Finite Group Databasebeta

← C7×#11553 #13188769

Subgroups — C7×#11553 #13188769

Order 1792.

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

normal subgroups only

This group has 440 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×#2504C211yesmaximal, characteristic
2C7×#1463C211yesmaximal, characteristic, complemented
3C2×C7×#324C211yesmaximal, characteristic, complemented
4C7×#1324C212yesmaximal, complemented
5C7×#1318C212yesmaximal, complemented
6C2×C7×#121C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#397—21no—
11C7×#397—21no—
12C7×#390C411yescharacteristic
13C7×#390C2211yescharacteristic
14C7×#386—21no—
15C7×#386—21no—
16C7×#382C411yescharacteristic
17C7×#382C2211yescharacteristic
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#324—21no—
20C7×#324—21no—
21C7×#324C2212yes—
22C7×#324C2212yes—
23#11553C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—41no—
30C22×C7×C8—41no—
31C7×#121—41no—
32C7×#121—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
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×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C7×C22⋊C8—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2504C2×C711yescharacteristic
58#1463C2×C711yescharacteristic, complemented
59C2×#324C2×C711yescharacteristic, complemented
60#1324C2×C712yescomplemented
61#1318C2×C712yescomplemented
62C24×C7—21no—
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—22no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C7×M16—82no—
91C2×C7×C8—42no—
92C2×C7×C8—42no—
93C2×C7×C8—82no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—81no—
102C7×C22⋊C4—81no—
103C7×C22⋊C4—81no—
104C7×C22⋊C4—81no—
105C42×C7—22no—
106C42×C7—41no—
107C2×#121C4×C711yescharacteristic
108C22×C4⋊C4C22×C711yescharacteristic
109#407—21no—
110C2×C22⋊C8—21no—
111#397—21no—
112#397—21no—
113#390C4×C711yescharacteristic
114#390C22×C711yescharacteristic
115#386—21no—
116#386—21no—
117#382C4×C711yescharacteristic
118#382C22×C711yescharacteristic
119C4×C22⋊C4C4×C711yescharacteristic
120#324—21no—
121#324—21no—
122#324C22×C712yes—
123#324C22×C712yes—
124C23×C7—22no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7#9911yescharacteristic
136C23×C7#9911yescharacteristic
137C23×C7C2×C22⋊C411yescharacteristic
138C7×D4—82no—
139C7×D4—82no—
140C7×D4—82no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7#13112yes—
170C7×C8—82no—
171C7×C8—82no—
172C22×D4—21no—
173C22×D4C2×C4×C711yescharacteristic
174C23×C4C2×C4×C711yescharacteristic
175C23×C4C7×D411yescharacteristic
176C23×C4C7×D411yescharacteristic
177C2×M16—41no—
178C22×C8—41no—
179#121—41no—
180#121—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C2×C4×C711yescharacteristic
184C2×C4⋊C4C2×C4×C711yescharacteristic
185C2×C4⋊C4C7×D411yescharacteristic
186C2×C4⋊C4C7×D411yescharacteristic
187C2×C4⋊C4C23×C711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C42C2×C4×C711yescharacteristic
195C2×C42C2×C4×C711yescharacteristic
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7#35512yes—
222C22×C7#40811yescharacteristic
223C22×C7C2×#9911yescharacteristic
224C22×C7#41311yescharacteristic
225C4×C7—22no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—81no—
234C24—21no—
235C24C7×C22⋊C411yescharacteristic
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C7×C22⋊C411yescharacteristic
258C22×C4C7×C22⋊C411yescharacteristic
259C22×C4C22×C4×C711yescharacteristic
260C22×C4C2×C7×D411yescharacteristic
261C22×C4C2×C7×D411yescharacteristic
262M16—82no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—82no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—22no—
278C42—41no—
279C2×C7—21no—
280C2×C7—21no—
281C2×C7—22no—
282C2×C7—41no—
283C2×C7—161no—
284C2×C7#146811yescharacteristic
285C2×C7#212711yescharacteristic
286C2×C7C2×#35511yescharacteristic
287C23—22no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23C7×#9911yescharacteristic
299C23C7×#9911yescharacteristic
300C23C2×C7×C22⋊C411yescharacteristic
301D4—82no—
302D4—82no—
303D4—82no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4C7×#13112yes—
333C8—82no—
334C8—82no—
335C7#1155311yescharacteristic, direct product factor, Sylow
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C7×#35512yes—
353C22C7×#40811yescharacteristic
354C22C2×C7×#9911yescharacteristic
355C22C7×#41311yescharacteristic
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—22no—
368C2—41no—
369C2—161no—
370C2C7×#146811yescharacteristic
371C2C7×#212711yescharacteristic
372C2C2×C7×#35511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1463C2semidirect
3. C2×C7×#324C2semidirect
4. C7×#1324C2semidirect
5. C7×#1318C2semidirect
23. #11553C7direct (N × H)
58. #1463C2×C7semidirect
59. C2×#324C2×C7semidirect
60. #1324C2×C7semidirect
61. #1318C2×C7semidirect
335. C7#11553direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25042
C7×#14632
C2×C7×#3242
C7×#13242
C7×#13182
#115537