Finite Group Databasebeta

← C7×#12187 #13189403

Subgroups — C7×#12187 #13189403

Order 1792.

508 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2760C211yesmaximal, characteristic, complemented
2C7×#2022C211yesmaximal, characteristic, complemented
3C7×#1601C211yesmaximal, characteristic, complemented
4C7×#1595C211yesmaximal, characteristic
5C7×#1351C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic, complemented
7C7×#1317C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C2×C7×#123—21no—
10C7×#448—21no—
11C7×#447—21no—
12C2×C7×#99—21no—
13C7×#401—21no—
14C7×#393—21no—
15C7×#393C2211yescharacteristic
16C7×#392C2211yescharacteristic
17C7×#386—21no—
18C7×#386—21no—
19C7×#386—21no—
20C7×#381—21no—
21C7×#381C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12187C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C23×C4×C7D411yescharacteristic
34C2×C7×SD16—41no—
35C2×C7×D8—41no—
36C7×#123—42no—
37C7×#123—41no—
38C7×#123—42no—
39C7×#121—21no—
40C7×#121—21no—
41C7×#120—41no—
42C2×C7×C4⋊C4—41no—
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—21no—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C7×C22⋊C4C2311yescharacteristic
52C7×#103—41no—
53C7×#102—41no—
54C7×#99—42no—
55C7×C22⋊C8—21no—
56C7×C22⋊C8—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95D411yescharacteristic
65C7×#95D411yescharacteristic
66#2760C2×C711yescharacteristic, complemented
67#2022C2×C711yescharacteristic, complemented
68#1601C2×C711yescharacteristic, complemented
69#1595C2×C711yescharacteristic
70#1351C2×C711yescharacteristic, complemented
71#1343C2×C711yescharacteristic, complemented
72#1317C2×C711yescharacteristic, complemented
73C24×C7—21no—
74C24×C7—21no—
75C24×C7C2×D411yescharacteristic
76C2×C7×Q8—41no—
77C2×C7×Q8—81no—
78C2×C7×D4—21no—
79C2×C7×D4—21no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—42no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—81no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—42no—
104C22×C4×C7—81no—
105C22×C4×C7—81no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C7×SD16—82no—
109C7×D8—82no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—44no—
115C7×C22⋊C4—22no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—44no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—81no—
123C7×C22⋊C4—81no—
124C7×C22⋊C4—81no—
125C7×C22⋊C4—81no—
126C7×C22⋊C4—81no—
127#534C22×C711yescharacteristic
128C2×#123—21no—
129#448—21no—
130#447—21no—
131C2×#99—21no—
132#401—21no—
133#393—21no—
134#393C22×C711yescharacteristic
135#392C22×C711yescharacteristic
136#386—21no—
137#386—21no—
138#386—21no—
139#381—21no—
140#381C22×C711yescharacteristic
141#342C22×C711yescharacteristic
142#327—21no—
143#327—21no—
144#323C22×C711yescharacteristic
145#323C22×C711yescharacteristic
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—22no—
149C23×C7—41no—
150C23×C7—41no—
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—81no—
161C23×C7—81no—
162C23×C7#12011yescharacteristic
163C23×C7#13511yescharacteristic
164C23×C7AGL(1,ℤ/8)11yescharacteristic
165C7×Q8—42no—
166C7×Q8—81no—
167C7×D4—42no—
168C7×D4—81no—
169C7×D4—81no—
170C7×D4—81no—
171C7×D4—81no—
172C7×D4—81no—
173C7×D4—82no—
174C7×D4—82no—
175C7×D4—81no—
176C7×D4—81no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
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—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—81no—
202C2×C4×C7—81no—
203C2×C4×C7—84no—
204C2×C4×C7—81no—
205C7×C8—81no—
206C7×C8—81no—
207C22×Q8C7×D411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C7×D411yescharacteristic
211C23×C4—21no—
212C23×C4C7×D411yescharacteristic
213C2×SD16—41no—
214C2×D8—41no—
215#123—42no—
216#123—41no—
217#123—42no—
218#121—21no—
219#121—21no—
220#120—41no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4C7×D411yescharacteristic
230C2×C22⋊C4C23×C711yescharacteristic
231#103—41no—
232#102—41no—
233#99—42no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C7×D411yescharacteristic
244#95C7×D411yescharacteristic
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—42no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—82no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—161no—
266C22×C7#44911yescharacteristic
267C22×C7#45311yescharacteristic
268C22×C7B(4,𝔽2)11yescharacteristic
269C4×C7—41no—
270C4×C7—41no—
271C4×C7—81no—
272C4×C7—81no—
273C4×C7—81no—
274C4×C7—82no—
275C4×C7—81no—
276C4×C7—81no—
277C4×C7—81no—
278C24—21no—
279C24—21no—
280C24C2×C7×D411yescharacteristic
281C2×Q8—41no—
282C2×Q8—81no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4C2×C7×D411yescharacteristic
312C22×C4C2×C7×D411yescharacteristic
313SD16—82no—
314D8—82no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—44no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—44no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C2×C7—21no—
333C2×C7—21no—
334C2×C7—41no—
335C2×C7—41no—
336C2×C7—82no—
337C2×C7—161no—
338C2×C7#160011yescharacteristic
339C2×C7#219711yescharacteristic
340C2×C7#220011yescharacteristic
341C23—21no—
342C23—21no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—81no—
356C23—81no—
357C23C7×#12011yescharacteristic
358C23C7×#13511yescharacteristic
359C23C7×AGL(1,ℤ/8)11yescharacteristic
360Q8—42no—
361Q8—81no—
362D4—42no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—82no—
369D4—82no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
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—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—84no—
399C2×C4—81no—
400C8—81no—
401C8—81no—
402C7#1218711yescharacteristic, direct product factor, Sylow
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—161no—
424C22C7×#44911yescharacteristic
425C22C7×#45311yescharacteristic
426C22C7×B(4,𝔽2)11yescharacteristic
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4—81no—
435C4—81no—
436C2—21no—
437C2—21no—
438C2—41no—
439C2—41no—
440C2—82no—
441C2—161no—
442C2C7×#160011yescharacteristic
443C2C7×#219711yescharacteristic
444C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#1595C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2760C2semidirect
2. C7×#2022C2semidirect
3. C7×#1601C2semidirect
5. C7×#1351C2semidirect
6. C7×#1343C2semidirect
7. C7×#1317C2semidirect
27. #12187C7direct (N × H)
66. #2760C2×C7semidirect
67. #2022C2×C7semidirect
68. #1601C2×C7semidirect
70. #1351C2×C7semidirect
71. #1343C2×C7semidirect
72. #1317C2×C7semidirect
402. C7#12187direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27602
C7×#20222
C7×#16012
C7×#15952
C7×#13512
C7×#13432
C7×#13172
#121877