Finite Group Databasebeta

← C7×#9204 #13186420

Subgroups — C7×#9204 #13186420

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2618C211yesmaximal, characteristic
2C7×#2316C211yesmaximal, characteristic, complemented
3C7×#1519C212yesmaximal
4C2×C7×#324C211yesmaximal, characteristic, complemented
5C7×#1289C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C2211yescharacteristic
8C2×C7×#117—21no—
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×C7×C22⋊C8C2212yes—
11C7×#398C412yescomplemented
12C7×#396—22no—
13C7×#394C412yescomplemented
14C7×#389—24no—
15C7×#386—21no—
16C7×#380—21no—
17C7×#380C2211yescharacteristic
18C7×#380C2212yes—
19C7×#327—24no—
20C7×#324—22no—
21#9204C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4—21no—
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C22×C7×C8—22no—
28C7×#123—42no—
29C7×#123—42no—
30C7×#121—22no—
31C7×#121—22no—
32C7×#117—42no—
33C7×#117—42no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4C2×C412yes—
43C2×C7×C4⋊C4D412yes—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—22no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—22no—
49C2×C42×C7—22no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#95—22no—
53C7×#95—22no—
54C7×#95C2×C412yes—
55C7×#95D414yes—
56#2618C2×C711yescharacteristic
57#2316C2×C711yescharacteristic, complemented
58#1519C2×C712yes—
59C2×#324C2×C711yescharacteristic, complemented
60#1289C2×C712yescomplemented
61C24×C7—21no—
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—41no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—81no—
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—21no—
78C22×C4×C7—22no—
79C22×C4×C7—24no—
80C22×C4×C7—81no—
81C22×C4×C7—81no—
82C22×C4×C7C22⋊C412yes—
83C22×C4×C7C22⋊C412yes—
84C22×C4×C7D812yes—
85C22×C4×C7SD1612yes—
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7#4112yes—
90C2×C7×C8—42no—
91C2×C7×C8—82no—
92C7×C4⋊C4—24no—
93C7×C4⋊C4—24no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—44no—
96C7×C4⋊C4—44no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—82no—
99C7×C4⋊C4—82no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—44no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—82no—
107C42×C7—44no—
108C2×#123—21no—
109C2×#121C22×C711yescharacteristic
110C2×#117—21no—
111C22×C4⋊C4C22×C711yescharacteristic
112C2×C22⋊C8C22×C712yes—
113#398C4×C712yescomplemented
114#396—22no—
115#394C4×C712yescomplemented
116#389—24no—
117#386—21no—
118#380—21no—
119#380C22×C711yescharacteristic
120#380C22×C712yes—
121#327—24no—
122#324—22no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—42no—
126C23×C7—42no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#12011yescharacteristic
132C23×C7#12311yescharacteristic
133C23×C7C2×D811yescharacteristic
134C23×C7C2×SD1611yescharacteristic
135C23×C7AGL(1,ℤ/8)11yescharacteristic
136C23×C7#13711yescharacteristic
137C7×D4—82no—
138C7×D4—81no—
139C7×D4—81no—
140C7×D4—81no—
141C7×D4—81no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—22no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—44no—
157C2×C4×C7—44no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—44no—
164C2×C4×C7—81no—
165C2×C4×C7#10214yes—
166C2×C4×C7C4×D412yes—
167C2×C4×C7#12112yes—
168C7×C8—82no—
169C22×D4—21no—
170C22×D4C2×C4×C711yescharacteristic
171C23×C4C7×D411yescharacteristic
172C23×C4C7×D411yescharacteristic
173C23×C4C23×C711yescharacteristic
174C22×C8—22no—
175#123—42no—
176#123—42no—
177#121—22no—
178#121—22no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×C4×C711yescharacteristic
189C2×C4⋊C4C2×C4×C712yes—
190C2×C4⋊C4C7×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C42—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#95—22no—
200#95—22no—
201#95C2×C4×C712yes—
202#95C7×D414yes—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—82no—
214C22×C7—82no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7#38611yescharacteristic
219C22×C7C2×#10211yescharacteristic
220C22×C7#41711yescharacteristic
221C22×C7#44711yescharacteristic
222C22×C7#44811yescharacteristic
223C22×C7#48011yescharacteristic
224C22×C7#48111yescharacteristic
225C4×C7—24no—
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—42no—
229C4×C7—84no—
230C4×C7—82no—
231C4×C7—81no—
232C24—21no—
233C24C2×C7×D411yescharacteristic
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4C7×C22⋊C412yes—
254C22×C4C7×C22⋊C412yes—
255C22×C4C7×D812yes—
256C22×C4C7×SD1612yes—
257C22×C4C22×C4×C711yescharacteristic
258C22×C4C2×C7×D412yes—
259C22×C4C2×C7×D411yescharacteristic
260C22×C4C7×#4112yes—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—81no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—82no—
278C42—44no—
279C2×C7—21no—
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—21no—
283C2×C7—21no—
284C2×C7—21no—
285C2×C7—161no—
286C2×C7#187411yescharacteristic
287C2×C7#191211yescharacteristic
288C2×C7#191311yescharacteristic
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C2×C7×C22⋊C411yescharacteristic
297C23C7×#12011yescharacteristic
298C23C7×#12311yescharacteristic
299C23C2×C7×D811yescharacteristic
300C23C2×C7×SD1611yescharacteristic
301C23C7×AGL(1,ℤ/8)11yescharacteristic
302C23C7×#13711yescharacteristic
303D4—82no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4C7×#10214yes—
332C2×C4C4×C7×D412yes—
333C2×C4C7×#12112yes—
334C8—82no—
335C7#920411yescharacteristic, direct product factor, Sylow
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—82no—
347C22—82no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C7×#38611yescharacteristic
352C22C2×C7×#10211yescharacteristic
353C22C7×#41711yescharacteristic
354C22C7×#44711yescharacteristic
355C22C7×#44811yescharacteristic
356C22C7×#48011yescharacteristic
357C22C7×#48111yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—84no—
363C4—82no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—161no—
372C2C7×#187411yescharacteristic
373C2C7×#191211yescharacteristic
374C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2316C2semidirect
4. C2×C7×#324C2semidirect
5. C7×#1289C2semidirect
11. C7×#398C4semidirect
13. C7×#394C4semidirect
21. #9204C7direct (N × H)
57. #2316C2×C7semidirect
59. C2×#324C2×C7semidirect
60. #1289C2×C7semidirect
113. #398C4×C7semidirect
115. #394C4×C7semidirect
335. C7#9204direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26182
C7×#23162
C7×#15192
C2×C7×#3242
C7×#12892
#92047