Finite Group Databasebeta

← C7×#12191 #13189407

Subgroups — C7×#12191 #13189407

Order 1792.

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

normal subgroups only

This group has 484 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×#2752C211yesmaximal, characteristic, complemented
2C7×#2022C211yesmaximal, characteristic, complemented
3C7×#1613C211yesmaximal, characteristic, complemented
4C7×#1595C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic, complemented
7C7×#1317C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C2×C7×#123—21no—
10C7×#481—21no—
11C7×#447—21no—
12C2×C7×#99—21no—
13C7×#399C2211yescharacteristic
14C7×#397—21no—
15C7×#394—21no—
16C7×#392C2211yescharacteristic
17C7×#390—21no—
18C7×#388—21no—
19C7×#387—21no—
20C7×#386—21no—
21C7×#381C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12191C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×D8—41no—
33C7×#123—42no—
34C7×#123—41no—
35C7×#123—42no—
36C7×#121—21no—
37C7×#121—21no—
38C7×#120—41no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—41no—
42C2×C7×C4⋊C4D411yescharacteristic
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×#106—41no—
55C7×#102—41no—
56C7×#102—41no—
57C7×#99—42no—
58C7×C22⋊C8—21no—
59C7×C22⋊C8—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95D411yescharacteristic
65C7×#95D411yescharacteristic
66#2752C2×C711yescharacteristic, complemented
67#2022C2×C711yescharacteristic, complemented
68#1613C2×C711yescharacteristic, complemented
69#1595C2×C711yescharacteristic
70#1352C2×C711yescharacteristic, complemented
71#1343C2×C711yescharacteristic, complemented
72#1317C2×C711yescharacteristic, complemented
73C24×C7—21no—
74C24×C7—21no—
75C24×C7C2×D411yescharacteristic
76C2×C7×D4—21no—
77C2×C7×D4—21no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—81no—
89C2×C7×D4—81no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
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—41no—
99C22×C4×C7—41no—
100C22×C4×C7—41no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C7×D8—82no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—44no—
111C7×C4⋊C4—81no—
112C7×C22⋊C4—22no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—44no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—81no—
121C7×C22⋊C4—81no—
122C7×C22⋊C4—81no—
123C7×C22⋊C4—81no—
124C7×C22⋊C4—81no—
125C42×C7—81no—
126#534C22×C711yescharacteristic
127C2×#123—21no—
128#481—21no—
129#447—21no—
130C2×#99—21no—
131#399C22×C711yescharacteristic
132#397—21no—
133#394—21no—
134#392C22×C711yescharacteristic
135#390—21no—
136#388—21no—
137#387—21no—
138#386—21no—
139#381C22×C711yescharacteristic
140#342C22×C711yescharacteristic
141#327—21no—
142#327—21no—
143#323C22×C711yescharacteristic
144#323C22×C711yescharacteristic
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—41no—
148C23×C7—41no—
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—81no—
160C23×C7—81no—
161C23×C7#12011yescharacteristic
162C23×C7#13511yescharacteristic
163C23×C7AGL(1,ℤ/8)11yescharacteristic
164C7×D4—42no—
165C7×D4—82no—
166C7×D4—82no—
167C7×D4—81no—
168C7×D4—81no—
169C7×D4—81no—
170C7×D4—81no—
171C7×D4—81no—
172C7×D4—81no—
173C7×D4—81no—
174C2×C4×C7—22no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
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—42no—
184C2×C4×C7—42no—
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—81no—
199C2×C4×C7—81no—
200C2×C4×C7—81no—
201C7×C8—81no—
202C7×C8—81no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4C7×D411yescharacteristic
206C23×C4C7×D411yescharacteristic
207C2×D8—41no—
208#123—42no—
209#123—41no—
210#123—42no—
211#121—21no—
212#121—21no—
213#120—41no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C7×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4C7×D411yescharacteristic
227C2×C22⋊C4C23×C711yescharacteristic
228C2×C42—21no—
229#106—41no—
230#102—41no—
231#102—41no—
232#99—42no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C7×D411yescharacteristic
240#95C7×D411yescharacteristic
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—82no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—161no—
262C22×C7#44911yescharacteristic
263C22×C7#45311yescharacteristic
264C22×C7B(4,𝔽2)11yescharacteristic
265C4×C7—42no—
266C4×C7—41no—
267C4×C7—41no—
268C4×C7—81no—
269C4×C7—81no—
270C4×C7—82no—
271C4×C7—81no—
272C4×C7—81no—
273C4×C7—81no—
274C24—21no—
275C24—21no—
276C24C2×C7×D411yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304D8—82no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—81no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—44no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C42—81no—
327C2×C7—21no—
328C2×C7—21no—
329C2×C7—41no—
330C2×C7—41no—
331C2×C7—161no—
332C2×C7—161no—
333C2×C7#160011yescharacteristic
334C2×C7#219811yescharacteristic
335C2×C7#220011yescharacteristic
336C23—21no—
337C23—21no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23—81no—
352C23C7×#12011yescharacteristic
353C23C7×#13511yescharacteristic
354C23C7×AGL(1,ℤ/8)11yescharacteristic
355D4—42no—
356D4—82no—
357D4—82no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
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—
392C8—81no—
393C8—81no—
394C7#1219111yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—161no—
416C22C7×#44911yescharacteristic
417C22C7×#45311yescharacteristic
418C22C7×B(4,𝔽2)11yescharacteristic
419C4—42no—
420C4—41no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4—81no—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—161no—
433C2—161no—
434C2C7×#160011yescharacteristic
435C2C7×#219811yescharacteristic
436C2C7×#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×#2752C2semidirect
2. C7×#2022C2semidirect
3. C7×#1613C2semidirect
5. C7×#1352C2semidirect
6. C7×#1343C2semidirect
7. C7×#1317C2semidirect
27. #12191C7direct (N × H)
66. #2752C2×C7semidirect
67. #2022C2×C7semidirect
68. #1613C2×C7semidirect
70. #1352C2×C7semidirect
71. #1343C2×C7semidirect
72. #1317C2×C7semidirect
394. C7#12191direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27522
C7×#20222
C7×#16132
C7×#15952
C7×#13522
C7×#13432
C7×#13172
#121917