Finite Group Databasebeta

← C7×#7825 #13185041

Subgroups — C7×#7825 #13185041

Order 1792.

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

normal subgroups only

This group has 516 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×#2682C211yesmaximal, characteristic
2C7×#2100C211yesmaximal, characteristic, complemented
3C7×#2094C211yesmaximal, characteristic, complemented
4C7×#2094C211yesmaximal, characteristic, complemented
5C7×#1271C211yesmaximal, characteristic, complemented
6C7×#1270C211yesmaximal, characteristic, complemented
7C7×#1270C211yesmaximal, characteristic, complemented
8C2×C7×#123—21no—
9C2×C7×#121—21no—
10C2×C7×#106—21no—
11C2×C7×#106—21no—
12C2×C7×#102—21no—
13C2×C7×#102—21no—
14C7×#396—21no—
15C7×#394—21no—
16C7×#392—21no—
17C7×#390C2211yescharacteristic
18C7×#390C2211yescharacteristic
19C7×#389C2211yescharacteristic
20C7×#385—21no—
21C7×#384C2211yescharacteristic
22C7×#340—21no—
23C7×#340—21no—
24C7×#336C2211yescharacteristic
25C7×#336C2211yescharacteristic
26C7×#336C2211yescharacteristic
27#7825C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4D411yescharacteristic
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C22×C7×C8—21no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C7×#123—42no—
37C7×#123—42no—
38C7×#121—42no—
39C7×#121—42no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C7×#106—42no—
58C7×#106—42no—
59C7×#102—42no—
60C7×#102—42no—
61C7×#95—21no—
62C7×#95—21no—
63#2682C2×C711yescharacteristic
64#2100C2×C711yescharacteristic, complemented
65#2094C2×C711yescharacteristic, complemented
66#2094C2×C711yescharacteristic, complemented
67#1271C2×C711yescharacteristic, complemented
68#1270C2×C711yescharacteristic, complemented
69#1270C2×C711yescharacteristic, complemented
70C24×C7—21no—
71C24×C7—21no—
72C2×C7×D4—21no—
73C2×C7×D4—21no—
74C2×C7×D4—21no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—81no—
83C2×C7×D4—81no—
84C2×C7×D4—81no—
85C2×C7×D4SD1612yes—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—41no—
93C22×C4×C7—81no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C2×C7×C8—22no—
98C2×C7×C8—22no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—42no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—42no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4SD1612yes—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—41no—
125C7×C22⋊C4—41no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—81no—
130C42×C7—41no—
131C2×#123—21no—
132C2×#121—21no—
133C2×#106—21no—
134C2×#106—21no—
135C2×#102—21no—
136C2×#102—21no—
137#396—21no—
138#394—21no—
139#392—21no—
140#390C22×C711yescharacteristic
141#390C22×C711yescharacteristic
142#389C22×C711yescharacteristic
143#385—21no—
144#384C22×C711yescharacteristic
145#340—21no—
146#340—21no—
147#336C22×C711yescharacteristic
148#336C22×C711yescharacteristic
149#336C22×C711yescharacteristic
150C23×C7—22no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7#12011yescharacteristic
165C7×D4—42no—
166C7×D4—42no—
167C7×D4—81no—
168C7×D4—81no—
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—21no—
176C2×C4×C7—21no—
177C2×C4×C7—22no—
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—84no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7C2×SD1611yescharacteristic
194C2×C4×C7C2×SD1611yescharacteristic
195C2×C4×C7#13511yescharacteristic
196C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C7#13711yescharacteristic
199C7×C8—42no—
200C7×C8—42no—
201C7×C8—81no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C7×D411yescharacteristic
205C22×D4C7×D411yescharacteristic
206C23×C4—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210#123—42no—
211#123—42no—
212#121—42no—
213#121—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C7×D411yescharacteristic
220C2×C4⋊C4C7×D411yescharacteristic
221C2×C4⋊C4C7×D411yescharacteristic
222C2×C4⋊C4C7×D411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42C23×C711yescharacteristic
231#106—42no—
232#106—42no—
233#102—42no—
234#102—42no—
235#95—21no—
236#95—21no—
237C22×C7—42no—
238C22×C7—42no—
239C22×C7—42no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7#45011yescharacteristic
252C22×C7#45011yescharacteristic
253C22×C7#45211yescharacteristic
254C22×C7#45311yescharacteristic
255C22×C7#45311yescharacteristic
256C22×C7#45411yescharacteristic
257C22×C7#45811yescharacteristic
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—42no—
265C4×C7—81no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7—161no—
269C24—21no—
270C24—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4C7×SD1612yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—81no—
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C7×SD1612yes—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—81no—
329C42—41no—
330C2×C7—82no—
331C2×C7—161no—
332C2×C7#161311yescharacteristic
333C2×C7#161611yescharacteristic
334C2×C7#161811yescharacteristic
335C2×C7#162911yescharacteristic
336C2×C7#163111yescharacteristic
337C2×C7#163511yescharacteristic
338C2×C7#165611yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C7×#12011yescharacteristic
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—84no—
359D4—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—84no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4C2×C7×SD1611yescharacteristic
383C2×C4C2×C7×SD1611yescharacteristic
384C2×C4C7×#13511yescharacteristic
385C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C7×#13711yescharacteristic
388C8—42no—
389C8—42no—
390C8—81no—
391C7#782511yescharacteristic, direct product factor, Sylow
392C22—42no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C7×#45011yescharacteristic
407C22C7×#45011yescharacteristic
408C22C7×#45211yescharacteristic
409C22C7×#45311yescharacteristic
410C22C7×#45311yescharacteristic
411C22C7×#45411yescharacteristic
412C22C7×#45811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—161no—
424C2—82no—
425C2—161no—
426C2C7×#161311yescharacteristic
427C2C7×#161611yescharacteristic
428C2C7×#161811yescharacteristic
429C2C7×#162911yescharacteristic
430C2C7×#163111yescharacteristic
431C2C7×#163511yescharacteristic
432C2C7×#165611yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2100C2semidirect
3. C7×#2094C2semidirect
4. C7×#2094C2semidirect
5. C7×#1271C2semidirect
6. C7×#1270C2semidirect
7. C7×#1270C2semidirect
27. #7825C7direct (N × H)
64. #2100C2×C7semidirect
65. #2094C2×C7semidirect
66. #2094C2×C7semidirect
67. #1271C2×C7semidirect
68. #1270C2×C7semidirect
69. #1270C2×C7semidirect
391. C7#7825direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26822
C7×#21002
C7×#20942
C7×#12712
C7×#12702
#78257