Finite Group Databasebeta

← C7×#11845 #13189061

Subgroups — C7×#11845 #13189061

Order 1792.

506 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2561C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic
3C2×C7×#323C211yesmaximal, characteristic
4C7×#1351C212yesmaximal, complemented
5C7×#1344C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic, complemented
7C22×C7×C22⋊C4C2211yescharacteristic
8C2×C7×C22⋊C8—21no—
9C7×#397C411yescharacteristic, complemented
10C7×#397C2211yescharacteristic
11C7×#393C411yescharacteristic, complemented
12C7×#393C2211yescharacteristic
13C7×#388—21no—
14C7×#388—21no—
15C7×#381—21no—
16C7×#379—21no—
17C7×#379—21no—
18C4×C7×C22⋊C4C411yescharacteristic, complemented
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11845C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7D411yescharacteristic
25C22×C7×Q8C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C22×C7×C8—41no—
29C7×C22⋊Q8—41no—
30C7×C22⋊Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—41no—
41C2×C7×C22⋊C4—41no—
42C2×C7×C22⋊C4—41no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—21no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2561C2×C711yescharacteristic
58#1819C2×C711yescharacteristic
59C2×#323C2×C711yescharacteristic
60#1351C2×C712yescomplemented
61#1344C2×C712yescomplemented
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—22no—
65C24×C7—41no—
66C24×C7—42no—
67C24×C7—81no—
68C24×C7C22⋊C411yescharacteristic
69C24×C7C22⋊C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×Q8—41no—
72C2×C7×Q8—81no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C2×C7×C8—42no—
89C2×C7×C8—82no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—81no—
104C7×C22⋊C4—81no—
105C7×C22⋊C4—82no—
106C7×C22⋊C4—82no—
107C42×C7—81no—
108C2×C22⋊Q8C4×C711yescharacteristic, complemented
109C22×C22⋊C4C22×C711yescharacteristic
110C2×C22⋊C8—21no—
111#397C4×C711yescharacteristic, complemented
112#397C22×C711yescharacteristic
113#393C4×C711yescharacteristic, complemented
114#393C22×C711yescharacteristic
115#388—21no—
116#388—21no—
117#381—21no—
118#379—21no—
119#379—21no—
120C4×C22⋊C4C4×C711yescharacteristic, complemented
121#342—21no—
122#342C22×C712yes—
123#323—21no—
124#323C22×C712yes—
125C23×C7—22no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—41no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—81no—
143C23×C7—81no—
144C23×C7—84no—
145C23×C7—84no—
146C23×C7#9911yescharacteristic
147C23×C7#9911yescharacteristic
148C23×C7C2×C22⋊C411yescharacteristic
149C23×C7#13512yes—
150C7×Q8—82no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—42no—
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—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—81no—
173C2×C4×C7—81no—
174C2×C4×C7—81no—
175C2×C4×C7—82no—
176C2×C4×C7—82no—
177C2×C4×C7—82no—
178C7×C8—82no—
179C25C7×D411yescharacteristic
180C22×Q8C2×C4×C711yescharacteristic
181C23×C4C2×C4×C711yescharacteristic
182C23×C4C7×D411yescharacteristic
183C22×C8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C2×C4×C711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4C2×C4×C711yescharacteristic
200C2×C22⋊C4C7×D411yescharacteristic
201C2×C22⋊C4C7×D411yescharacteristic
202C2×C22⋊C4C23×C711yescharacteristic
203C2×C42—21no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95C2×C4×C711yescharacteristic
211#95C2×C4×C711yescharacteristic
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—22no—
219C22×C7—21no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—42no—
223C22×C7—42no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—81no—
230C22×C7—84no—
231C22×C7—81no—
232C22×C7—84no—
233C22×C7#35412yes—
234C22×C7C2×#9911yescharacteristic
235C22×C7#41611yescharacteristic
236C22×C7#42111yescharacteristic
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241C4×C7—81no—
242C4×C7—81no—
243C4×C7—81no—
244C4×C7—81no—
245C4×C7—162no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—41no—
250C24—42no—
251C24—81no—
252C24C7×C22⋊C411yescharacteristic
253C24C7×C22⋊C411yescharacteristic
254C24C2×C7×D411yescharacteristic
255C2×Q8—41no—
256C2×Q8—81no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4C7×C22⋊C411yescharacteristic
269C22×C4C7×C22⋊C411yescharacteristic
270C22×C4C22×C4×C711yescharacteristic
271C22×C4C2×C7×D411yescharacteristic
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C42—81no—
292C2×C7—21no—
293C2×C7—21no—
294C2×C7—22no—
295C2×C7—42no—
296C2×C7—41no—
297C2×C7—81no—
298C2×C7#151611yescharacteristic
299C2×C7#212111yescharacteristic
300C2×C7C2×#35411yescharacteristic
301C23—22no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23C7×#9911yescharacteristic
323C23C7×#9911yescharacteristic
324C23C2×C7×C22⋊C411yescharacteristic
325C23C7×#13512yes—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C8—82no—
355C7#1184511yescharacteristic, direct product factor, Sylow
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22C7×#35412yes—
378C22C2×C7×#9911yescharacteristic
379C22C7×#41611yescharacteristic
380C22C7×#42111yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—162no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2C7×#151611yescharacteristic
397C2C7×#212111yescharacteristic
398C2C2×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. C7×#1344C2semidirect
6. C2×C7×C22⋊Q8C4semidirect
9. C7×#397C4semidirect
11. C7×#393C4semidirect
18. C4×C7×C22⋊C4C4semidirect
23. #11845C7direct (N × H)
60. #1351C2×C7semidirect
61. #1344C2×C7semidirect
108. C2×C22⋊Q8C4×C7semidirect
111. #397C4×C7semidirect
113. #393C4×C7semidirect
120. C4×C22⋊C4C4×C7semidirect
355. C7#11845direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25612
C7×#18192
C2×C7×#3232
C7×#13512
C7×#13442
#118457