4×C7—42no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—81no—
266C4×C7—162no—
267C4×C7AΣL(1,𝔽16)12yes—
268C24—21no—
269C24C7×C22⋊C411yescharacteristic
270C24C2×C7×D411yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4C7×C22⋊C411yescharacteristic
305C22×C4C7×C22⋊C411yescharacteristic
306C22×C4C7×C22⋊C411yescharacteristic
307C22×C4C22×C4×C711yescharacteristic
308C22×C4C2×C7×D411yescharacteristic
309C2×C8—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—82no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C42—21no—
323C42—21no—
324C42—42no—
325C2×C7—21no—
326C2×C7—21no—
327C2×C7—41no—
328C2×C7—41no—
329C2×C7—81no—
330C2×C7—82no—
331C2×C7#151311yescharacteristic
332C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
333C2×C7#212611yescharacteristic
334C23—21no—
335C23—21no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23C2×C7×C22⋊C411yescharacteristic
351C23C7×AGL(1,ℤ/8)11yescharacteristic
352C23C7×#13711yescharacteristic
353Q8—42no—
354Q8—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—44no—
360D4—48no—
361D4—42no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4C7×#9911yescharacteristic
391C2×C4C7×#9911yescharacteristic
392C8—82no—
393C7#1182011yescharacteristic, direct product factor, Sylow
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C2×C7×#9911yescharacteristic
416C22C7×#41711yescharacteristic
417C22C7×#42111yescharacteristic
418C4—21no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—162no—
427C4C7×AΣL(1,𝔽16)12yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—82no—
433C2—81no—
434C2C7×#151311yescharacteristic
435C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
436C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2388C2
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. C7×#1799C2semidirect
3. C7×#1466C2semidirect
4. C7×#1351C2semidirect
5. C7×#1343C2semidirect
6. C22×C7×#41C4semidirect
11. C7×#395C4semidirect
13. C7×#394C4semidirect
19. C4×C7×C22⋊C4C4semidirect
23. #11820C7direct (N × H)
61. #1799C2×C7semidirect
62. #1466C2×C7semidirect
63. #1351C2×C7semidirect
64. #1343C2×C7semidirect
122. C22×#41C4×C7semidirect
127. #395C4×C7semidirect
129. #394C4×C7semidirect
135. C4×C22⋊C4C4×C7semidirect
393. C7#11820direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23882
C7×#17992
C7×#14662
C7×#13512
C7×#13432
#118207