Finite Group Databasebeta

← C7×#12192 #13189408

Subgroups — C7×#12192 #13189408

Order 1792.

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

normal subgroups only

This group has 544 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×#2735C211yesmaximal, characteristic, complemented
2C7×#2021C211yesmaximal, characteristic
3C7×#1613C211yesmaximal, characteristic, complemented
4C7×#1607C211yesmaximal, characteristic, complemented
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1344C211yesmaximal, characteristic, complemented
7C7×#1316C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C2×C7×#120—21no—
10C7×#481—21no—
11C7×#480—21no—
12C2×C7×#99—21no—
13C7×#399C2211yescharacteristic
14C7×#397—21no—
15C7×#397C2211yescharacteristic
16C7×#396—21no—
17C7×#388—21no—
18C7×#388—21no—
19C7×#388—21no—
20C7×#379—21no—
21C7×#379C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12192C711yesmaximal, characteristic, direct product factor, Sylow
28C25×C7D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4—41no—
31C7×#123—41no—
32C7×#121—21no—
33C7×#121—21no—
34C7×#120—42no—
35C7×#120—41no—
36C7×#120—42no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D411yescharacteristic, complemented
40C2×C7×C4⋊C4D411yescharacteristic, complemented
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
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—21no—
50C2×C7×C22⋊C4—41no—
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C7×C22⋊C4C2311yescharacteristic
53C2×C42×C7—21no—
54C7×#107—41no—
55C7×#106—41no—
56C7×#102—41no—
57C7×#102—41no—
58C7×#99—42no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95D411yescharacteristic, complemented
65C7×#95D411yescharacteristic, complemented
66#2735C2×C711yescharacteristic, complemented
67#2021C2×C711yescharacteristic
68#1613C2×C711yescharacteristic, complemented
69#1607C2×C711yescharacteristic, complemented
70#1352C2×C711yescharacteristic, complemented
71#1344C2×C711yescharacteristic, complemented
72#1316C2×C711yescharacteristic, complemented
73C24×C7—21no—
74C24×C7—21no—
75C24×C7—21no—
76C24×C7—21no—
77C24×C7—41no—
78C24×C7—41no—
79C24×C7—42no—
80C24×C7—81no—
81C24×C7C2×D411yescharacteristic
82C2×C7×D4—21no—
83C2×C7×D4—21no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—44no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—82no—
92C2×C7×D4—82no—
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—41no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—81no—
109C7×C4⋊C4—81no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—44no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—81no—
122C7×C22⋊C4—81no—
123C7×C22⋊C4—81no—
124C7×C22⋊C4—81no—
125C7×C22⋊C4—81no—
126C42×C7—81no—
127#534C22×C711yescharacteristic
128C2×#120—21no—
129#481—21no—
130#480—21no—
131C2×#99—21no—
132#399C22×C711yescharacteristic
133#397—21no—
134#397C22×C711yescharacteristic
135#396—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#379—21no—
140#379C22×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—42no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—42no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—81no—
164C23×C7—82no—
165C23×C7—81no—
166C23×C7—82no—
167C23×C7—84no—
168C23×C7—81no—
169C23×C7—81no—
170C23×C7—81no—
171C23×C7#12011yescharacteristic
172C23×C7#13511yescharacteristic
173C23×C7AGL(1,ℤ/8)11yescharacteristic
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—88no—
179C7×D4—81no—
180C7×D4—81no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
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—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—81no—
205C2×C4×C7—81no—
206C7×C8—81no—
207C7×C8—81no—
208C25C7×D411yescharacteristic
209C22×D4—21no—
210C22×D4—41no—
211#123—41no—
212#121—21no—
213#121—21no—
214#120—42no—
215#120—41no—
216#120—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C7×D411yescharacteristic, complemented
220C2×C4⋊C4C7×D411yescharacteristic, complemented
221C2×C22⋊C4—21no—
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—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4C7×D411yescharacteristic
232C2×C22⋊C4C23×C711yescharacteristic
233C2×C42—21no—
234#107—41no—
235#106—41no—
236#102—41no—
237#102—41no—
238#99—42no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95C7×D411yescharacteristic, complemented
245#95C7×D411yescharacteristic, complemented
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—42no—
253C22×C7—42no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—44no—
257C22×C7—42no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—82no—
262C22×C7—82no—
263C22×C7—82no—
264C22×C7—82no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7—161no—
271C22×C7#44911yescharacteristic
272C22×C7#45311yescharacteristic
273C22×C7B(4,𝔽2)11yescharacteristic
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—81no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7—82no—
280C4×C7—81no—
281C4×C7—81no—
282C4×C7—81no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—81no—
291C24C2×C7×D411yescharacteristic
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C2×D4—82no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4C2×C7×D411yescharacteristic
311C22×C4C2×C7×D411yescharacteristic
312C2×C8—41no—
313C2×C8—41no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C42—81no—
337C2×C7—21no—
338C2×C7—21no—
339C2×C7—41no—
340C2×C7—41no—
341C2×C7—82no—
342C2×C7—161no—
343C2×C7#160011yescharacteristic
344C2×C7#219711yescharacteristic
345C2×C7#220011yescharacteristic
346C23—21no—
347C23—21no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23—82no—
365C23—81no—
366C23—82no—
367C23—84no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23C7×#12011yescharacteristic
372C23C7×#13511yescharacteristic
373C23C7×AGL(1,ℤ/8)11yescharacteristic
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—88no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C8—81no—
407C8—81no—
408C7#1219211yescharacteristic, direct product factor, Sylow
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—44no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—161no—
434C22C7×#44911yescharacteristic
435C22C7×#45311yescharacteristic
436C22C7×B(4,𝔽2)11yescharacteristic
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4—81no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—41no—
450C2—82no—
451C2—161no—
452C2C7×#160011yescharacteristic
453C2C7×#219711yescharacteristic
454C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2021C2
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×#2735C2semidirect
3. C7×#1613C2semidirect
4. C7×#1607C2semidirect
5. C7×#1352C2semidirect
6. C7×#1344C2semidirect
7. C7×#1316C2semidirect
27. #12192C7direct (N × H)
39. C2×C7×C4⋊C4D4semidirect
40. C2×C7×C4⋊C4D4semidirect
64. C7×#95D4semidirect
65. C7×#95D4semidirect
66. #2735C2×C7semidirect
68. #1613C2×C7semidirect
69. #1607C2×C7semidirect
70. #1352C2×C7semidirect
71. #1344C2×C7semidirect
72. #1316C2×C7semidirect
219. C2×C4⋊C4C7×D4semidirect
220. C2×C4⋊C4C7×D4semidirect
244. #95C7×D4semidirect
245. #95C7×D4semidirect
408. C7#12192direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27352
C7×#20212
C7×#16132
C7×#16072
C7×#13522
C7×#13442
C7×#13162
#121927