Finite Group Databasebeta

← C7×#11885 #13189101

Subgroups — C7×#11885 #13189101

Order 1792.

422 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3050C211yesmaximal, characteristic
2C2×C7×C22⋊C16C211yesmaximal, characteristic, complemented
3C7×#1807C211yesmaximal, characteristic, complemented
4C7×#1348C212yesmaximal, complemented
5C7×#1347C212yesmaximal, complemented
6C2×C7×#135C411yescharacteristic, complemented
7C23×C7×C8C2211yescharacteristic
8C7×#525—21no—
9C22×C7×C16—21no—
10C7×#467C411yescharacteristic, complemented
11C7×#467C2211yescharacteristic
12C7×#466C411yescharacteristic, complemented
13C7×#466C2211yescharacteristic
14C7×#465—21no—
15C7×#465—21no—
16C7×#427C411yescharacteristic, complemented
17C2×C7×#106—21no—
18C7×#416—21no—
19C7×#380—21no—
20C7×C22⋊C16—21no—
21C7×C22⋊C16C2212yes—
22C7×#340C2212yes—
23#11885C711yesmaximal, characteristic, direct product factor, Sylow
24C2×C7×#41—21no—
25C23×C4×C7D411yescharacteristic
26C7×#135—42no—
27C2×C7×Q16C2×C411yescharacteristic
28C2×C7×SD16—21no—
29C2×C7×D8C2×C411yescharacteristic
30C22×C7×C8—42no—
31C22×C7×C8C2×C411yescharacteristic
32C22×C7×C8C2×C411yescharacteristic
33C22×C7×C8D411yescharacteristic
34C22×C7×C8D411yescharacteristic
35C22×C7×C8D411yescharacteristic
36C22×C7×C8C2311yescharacteristic
37C7×#123—41no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×#121—21no—
41C7×#121—21no—
42C7×#120—41no—
43C4×C7×D4—41no—
44C4×C7×D4—41no—
45C7×#117—21no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C16—22no—
49C2×C7×C16—42no—
50C7×#107C2×C411yescharacteristic
51C7×#107C2×C411yescharacteristic
52C7×#106—21no—
53C7×#106—42no—
54C7×#103—21no—
55C7×#103—21no—
56C7×#102—21no—
57C7×#102—21no—
58C7×#95—41no—
59#3050C2×C711yescharacteristic
60C2×C22⋊C16C2×C711yescharacteristic, complemented
61#1807C2×C711yescharacteristic, complemented
62#1348C2×C712yescomplemented
63#1347C2×C712yescomplemented
64C24×C7D811yescharacteristic
65C7×#41—42no—
66C7×#41—42no—
67C2×C7×Q8—21no—
68C2×C7×D4—21no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7—42no—
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7D811yescharacteristic
79C22×C4×C7SD1611yescharacteristic
80C22×C4×C7SD1611yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×Q16—41no—
83C7×SD16—42no—
84C7×D8—41no—
85C2×C7×C8—22no—
86C2×C7×C8—22no—
87C2×C7×C8—22no—
88C2×C7×C8—41no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C2×C7×C8C22⋊C411yescharacteristic
92C2×C7×C8C22⋊C411yescharacteristic
93C2×C7×C8C22×C411yescharacteristic
94C2×C7×C8C2×D411yescharacteristic
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—82no—
107C42×C7—41no—
108C7×C16—42no—
109C2×#135C4×C711yescharacteristic, complemented
110C23×C8C22×C711yescharacteristic
111#525—21no—
112C22×C16—21no—
113#467C4×C711yescharacteristic, complemented
114#467C22×C711yescharacteristic
115#466C4×C711yescharacteristic, complemented
116#466C22×C711yescharacteristic
117#465—21no—
118#465—21no—
119#427C4×C711yescharacteristic, complemented
120C2×#106—21no—
121#416—21no—
122#380—21no—
123C22⋊C16—21no—
124C22⋊C16C22×C712yes—
125#340C22×C712yes—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7#10211yescharacteristic
131C23×C7#10211yescharacteristic
132C23×C7C2×D811yescharacteristic
133C7×Q8—41no—
134C7×Q8—41no—
135C7×D4—41no—
136C7×D4—41no—
137C7×D4—82no—
138C7×D4—82no—
139C7×D4—82no—
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—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—82no—
153C2×C4×C7—82no—
154C2×C4×C7—82no—
155C2×C4×C7—82no—
156C2×C4×C7#10211yescharacteristic
157C2×C4×C7#10211yescharacteristic
158C2×C4×C7C2×C22⋊C411yescharacteristic
159C2×C4×C7C2×SD1611yescharacteristic
160C7×C8—21no—
161C7×C8—21no—
162C7×C8—21no—
