Finite Group Databasebeta

← C7×#11824 #13189040

Subgroups — C7×#11824 #13189040

Order 1792.

464 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2635C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic, complemented
3C7×#1466C211yesmaximal, characteristic, complemented
4C7×#1352C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#121C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#432—21no—
9C2×C7×#99—21no—
10C7×#400C411yescharacteristic, complemented
11C7×#399C2211yescharacteristic
12C7×#394—21no—
13C7×#394C411yescharacteristic, complemented
14C7×#392—21no—
15C7×#392C2211yescharacteristic
16C7×#388—21no—
17C7×#386—21no—
18C7×#381C411yescharacteristic, complemented
19C7×#380—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11824C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C7×#121—41no—
31C7×#121—41no—
32C4×C7×D4—41no—
33C4×C7×D4—41no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—41no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
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⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C2×C42×C7D411yescharacteristic
48C7×C4⋊C8—41no—
49C7×#99—42no—
50C7×C22⋊C8—22no—
51C7×#97—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—41no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2635C2×C711yescharacteristic
59#1799C2×C711yescharacteristic, complemented
60#1466C2×C711yescharacteristic, complemented
61#1352C2×C712yescomplemented
62#1343C2×C712yescomplemented
63C24×C7—21no—
64C24×C7C22⋊C411yescharacteristic
65C24×C7C2×D411yescharacteristic
66C2×C7×D4—21no—
67C2×C7×D4—21no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—42no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—81no—
75C2×C7×D4—81no—
76C2×C7×D4—81no—
77C2×C7×D4—81no—
78C2×C7×D4—81no—
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—21no—
86C22×C4×C7—42no—
87C22×C4×C7—81no—
88C22×C4×C7—81no—
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22×C411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C2×C7×C8—42no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—82no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4—81no—
110C42×C7—21no—
111C42×C7—21no—
112C2×#121C4×C711yescharacteristic, complemented
113C2×C4×D4C22×C711yescharacteristic
114#432—21no—
115C2×#99—21no—
116#400C4×C711yescharacteristic, complemented
117#399C22×C711yescharacteristic
118#394—21no—
119#394C4×C711yescharacteristic, complemented
120#392—21no—
121#392C22×C711yescharacteristic
122#388—21no—
123#386—21no—
124#381C4×C711yescharacteristic, complemented
125#380—21no—
126#342C22×C712yes—
127#323—21no—
128#323C22×C712yes—
129C23×C7—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×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7C2×C22⋊C411yescharacteristic
146C23×C7AGL(1,ℤ/8)11yescharacteristic
147C23×C7#13711yescharacteristic
148C7×D4—42no—
149C7×D4—42no—
150C7×D4—82no—
151C7×D4—82no—
152C7×D4—82no—
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—21no—
160C2×C4×C7—41no—
161C2×C4×C7—42no—
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—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7—82no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C2×C4×C7#9911yescharacteristic
181C2×C4×C7#9911yescharacteristic
182C7×C8—82no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C2×C4×C711yescharacteristic
186C22×D4C7×D411yescharacteristic
187C23×C4C2×C4×C711yescharacteristic
188C23×C4C7×D411yescharacteristic
189#121—41no—
190#121—41no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C2×C4×C711yescharacteristic
196C2×C4⋊C4C2×C4×C711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C7×D411yescharacteristic
205C2×C22⋊C4C23×C711yescharacteristic
206C2×C42C7×D411yescharacteristic
207C4⋊C8—41no—
208#99—42no—
209C22⋊C8—22no—
210#97—41no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—41no—
215#95C2×C4×C711yescharacteristic
216#95C2×C4×C711yescharacteristic
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7—81no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7C2×#9911yescharacteristic
239C22×C7#41711yescharacteristic
240C22×C7#42111yescharacteristic
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—81no—
250C4×C7—162no—
251C24—21no—
252C24C7×C22⋊C411yescharacteristic
253C24C2×C7×D411yescharacteristic
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4C7×C22⋊C411yescharacteristic
278C22×C4C7×C22⋊C411yescharacteristic
279C22×C4C7×C22⋊C411yescharacteristic
280C22×C4C22×C4×C711yescharacteristic
281C22×C4C2×C7×D411yescharacteristic
282C2×C8—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C4⋊C4—82no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C42—21no—
299C42—21no—
300C2×C7—21no—
301C2×C7—21no—
302C2×C7—41no—
303C2×C7—41no—
304C2×C7—81no—
305C2×C7—161no—
306C2×C7#151311yescharacteristic
307C2×C7#212111yescharacteristic
308C2×C7#212811yescharacteristic
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
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)11yescharacteristic
327C23C7×#13711yescharacteristic
328D4—42no—
329D4—42no—
330D4—82no—
331D4—82no—
332D4—82no—
333D4—84no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
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—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4C7×#9911yescharacteristic
361C2×C4C7×#9911yescharacteristic
362C8—82no—
363C7#1182411yescharacteristic, direct product factor, Sylow
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×C7×#9911yescharacteristic
386C22C7×#41711yescharacteristic
387C22C7×#42111yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—162no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—81no—
403C2—161no—
404C2C7×#151311yescharacteristic
405C2C7×#212111yescharacteristic
406C2C7×#212811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2635C2
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×#1352C2semidirect
5. C7×#1343C2semidirect
6. C2×C7×#121C4semidirect
10. C7×#400C4semidirect
13. C7×#394C4semidirect
18. C7×#381C4semidirect
23. #11824C7direct (N × H)
59. #1799C2×C7semidirect
60. #1466C2×C7semidirect
61. #1352C2×C7semidirect
62. #1343C2×C7semidirect
112. C2×#121C4×C7semidirect
116. #400C4×C7semidirect
119. #394C4×C7semidirect
124. #381C4×C7semidirect
363. C7#11824direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26352
C7×#17992
C7×#14662
C7×#13522
C7×#13432
#118247