Finite Group Databasebeta

← C7×#11844 #13189060

Subgroups — C7×#11844 #13189060

Order 1792.

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

normal subgroups only

This group has 432 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×#2610C211yesmaximal, characteristic
2C7×#1820C211yesmaximal, characteristic, complemented
3C7×#1458C211yesmaximal, characteristic, complemented
4C7×#1351C212yesmaximal, complemented
5C7×#1343C212yesmaximal
6C2×C7×#124C411yescharacteristic, complemented
7C2×C7×#117C2211yescharacteristic
8C7×#431—21no—
9C7×#395C411yescharacteristic, complemented
10C7×#394C411yescharacteristic, complemented
11C7×#393C2211yescharacteristic
12C7×#392C2211yescharacteristic
13C7×#390—21no—
14C7×#388—21no—
15C7×#388—21no—
16C7×#386—21no—
17C7×#381—21no—
18C7×#376C411yescharacteristic, complemented
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11844C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×#124—41no—
29C7×#124—41no—
30C7×#117—41no—
31C7×#117—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—21no—
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×C7—21no—
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C7×C4⋊C8—41no—
48C7×C22⋊C8—22no—
49C4×C7×C8—41no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—41no—
54C7×#95—41no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2610C2×C711yescharacteristic
58#1820C2×C711yescharacteristic, complemented
59#1458C2×C711yescharacteristic, complemented
60#1351C2×C712yescomplemented
61#1343C2×C712yes—
62C24×C7—21no—
63C24×C7C2×D411yescharacteristic
64C2×C7×Q8—41no—
65C2×C7×Q8—81no—
66C2×C7×D4—42no—
67C2×C7×D4—41no—
68C2×C7×D4—42no—
69C2×C7×D4—81no—
70C2×C7×D4—81no—
71C22×C4×C7—21no—
72C22×C4×C7—22no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—41no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C2×C7×C8—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—81no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—81no—
99C7×C22⋊C4—81no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—42no—
105C2×#124C4×C711yescharacteristic, complemented
106C2×#117C22×C711yescharacteristic
107#431—21no—
108#395C4×C711yescharacteristic, complemented
109#394C4×C711yescharacteristic, complemented
110#393C22×C711yescharacteristic
111#392C22×C711yescharacteristic
112#390—21no—
113#388—21no—
114#388—21no—
115#386—21no—
116#381—21no—
117#376C4×C711yescharacteristic, complemented
118#342—21no—
119#342C22×C712yes—
120#324—21no—
121#323C22×C712yes—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
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×Q8—82no—
136C7×D4—84no—
137C7×D4—82no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—22no—
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—42no—
149C2×C4×C7—42no—
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—82no—
163C2×C4×C7—82no—
164C2×C4×C7—82no—
165C2×C4×C7#9911yescharacteristic
166C2×C4×C7#9911yescharacteristic
167C7×C8—82no—
168C22×Q8C2×C4×C711yescharacteristic
169C22×D4—21no—
170C22×D4C2×C4×C711yescharacteristic
171C23×C4C7×D411yescharacteristic
172#124—41no—
173#124—41no—
174#117—41no—
175#117—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C2×C4×C711yescharacteristic
178C2×C4⋊C4C7×D411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4C7×D411yescharacteristic
187C2×C22⋊C4C23×C711yescharacteristic
188C2×C42—21no—
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—41no—
198#95—41no—
199#95C2×C4×C711yescharacteristic
200#95C2×C4×C711yescharacteristic
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—22no—
204C22×C7—42no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7C2×#9911yescharacteristic
216C22×C7#41611yescharacteristic
217C22×C7#42111yescharacteristic
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—42no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—162no—
229C24—21no—
230C24C2×C7×D411yescharacteristic
231C2×Q8—41no—
232C2×Q8—81no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—81no—
237C2×D4—81no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4C7×C22⋊C411yescharacteristic
247C22×C4C7×C22⋊C411yescharacteristic
248C22×C4C7×C22⋊C411yescharacteristic
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C22×C4×C711yescharacteristic
251C22×C4C2×C7×D411yescharacteristic
252C2×C8—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—42no—
272C2×C7—21no—
273C2×C7—21no—
274C2×C7—42no—
275C2×C7—161no—
276C2×C7#151611yescharacteristic
277C2×C7#212211yescharacteristic
278C2×C7#212611yescharacteristic
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×C7×C22⋊C411yescharacteristic
291C23C7×#13512yes—
292Q8—82no—
293D4—84no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
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—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4C7×#9911yescharacteristic
323C2×C4C7×#9911yescharacteristic
324C8—82no—
325C7#1184411yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C2×C7×#9911yescharacteristic
341C22C7×#41611yescharacteristic
342C22C7×#42111yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—162no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—161no—
358C2C7×#151611yescharacteristic
359C2C7×#212211yescharacteristic
360C2C7×#212611yescharacteristic
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×#1351C2semidirect
6. C2×C7×#124C4semidirect
9. C7×#395C4semidirect
10. C7×#394C4semidirect
18. C7×#376C4semidirect
23. #11844C7direct (N × H)
58. #1820C2×C7semidirect
59. #1458C2×C7semidirect
60. #1351C2×C7semidirect
105. C2×#124C4×C7semidirect
108. #395C4×C7semidirect
109. #394C4×C7semidirect
117. #376C4×C7semidirect
325. C7#11844direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26102
C7×#18202
C7×#14582
C7×#13512
C7×#13432
#118447