Finite Group Databasebeta

← C7×#12189 #13189405

Subgroups — C7×#12189 #13189405

Order 1792.

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

normal subgroups only

This group has 432 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×#2771C211yesmaximal, characteristic, complemented
2C7×#2022C211yesmaximal, characteristic
3C7×#1613C211yesmaximal, characteristic
4C7×#1607C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1344C211yesmaximal, characteristic, complemented
7C7×#1317C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C2×C7×#123—21no—
10C7×#481—21no—
11C7×#480—21no—
12C2×C7×#99—21no—
13C7×#400—21no—
14C7×#400—21no—
15C7×#399C2211yescharacteristic
16C7×#397C2211yescharacteristic
17C7×#389—21no—
18C7×#388—21no—
19C7×#387—21no—
20C7×#385—21no—
21C7×#381C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12189C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C23×C4×C7D411yescharacteristic
30C7×#123—42no—
31C7×#123—41no—
32C7×#123—42no—
33C7×#121—21no—
34C7×#121—21no—
35C7×#120—41no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—41no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C2×C42×C7—21no—
50C7×#107—41no—
51C7×#106—41no—
52C7×#102—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#2771C2×C711yescharacteristic, complemented
67#2022C2×C711yescharacteristic
68#1613C2×C711yescharacteristic
69#1607C2×C711yescharacteristic
70#1352C2×C711yescharacteristic, complemented
71#1344C2×C711yescharacteristic, complemented
72#1317C2×C711yescharacteristic, complemented
73C24×C7—21no—
74C24×C7C2×D411yescharacteristic
75C2×C7×D4—21no—
76C2×C7×D4—21no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
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—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—42no—
105C7×C4⋊C4—44no—
106C7×C4⋊C4—81no—
107C7×C4⋊C4—81no—
108C7×C22⋊C4—22no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—44no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C42×C7—81no—
119#534C22×C711yescharacteristic
120C2×#123—21no—
121#481—21no—
122#480—21no—
123C2×#99—21no—
124#400—21no—
125#400—21no—
126#399C22×C711yescharacteristic
127#397C22×C711yescharacteristic
128#389—21no—
129#388—21no—
130#387—21no—
131#385—21no—
132#381C22×C711yescharacteristic
133#342C22×C711yescharacteristic
134#327—21no—
135#327—21no—
136#323C22×C711yescharacteristic
137#323C22×C711yescharacteristic
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—81no—
146C23×C7—81no—
147C23×C7#12011yescharacteristic
148C23×C7#13511yescharacteristic
149C23×C7AGL(1,ℤ/8)11yescharacteristic
150C7×D4—81no—
151C7×D4—81no—
152C7×D4—81no—
153C7×D4—81no—
154C7×D4—81no—
155C7×D4—81no—
156C2×C4×C7—22no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
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—81no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C7×C8—81no—
187C7×C8—81no—
188C22×D4—21no—
189C23×C4C7×D411yescharacteristic
190#123—42no—
191#123—41no—
192#123—42no—
193#121—21no—
194#121—21no—
195#120—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C7×D411yescharacteristic
202C2×C4⋊C4C7×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4C7×D411yescharacteristic
208C2×C22⋊C4C23×C711yescharacteristic
209C2×C42—21no—
210#107—41no—
211#106—41no—
212#102—41no—
213#102—41no—
214#99—42no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C7×D411yescharacteristic
225#95C7×D411yescharacteristic
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—21no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7—81no—
235C22×C7—82no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—161no—
240C22×C7#44911yescharacteristic
241C22×C7#45311yescharacteristic
242C22×C7B(4,𝔽2)11yescharacteristic
243C4×C7—42no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—81no—
247C4×C7—81no—
248C4×C7—82no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—81no—
253C24—21no—
254C24C2×C7×D411yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×C7×D411yescharacteristic
276C22×C4C2×C7×D411yescharacteristic
277C2×C8—41no—
278C2×C8—41no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C42—81no—
299C2×C7—21no—
300C2×C7—21no—
301C2×C7—41no—
302C2×C7—41no—
303C2×C7—161no—
304C2×C7#160011yescharacteristic
305C2×C7#219811yescharacteristic
306C2×C7#220111yescharacteristic
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—81no—
315C23—81no—
316C23C7×#12011yescharacteristic
317C23C7×#13511yescharacteristic
318C23C7×AGL(1,ℤ/8)11yescharacteristic
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C8—81no—
356C8—81no—
357C7#1218911yescharacteristic, direct product factor, Sylow
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—161no—
372C22C7×#44911yescharacteristic
373C22C7×#45311yescharacteristic
374C22C7×B(4,𝔽2)11yescharacteristic
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2C7×#160011yescharacteristic
391C2C7×#219811yescharacteristic
392C2C7×#220111yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#534C22
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×#2771C2semidirect
5. C7×#1352C2semidirect
6. C7×#1344C2semidirect
7. C7×#1317C2semidirect
27. #12189C7direct (N × H)
66. #2771C2×C7semidirect
70. #1352C2×C7semidirect
71. #1344C2×C7semidirect
72. #1317C2×C7semidirect
357. C7#12189direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27712
C7×#20222
C7×#16132
C7×#16072
C7×#13522
C7×#13442
C7×#13172
#121897