163C7×C8#9912yes—
164C2×#41—21no—
165C23×C4C7×D411yescharacteristic
166#135—42no—
167C2×Q16C2×C4×C711yescharacteristic
168C2×SD16—21no—
169C2×D8C2×C4×C711yescharacteristic
170C22×C8—42no—
171C22×C8C2×C4×C711yescharacteristic
172C22×C8C2×C4×C711yescharacteristic
173C22×C8C7×D411yescharacteristic
174C22×C8C7×D411yescharacteristic
175C22×C8C7×D411yescharacteristic
176C22×C8C23×C711yescharacteristic
177#123—41no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180#121—21no—
181#121—21no—
182#120—41no—
183C4×D4—41no—
184C4×D4—41no—
185#117—21no—
186C2×C4⋊C4—22no—
187C2×C22⋊C4—41no—
188C2×C16—22no—
189C2×C16—42no—
190#107C2×C4×C711yescharacteristic
191#107C2×C4×C711yescharacteristic
192#106—21no—
193#106—42no—
194#103—21no—
195#103—21no—
196#102—21no—
197#102—21no—
198#95—41no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—44no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7#36012yes—
211C22×C7C2×#10211yescharacteristic
212C22×C7#50812yes—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—81no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—162no—
220C4×C7C2×#9911yescharacteristic
221C4×C7#42111yescharacteristic
222C24C7×D811yescharacteristic
223#41—42no—
224#41—42no—
225C2×Q8—21no—
226C2×D4—21no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4C7×C22⋊C411yescharacteristic
235C22×C4C7×C22⋊C411yescharacteristic
236C22×C4C7×D811yescharacteristic
237C22×C4C7×SD1611yescharacteristic
238C22×C4C7×SD1611yescharacteristic
239C22×C4C2×C7×D411yescharacteristic
240Q16—41no—
241SD16—42no—
242D8—41no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8C7×C22⋊C411yescharacteristic
250C2×C8C7×C22⋊C411yescharacteristic
251C2×C8C22×C4×C711yescharacteristic
252C2×C8C2×C7×D411yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—82no—
265C42—41no—
266C16—42no—
267C2×C7—22no—
268C2×C7—22no—
269C2×C7—41no—
270C2×C7—161no—
271C2×C7#151911yescharacteristic
272C2×C7#213811yescharacteristic
273C2×C7C2×#36011yescharacteristic
274C23—22no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23C7×#10211yescharacteristic
279C23C7×#10211yescharacteristic
280C23C2×C7×D811yescharacteristic
281Q8—41no—
282Q8—41no—
283D4—41no—
284D4—41no—
285D4—82no—
286D4—82no—
287D4—82no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4C7×#10211yescharacteristic
305C2×C4C7×#10211yescharacteristic
306C2×C4C2×C7×C22⋊C411yescharacteristic
307C2×C4C2×C7×SD1611yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8C7×#9912yes—
312C7#1188511yescharacteristic, direct product factor, Sylow
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—44no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C7×#36012yes—
325C22C2×C7×#10211yescharacteristic
326C22C7×#50812yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—162no—
334C4C2×C7×#9911yescharacteristic
335C4C7×#42111yescharacteristic
336C2—22no—
337C2—22no—
338C2—41no—
339C2—161no—
340C2C7×#151911yescharacteristic
341C2C7×#213811yescharacteristic
342C2C2×C7×#36011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#102
commutatorC2×C8C22×C4×C7
Frattini, cosocleC22×C8C23×C7
prime radicalC7×#3050C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×C22⋊C16C2semidirect
3. C7×#1807C2semidirect
4. C7×#1348C2semidirect
5. C7×#1347C2semidirect
6. C2×C7×#135C4semidirect
10. C7×#467C4semidirect
12. C7×#466C4semidirect
16. C7×#427C4semidirect
23. #11885C7direct (N × H)
60. C2×C22⋊C16C2×C7semidirect
61. #1807C2×C7semidirect
62. #1348C2×C7semidirect
63. #1347C2×C7semidirect
109. C2×#135C4×C7semidirect
113. #467C4×C7semidirect
115. #466C4×C7semidirect
119. #427C4×C7semidirect
312. C7#11885direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30502
C2×C7×C22⋊C162
C7×#18072
C7×#13482
C7×#13472
#118857