Finite Group Databasebeta

← C7×#11808 #13189024

Subgroups — C7×#11808 #13189024

Order 1792.

680 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×C7×#393C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic
3C7×#1462C211yesmaximal, characteristic
4C7×#1351C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic, complemented
6C22×C7×C22⋊C4—21no—
7C22×C7×C22⋊C4C2211yescharacteristic
8C7×#407—21no—
9C7×#393—21no—
10C7×#393—21no—
11C7×#393C412yescomplemented
12C7×#393C2212yes—
13C7×#379—21no—
14C2×C7×#95—21no—
15C2×C7×#95C411yescharacteristic, complemented
16C7×#342—21no—
17C7×#342C2212yes—
18C7×#323—21no—
19C7×#323C2212yes—
20#11808C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7D411yescharacteristic
22C22×C7×Q8—21no—
23C22×C7×Q8—44no—
24C22×C7×Q8—41no—
25C22×C7×Q8—41no—
26C22×C7×Q8C2×C412yes—
27C23×C4×C7—21no—
28C23×C4×C7—21no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—41no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—41no—
36C2×C7×C22⋊C4—41no—
37C2×C7×C22⋊C4—44no—
38C2×C7×C22⋊C4—41no—
39C2×C7×C22⋊C4—41no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C7×C22⋊C8—22no—
45C7×C22⋊C8—41no—
46C7×#95—22no—
47C7×#95—21no—
48C7×#95—22no—
49C7×#95C2×C412yes—
50C2×#393C2×C711yescharacteristic
51#1819C2×C711yescharacteristic
52#1462C2×C711yescharacteristic
53#1351C2×C714yescomplemented
54C24×C7—22no—
55C24×C7—21no—
56C24×C7—21no—
57C24×C7—41no—
58C24×C7—42no—
59C24×C7—42no—
60C24×C7C22⋊C411yescharacteristic
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×Q8—44no—
64C2×C7×Q8—42no—
65C2×C7×Q8—44no—
66C2×C7×Q8—42no—
67C2×C7×Q8—48no—
68C2×C7×Q8—44no—
69C2×C7×Q8—44no—
70C22×C4×C7—24no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—42no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C7×M16—82no—
91C2×C7×C8—42no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—44no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—82no—
101C7×C22⋊C4—82no—
102C23×Q8C4×C711yescharacteristic, complemented
103C22×C22⋊C4—21no—
104C22×C22⋊C4C22×C711yescharacteristic
105#407—21no—
106#393—21no—
107#393—21no—
108#393C4×C712yescomplemented
109#393C22×C712yes—
110#379—21no—
111C2×#95—21no—
112C2×#95C4×C711yescharacteristic, complemented
113#342—21no—
114#342C22×C712yes—
115#323—21no—
116#323C22×C712yes—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—42no—
127C23×C7—42no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—48no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—81no—
136C23×C7—81no—
137C23×C7—84no—
138C23×C7#9911yescharacteristic
139C23×C7#9911yescharacteristic
140C23×C7C2×C22⋊C411yescharacteristic
141C23×C7#13712yes—
142C7×Q8—44no—
143C7×Q8—48no—
144C7×Q8—44no—
145C2×C4×C7—24no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—42no—
151C2×C4×C7—44no—
152C2×C4×C7—42no—
153C2×C4×C7—44no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—81no—
165C2×C4×C7—84no—
166C2×C4×C7—82no—
167C2×C4×C7—82no—
168C2×C4×C7—82no—
169C7×C8—82no—
170C25C7×D411yescharacteristic
171C22×Q8—21no—
172C22×Q8—44no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8C2×C4×C712yes—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C2×C4×C711yescharacteristic
179C23×C4C7×D411yescharacteristic
180C2×M16—41no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4C2×C4×C711yescharacteristic
190C2×C22⋊C4C7×D411yescharacteristic
191C2×C22⋊C4C7×D411yescharacteristic
192C2×C22⋊C4C23×C711yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195#95—22no—
196#95—21no—
197#95—22no—
198#95C2×C4×C712yes—
199C22×C7—24no—
200C22×C7—24no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—22no—
208C22×C7—42no—
209C22×C7—42no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—81no—
216C22×C7—84no—
217C22×C7—84no—
218C22×C7—81no—
219C22×C7AΣL(1,𝔽16)12yes—
220C22×C7#35412yes—
221C22×C7C2×#9911yescharacteristic
222C22×C7#41911yescharacteristic
223C22×C7#42111yescharacteristic
224C4×C7—44no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—82no—
230C4×C7—162no—
231C24—22no—
232C24—21no—
233C24—21no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24C7×C22⋊C411yescharacteristic
238C24C7×C22⋊C411yescharacteristic
239C24C2×C7×D411yescharacteristic
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—48no—
245C2×Q8—44no—
246C2×Q8—44no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C7×C22⋊C411yescharacteristic
264C22×C4C7×C22⋊C411yescharacteristic
265C22×C4C22×C4×C711yescharacteristic
266C22×C4C2×C7×D411yescharacteristic
267M16—82no—
268C2×C8—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C2×C7—21no—
280C2×C7—21no—
281C2×C7—22no—
282C2×C7—22no—
283C2×C7—42no—
284C2×C7—81no—
285C2×C7#151111yescharacteristic
286C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
287C2×C7C2×#35411yescharacteristic
288C23—22no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—48no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—84no—
309C23C7×#9911yescharacteristic
310C23C7×#9911yescharacteristic
311C23C2×C7×C22⋊C411yescharacteristic
312C23C7×#13712yes—
313Q8—44no—
314Q8—44no—
315Q8—48no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C8—82no—
341C7#1180811yescharacteristic, direct product factor, Sylow
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—84no—
360C22—84no—
361C22—81no—
362C22C7×AΣL(1,𝔽16)12yes—
363C22C7×#35412yes—
364C22C2×C7×#9911yescharacteristic
365C22C7×#41911yescharacteristic
366C22C7×#42111yescharacteristic
367C4—44no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C4—162no—
374C2—21no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—42no—
379C2—81no—
380C2C7×#151111yescharacteristic
381C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
382C2C2×C7×#35411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC25×C7D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1351C2semidirect
5. C23×C7×Q8C4semidirect
11. C7×#393C4semidirect
15. C2×C7×#95C4semidirect
20. #11808C7direct (N × H)
53. #1351C2×C7semidirect
102. C23×Q8C4×C7semidirect
108. #393C4×C7semidirect
112. C2×#95C4×C7semidirect
341. C7#11808direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C7×#18192
C7×#14622
C7×#13512
#118087