Finite Group Databasebeta

← C7×#11840 #13189056

Subgroups — C7×#11840 #13189056

Order 1792.

446 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2598C211yesmaximal, characteristic
2C7×#1820C211yesmaximal, characteristic, complemented
3C7×#1458C211yesmaximal, characteristic, complemented
4C7×#1352C212yesmaximal, complemented
5C7×#1343C212yesmaximal
6C2×C4×C7×D4C411yescharacteristic, complemented
7C2×C7×#117C2211yescharacteristic
8C7×#431—21no—
9C7×#400C411yescharacteristic, complemented
10C7×#399C2211yescharacteristic
11C7×#394C411yescharacteristic, complemented
12C7×#392C2211yescharacteristic
13C7×#388—21no—
14C7×#386—21no—
15C7×#386—21no—
16C7×#385—21no—
17C7×#383C411yescharacteristic, complemented
18C7×#381—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11840C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7D411yescharacteristic
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C7×#117—41no—
31C7×#117—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×C22⋊C8—22no—
48C4×C7×C8—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—41no—
54C7×#95—41no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2598C2×C711yescharacteristic
58#1820C2×C711yescharacteristic, complemented
59#1458C2×C711yescharacteristic, complemented
60#1352C2×C712yescomplemented
61#1343C2×C712yes—
62C24×C7—21no—
63C24×C7C2×D411yescharacteristic
64C2×C7×D4—41no—
65C2×C7×D4—42no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—81no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—41no—
76C22×C4×C7—42no—
77C22×C4×C7—42no—
78C22×C4×C7—81no—
79C22×C4×C7—81no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22×C411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C2×C7×C8—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—81no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—81no—
100C7×C22⋊C4—81no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C2×C4×D4C4×C711yescharacteristic, complemented
106C2×#117C22×C711yescharacteristic
107#431—21no—
108#400C4×C711yescharacteristic, complemented
109#399C22×C711yescharacteristic
110#394C4×C711yescharacteristic, complemented
111#392C22×C711yescharacteristic
112#388—21no—
113#386—21no—
114#386—21no—
115#385—21no—
116#383C4×C711yescharacteristic, complemented
117#381—21no—
118#342—21no—
119#342C22×C712yes—
120#324—21no—
121#323C22×C712yes—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—22no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7C2×C22⋊C411yescharacteristic
134C23×C7#13512yes—
135C7×D4—42no—
136C7×D4—42no—
137C7×D4—84no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
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—42no—
148C2×C4×C7—42no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
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—84no—
163C2×C4×C7—82no—
164C2×C4×C7—82no—
165C2×C4×C7—82no—
166C2×C4×C7#9911yescharacteristic
167C2×C4×C7#9911yescharacteristic
168C7×C8—82no—
169C22×D4—21no—
170C22×D4C2×C4×C711yescharacteristic
171C23×C4—21no—
172C23×C4C7×D411yescharacteristic
173C4×D4—41no—
174C4×D4—41no—
175#117—41no—
176#117—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×C4×C711yescharacteristic
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C4⋊C4C7×D411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4C7×D411yescharacteristic
188C2×C22⋊C4C23×C711yescharacteristic
189C2×C42C2×C4×C711yescharacteristic
190C2×C42C7×D411yescharacteristic
191C4⋊C8—41no—
192C22⋊C8—22no—
193C4×C8—41no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—41no—
199#95—41no—
200#95C2×C4×C711yescharacteristic
201#95C2×C4×C711yescharacteristic
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—22no—
205C22×C7—42no—
206C22×C7—42no—
207C22×C7—42no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—42no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7C2×#9911yescharacteristic
217C22×C7#41611yescharacteristic
218C22×C7#42111yescharacteristic
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—162no—
230C24—21no—
231C24C2×C7×D411yescharacteristic
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—81no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4C7×C22⋊C411yescharacteristic
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C7×C22⋊C411yescharacteristic
252C22×C4C22×C4×C711yescharacteristic
253C22×C4C2×C7×D411yescharacteristic
254C2×C8—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C2×C7—21no—
274C2×C7—21no—
275C2×C7—42no—
276C2×C7—82no—
277C2×C7#151611yescharacteristic
278C2×C7#212011yescharacteristic
279C2×C7#212811yescharacteristic
280C23—21no—
281C23—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C7×C22⋊C411yescharacteristic
292C23C7×#13512yes—
293D4—42no—
294D4—42no—
295D4—84no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C7×#9911yescharacteristic
325C2×C4C7×#9911yescharacteristic
326C8—82no—
327C7#1184011yescharacteristic, direct product factor, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C2×C7×#9911yescharacteristic
343C22C7×#41611yescharacteristic
344C22C7×#42111yescharacteristic
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—162no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—82no—
360C2C7×#151611yescharacteristic
361C2C7×#212011yescharacteristic
362C2C7×#212811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1820C2semidirect
3. C7×#1458C2semidirect
4. C7×#1352C2semidirect
6. C2×C4×C7×D4C4semidirect
9. C7×#400C4semidirect
11. C7×#394C4semidirect
17. C7×#383C4semidirect
23. #11840C7direct (N × H)
58. #1820C2×C7semidirect
59. #1458C2×C7semidirect
60. #1352C2×C7semidirect
105. C2×C4×D4C4×C7semidirect
108. #400C4×C7semidirect
110. #394C4×C7semidirect
116. #383C4×C7semidirect
327. C7#11840direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25982
C7×#18202
C7×#14582
C7×#13522
C7×#13432
#118407