Finite Group Databasebeta

← C7×#11821 #13189037

Subgroups — C7×#11821 #13189037

Order 1792.

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

normal subgroups only

This group has 582 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×#2397C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic, complemented
3C7×#1465C211yesmaximal, characteristic, complemented
4C7×#1352C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C411yescharacteristic, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×C7×#99—21no—
11C7×#407—21no—
12C7×#399—21no—
13C7×#399C411yescharacteristic, complemented
14C7×#399C2211yescharacteristic
15C7×#392—21no—
16C7×#392C411yescharacteristic, complemented
17C7×#392C2211yescharacteristic
18C7×#380—21no—
19C7×#376C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11821C711yesmaximal, 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—
32C4×C7×D4—41no—
33C4×C7×D4—41no—
34C4×C7×D4—44no—
35C4×C7×D4—44no—
36C4×C7×D4—41no—
37C4×C7×D4—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—41no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—22no—
49C2×C7×C22⋊C4C2×C411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C2×C42×C7—21no—
52C2×C42×C7C2×C411yescharacteristic
53C7×#99—42no—
54C7×C22⋊C8—22no—
55C7×C22⋊C8—41no—
56C7×#95—21no—
57C7×#95—41no—
58C7×#95C2×C411yescharacteristic
59C7×#95C2×C411yescharacteristic
60#2397C2×C711yescharacteristic
61#1799C2×C711yescharacteristic, complemented
62#1465C2×C711yescharacteristic, complemented
63#1352C2×C712yescomplemented
64#1343C2×C712yescomplemented
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—41no—
75C2×C7×D4—42no—
76C2×C7×D4—42no—
77C2×C7×D4—42no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—42no—
89C22×C4×C7—42no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—42no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C7×M16—82no—
101C2×C7×C8—42no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—42no—
105C7×C4⋊C4—42no—
106C7×C4⋊C4—82no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C42×C7—24no—
117C42×C7—21no—
118C42×C7—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C4×C711yescharacteristic, complemented
122C2×C4×D4C22×C711yescharacteristic
123C2×#99—21no—
124#407—21no—
125#399—21no—
126#399C4×C711yescharacteristic, complemented
127#399C22×C711yescharacteristic
128#392—21no—
129#392C4×C711yescharacteristic, complemented
130#392C22×C711yescharacteristic
131#380—21no—
132#376C4×C711yescharacteristic, complemented
133#342C22×C712yes—
134#324—21no—
135#323C22×C712yes—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—22no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7C2×C22⋊C411yescharacteristic
153C23×C7AGL(1,ℤ/8)11yescharacteristic
154C23×C7#13711yescharacteristic
155C7×D4—42no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—44no—
159C7×D4—44no—
160C7×D4—48no—
161C7×D4—42no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—82no—
186C2×C4×C7—82no—
187C2×C4×C7—82no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7#9911yescharacteristic
191C2×C4×C7#9911yescharacteristic
192C7×C8—82no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C2×C4×C711yescharacteristic
196C22×D4C7×D411yescharacteristic
197C23×C4—21no—
198C23×C4C7×D411yescharacteristic
199C23×C4C7×D411yescharacteristic
200C2×M16—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×C4×C711yescharacteristic
211C2×C4⋊C4C7×D411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C2×C4×C711yescharacteristic
219C2×C22⋊C4C23×C711yescharacteristic
220C2×C42—21no—
221C2×C42C2×C4×C711yescharacteristic
222#99—42no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225#95—21no—
226#95—41no—
227#95C2×C4×C711yescharacteristic
228#95C2×C4×C711yescharacteristic
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—42no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—42no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7C2×#9911yescharacteristic
251C22×C7#41711yescharacteristic
252C22×C7#42111yescharacteristic
253C4×C7—21no—
254C4×C7—42no—
255C4×C7—42no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—41no—
260C4×C7—81no—
261C4×C7—162no—
262C4×C7#35312yes—
263C24—21no—
264C24C7×C22⋊C411yescharacteristic
265C24C2×C7×D411yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4C7×C22⋊C411yescharacteristic
294C22×C4C7×C22⋊C411yescharacteristic
295C22×C4C7×C22⋊C411yescharacteristic
296C22×C4C22×C4×C711yescharacteristic
297C22×C4C2×C7×D411yescharacteristic
298M16—82no—
299C2×C8—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C42—24no—
315C42—21no—
316C42—21no—
317C2×C7—21no—
318C2×C7—21no—
319C2×C7—41no—
320C2×C7—41no—
321C2×C7—81no—
322C2×C7—82no—
323C2×C7#151311yescharacteristic
324C2×C7#212011yescharacteristic
325C2×C7C2×#35311yescharacteristic
326C23—21no—
327C23—21no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23C2×C7×C22⋊C411yescharacteristic
343C23C7×AGL(1,ℤ/8)11yescharacteristic
344C23C7×#13711yescharacteristic
345D4—42no—
346D4—42no—
347D4—42no—
348D4—44no—
349D4—44no—
350D4—48no—
351D4—42no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4C7×#9911yescharacteristic
381C2×C4C7×#9911yescharacteristic
382C8—82no—
383C7#1182111yescharacteristic, direct product factor, Sylow
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C2×C7×#9911yescharacteristic
406C22C7×#41711yescharacteristic
407C22C7×#42111yescharacteristic
408C4—21no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—81no—
416C4—162no—
417C4C7×#35312yes—
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—81no—
423C2—82no—
424C2C7×#151311yescharacteristic
425C2C7×#212011yescharacteristic
426C2C2×C7×#35311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2397C2
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×#1465C2semidirect
4. C7×#1352C2semidirect
5. C7×#1343C2semidirect
8. C2×C4×C7×D4C4semidirect
13. C7×#399C4semidirect
16. C7×#392C4semidirect
19. C7×#376C4semidirect
23. #11821C7direct (N × H)
61. #1799C2×C7semidirect
62. #1465C2×C7semidirect
63. #1352C2×C7semidirect
64. #1343C2×C7semidirect
121. C2×C4×D4C4×C7semidirect
126. #399C4×C7semidirect
129. #392C4×C7semidirect
132. #376C4×C7semidirect
383. C7#11821direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23972
C7×#17992
C7×#14652
C7×#13522
C7×#13432
#118217