Finite Group Databasebeta

← C7×#11811 #13189027

Subgroups — C7×#11811 #13189027

Order 1792.

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

normal subgroups only

This group has 496 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×#2626C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic
3C7×#1462C211yesmaximal, characteristic
4C7×#1352C212yesmaximal, complemented
5C7×#1351C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic, complemented
7C22×C7×C22⋊C4C2211yescharacteristic
8C7×#407—21no—
9C7×#399C411yescharacteristic, complemented
10C7×#399C2211yescharacteristic
11C7×#396—21no—
12C7×#396—21no—
13C7×#393C411yescharacteristic, complemented
14C7×#393C2211yescharacteristic
15C7×#385—21no—
16C7×#380C411yescharacteristic, complemented
17C7×#379—21no—
18C7×#379—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11811C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7D411yescharacteristic
25C22×C7×Q8C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—41no—
29C7×C22⋊Q8—41no—
30C7×C22⋊Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C7×C22⋊C8—22no—
52C7×C22⋊C8—41no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2626C2×C711yescharacteristic
58#1819C2×C711yescharacteristic
59#1462C2×C711yescharacteristic
60#1352C2×C712yescomplemented
61#1351C2×C712yescomplemented
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—22no—
65C24×C7—41no—
66C24×C7—81no—
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—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C7×M16—82no—
89C2×C7×C8—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—81no—
93C7×C4⋊C4—81no—
94C7×C4⋊C4—81no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—81no—
105C7×C22⋊C4—81no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—82no—
109C7×C22⋊C4—82no—
110C2×C22⋊Q8C4×C711yescharacteristic, complemented
111C22×C22⋊C4C22×C711yescharacteristic
112#407—21no—
113#399C4×C711yescharacteristic, complemented
114#399C22×C711yescharacteristic
115#396—21no—
116#396—21no—
117#393C4×C711yescharacteristic, complemented
118#393C22×C711yescharacteristic
119#385—21no—
120#380C4×C711yescharacteristic, complemented
121#379—21no—
122#379—21no—
123#342—21no—
124#342C22×C712yes—
125#323—21no—
126#323C22×C712yes—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—81no—
145C23×C7—81no—
146C23×C7—84no—
147C23×C7—84no—
148C23×C7#9911yescharacteristic
149C23×C7#9911yescharacteristic
150C23×C7C2×C22⋊C411yescharacteristic
151C23×C7#13712yes—
152C7×Q8—82no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—42no—
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—41no—
173C2×C4×C7—41no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7—82no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C7×C8—82no—
181C25C7×D411yescharacteristic
182C22×Q8C2×C4×C711yescharacteristic
183C23×C4C2×C4×C711yescharacteristic
184C23×C4C7×D411yescharacteristic
185C2×M16—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×C4×C711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4C2×C4×C711yescharacteristic
205C2×C22⋊C4C7×D411yescharacteristic
206C2×C22⋊C4C7×D411yescharacteristic
207C2×C22⋊C4C23×C711yescharacteristic
208C22⋊C8—22no—
209C22⋊C8—41no—
210#95—21no—
211#95—21no—
212#95C2×C4×C711yescharacteristic
213#95C2×C4×C711yescharacteristic
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—22no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—42no—
224C22×C7—42no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—84no—
235C22×C7—84no—
236C22×C7C2×#9911yescharacteristic
237C22×C7#41911yescharacteristic
238C22×C7#42111yescharacteristic
239C4×C7—41no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—81no—
244C4×C7—81no—
245C4×C7—81no—
246C4×C7—81no—
247C4×C7—162no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—41no—
252C24—81no—
253C24—81no—
254C24C7×C22⋊C411yescharacteristic
255C24C7×C22⋊C411yescharacteristic
256C24C2×C7×D411yescharacteristic
257C2×Q8—41no—
258C2×Q8—81no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4C7×C22⋊C411yescharacteristic
271C22×C4C7×C22⋊C411yescharacteristic
272C22×C4C22×C4×C711yescharacteristic
273C22×C4C2×C7×D411yescharacteristic
274M16—82no—
275C2×C8—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C2×C7—21no—
297C2×C7—21no—
298C2×C7—42no—
299C2×C7—41no—
300C2×C7—41no—
301C2×C7—81no—
302C2×C7#151111yescharacteristic
303C2×C7#212111yescharacteristic
304C2×C7#212711yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—84no—
325C23—84no—
326C23C7×#9911yescharacteristic
327C23C7×#9911yescharacteristic
328C23C2×C7×C22⋊C411yescharacteristic
329C23C7×#13712yes—
330Q8—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
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—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C8—82no—
359C7#1181111yescharacteristic, direct product factor, Sylow
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×C7×#9911yescharacteristic
383C22C7×#41911yescharacteristic
384C22C7×#42111yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—162no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2C7×#151111yescharacteristic
401C2C7×#212111yescharacteristic
402C2C7×#212711yescharacteristic
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×#1352C2semidirect
5. C7×#1351C2semidirect
6. C2×C7×C22⋊Q8C4semidirect
9. C7×#399C4semidirect
13. C7×#393C4semidirect
16. C7×#380C4semidirect
23. #11811C7direct (N × H)
60. #1352C2×C7semidirect
61. #1351C2×C7semidirect
110. C2×C22⋊Q8C4×C7semidirect
113. #399C4×C7semidirect
117. #393C4×C7semidirect
120. #380C4×C7semidirect
359. C7#11811direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26262
C7×#18192
C7×#14622
C7×#13522
C7×#13512
#118117