Finite Group Databasebeta

← C7×#11799 #13189015

Subgroups — C7×#11799 #13189015

Order 1792.

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

normal subgroups only

This group has 574 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×#2617C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic, complemented
3C7×#1462C211yesmaximal, characteristic, complemented
4C7×#1344C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#121C411yescharacteristic, complemented
7C2×C7×#120—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C7×#407—21no—
10C7×#397C411yescharacteristic, complemented
11C7×#397C2211yescharacteristic
12C7×#394—21no—
13C7×#394—21no—
14C7×#392C411yescharacteristic, complemented
15C7×#392C2211yescharacteristic
16C7×#386—21no—
17C7×#380C411yescharacteristic, complemented
18C7×#379—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11799C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7D411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—41no—
32C7×#121—41no—
33C7×#121—41no—
34C7×#120—44no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C7×C22⋊C4C2311yescharacteristic
52C7×C22⋊C8—22no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2617C2×C711yescharacteristic
59#1819C2×C711yescharacteristic, complemented
60#1462C2×C711yescharacteristic, complemented
61#1344C2×C712yescomplemented
62#1343C2×C712yescomplemented
63C24×C7—21no—
64C24×C7—21no—
65C24×C7—22no—
66C24×C7—21no—
67C24×C7—41no—
68C24×C7—81no—
69C24×C7—81no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7C22⋊C411yescharacteristic
72C24×C7C2×D411yescharacteristic
73C2×C7×D4—41no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—42no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—81no—
81C2×C7×D4—81no—
82C2×C7×D4—81no—
83C2×C7×D4—81no—
84C2×C7×D4—81no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—42no—
93C22×C4×C7—81no—
94C22×C4×C7—81no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×M16—82no—
100C2×C7×C8—42no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—81no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—81no—
114C7×C22⋊C4—81no—
115C7×C22⋊C4—82no—
116C7×C22⋊C4—82no—
117C2×#121C4×C711yescharacteristic, complemented
118C2×#120—21no—
119C22×C22⋊C4C22×C711yescharacteristic
120#407—21no—
121#397C4×C711yescharacteristic, complemented
122#397C22×C711yescharacteristic
123#394—21no—
124#394—21no—
125#392C4×C711yescharacteristic, complemented
126#392C22×C711yescharacteristic
127#386—21no—
128#380C4×C711yescharacteristic, complemented
129#379—21no—
130#342—21no—
131#342C22×C712yes—
132#323—21no—
133#323C22×C712yes—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—42no—
144C23×C7—42no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—42no—
150C23×C7—42no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—81no—
159C23×C7—81no—
160C23×C7—84no—
161C23×C7—84no—
162C23×C7#9911yescharacteristic
163C23×C7#9911yescharacteristic
164C23×C7C2×C22⋊C411yescharacteristic
165C23×C7AGL(1,ℤ/8)12yes—
166C7×D4—82no—
167C7×D4—82no—
168C7×D4—82no—
169C7×D4—84no—
170C7×D4—84no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
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—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7—82no—
194C2×C4×C7—82no—
195C7×C8—82no—
196C25C7×D411yescharacteristic
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C2×C4×C711yescharacteristic
201C23×C4C2×C4×C711yescharacteristic
202C23×C4C7×D411yescharacteristic
203C2×M16—41no—
204#121—41no—
205#121—41no—
206#120—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×C4×C711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4C2×C4×C711yescharacteristic
221C2×C22⋊C4C7×D411yescharacteristic
222C2×C22⋊C4C7×D411yescharacteristic
223C2×C22⋊C4C23×C711yescharacteristic
224C22⋊C8—22no—
225C22⋊C8—41no—
226#95—21no—
227#95—21no—
228#95C2×C4×C711yescharacteristic
229#95C2×C4×C711yescharacteristic
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—22no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—42no—
242C22×C7—42no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
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—84no—
255C22×C7—84no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7C2×#9911yescharacteristic
260C22×C7#41811yescharacteristic
261C22×C7#42111yescharacteristic
262C4×C7—41no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—41no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7—81no—
269C4×C7—162no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—21no—
274C24—41no—
275C24—81no—
276C24—81no—
277C24C7×C22⋊C411yescharacteristic
278C24C7×C22⋊C411yescharacteristic
279C24C2×C7×D411yescharacteristic
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—81no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4—81no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4C7×C22⋊C411yescharacteristic
303C22×C4C7×C22⋊C411yescharacteristic
304C22×C4C22×C4×C711yescharacteristic
305C22×C4C2×C7×D411yescharacteristic
306M16—82no—
307C2×C8—42no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C2×C7—21no—
325C2×C7—21no—
326C2×C7—42no—
327C2×C7—41no—
328C2×C7—41no—
329C2×C7—81no—
330C2×C7—161no—
331C2×C7#151011yescharacteristic
332C2×C7#212111yescharacteristic
333C2×C7#212711yescharacteristic
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—81no—
359C23—81no—
360C23—84no—
361C23—84no—
362C23C7×#9911yescharacteristic
363C23C7×#9911yescharacteristic
364C23C2×C7×C22⋊C411yescharacteristic
365C23C7×AGL(1,ℤ/8)12yes—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—84no—
370D4—84no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C8—82no—
396C7#1179911yescharacteristic, direct product factor, Sylow
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C2×C7×#9911yescharacteristic
427C22C7×#41811yescharacteristic
428C22C7×#42111yescharacteristic
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—162no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—41no—
441C2—41no—
442C2—81no—
443C2—161no—
444C2C7×#151011yescharacteristic
445C2C7×#212111yescharacteristic
446C2C7×#212711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2617C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1819C2semidirect
3. C7×#1462C2semidirect
4. C7×#1344C2semidirect
5. C7×#1343C2semidirect
6. C2×C7×#121C4semidirect
10. C7×#397C4semidirect
14. C7×#392C4semidirect
17. C7×#380C4semidirect
23. #11799C7direct (N × H)
59. #1819C2×C7semidirect
60. #1462C2×C7semidirect
61. #1344C2×C7semidirect
62. #1343C2×C7semidirect
117. C2×#121C4×C7semidirect
121. #397C4×C7semidirect
125. #392C4×C7semidirect
128. #380C4×C7semidirect
396. C7#11799direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26172
C7×#18192
C7×#14622
C7×#13442
C7×#13432
#117997