Finite Group Databasebeta

← C7×#11547 #13188763

Subgroups — C7×#11547 #13188763

Order 1792.

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

normal subgroups only

This group has 492 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×#2483C211yesmaximal, characteristic
2C7×#1462C211yesmaximal, characteristic
3C2×C7×#323C211yesmaximal, characteristic
4C7×#1326C212yesmaximal, complemented
5C7×#1317C212yesmaximal, complemented
6C22×C7×C22⋊C4C411yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#397—21no—
11C7×#397—21no—
12C7×#385—21no—
13C7×#385C411yescharacteristic
14C7×#385C2211yescharacteristic
15C7×#381—21no—
16C7×#381C411yescharacteristic
17C7×#381C2211yescharacteristic
18C7×#379C411yescharacteristic
19C7×#323—21no—
20C7×#323—21no—
21C7×#323C2212yes—
22C7×#323C2212yes—
23#11547C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—41no—
30C22×C7×C8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—41no—
38C2×C7×C22⋊C4—41no—
39C2×C7×C22⋊C4—41no—
40C2×C7×C22⋊C4—41no—
41C2×C7×C22⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—41no—
50C7×C22⋊C8—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2483C2×C711yescharacteristic
58#1462C2×C711yescharacteristic
59C2×#323C2×C711yescharacteristic
60#1326C2×C712yescomplemented
61#1317C2×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×C7C22×C411yescharacteristic
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—21no—
79C22×C4×C7—22no—
80C22×C4×C7—42no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—81no—
85C22×C4×C7—81no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C7×M16—82no—
91C2×C7×C8—42no—
92C2×C7×C8—42no—
93C2×C7×C8—82no—
94C7×C4⋊C4—81no—
95C7×C4⋊C4—81no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—81no—
107C22×C22⋊C4C4×C711yescharacteristic
108C22×C22⋊C4C22×C711yescharacteristic
109#407—21no—
110C2×C22⋊C8—21no—
111#397—21no—
112#397—21no—
113#385—21no—
114#385C4×C711yescharacteristic
115#385C22×C711yescharacteristic
116#381—21no—
117#381C4×C711yescharacteristic
118#381C22×C711yescharacteristic
119#379C4×C711yescharacteristic
120#323—21no—
121#323—21no—
122#323C22×C712yes—
123#323C22×C712yes—
124C23×C7—22no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—41no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—42no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—42no—
141C23×C7—81no—
142C23×C7—81no—
143C23×C7—84no—
144C23×C7—84no—
145C23×C7#9911yescharacteristic
146C23×C7#9911yescharacteristic
147C23×C7C2×C22⋊C411yescharacteristic
148C23×C7#13112yes—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
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—81no—
172C2×C4×C7—81no—
173C2×C4×C7—82no—
174C2×C4×C7—82no—
175C7×C8—82no—
176C7×C8—82no—
177C25C2×C4×C711yescharacteristic
178C23×C4C2×C4×C711yescharacteristic
179C23×C4C2×C4×C711yescharacteristic
180C23×C4C7×D411yescharacteristic
181C23×C4C7×D411yescharacteristic
182C2×M16—41no—
183C22×C8—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4C2×C4×C711yescharacteristic
195C2×C22⋊C4C2×C4×C711yescharacteristic
196C2×C22⋊C4C2×C4×C711yescharacteristic
197C2×C22⋊C4C7×D411yescharacteristic
198C2×C22⋊C4C7×D411yescharacteristic
199C2×C22⋊C4C23×C711yescharacteristic
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—21no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—84no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7—84no—
231C22×C7#35212yes—
232C22×C7#40811yescharacteristic
233C22×C7C2×#9911yescharacteristic
234C22×C7#41311yescharacteristic
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—81no—
240C4×C7—81no—
241C4×C7—81no—
242C4×C7—81no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—41no—
247C24—42no—
248C24—81no—
249C24C7×C22⋊C411yescharacteristic
250C24C7×C22⋊C411yescharacteristic
251C24C22×C4×C711yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C7×C22⋊C411yescharacteristic
268C22×C4C7×C22⋊C411yescharacteristic
269C22×C4C2×C7×D411yescharacteristic
270C22×C4C2×C7×D411yescharacteristic
271M16—82no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—82no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C2×C7—21no—
289C2×C7—21no—
290C2×C7—22no—
291C2×C7—41no—
292C2×C7—42no—
293C2×C7—81no—
294C2×C7#146811yescharacteristic
295C2×C7C2×#35211yescharacteristic
296C2×C7#212111yescharacteristic
297C23—22no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—84no—
317C23—84no—
318C23C7×#9911yescharacteristic
319C23C7×#9911yescharacteristic
320C23C2×C7×C22⋊C411yescharacteristic
321C23C7×#13112yes—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
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—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C8—82no—
349C8—82no—
350C7#1154711yescharacteristic, direct product factor, Sylow
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—84no—
372C22C7×#35212yes—
373C22C7×#40811yescharacteristic
374C22C2×C7×#9911yescharacteristic
375C22C7×#41311yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C2—21no—
385C2—21no—
386C2—22no—
387C2—41no—
388C2—42no—
389C2—81no—
390C2C7×#146811yescharacteristic
391C2C2×C7×#35211yescharacteristic
392C2C7×#212111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC25×C7C2×C4
LCSG ▷ C24 ▷ C22 ▷ 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×#1326C2semidirect
5. C7×#1317C2semidirect
23. #11547C7direct (N × H)
60. #1326C2×C7semidirect
61. #1317C2×C7semidirect
350. C7#11547direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24832
C7×#14622
C2×C7×#3232
C7×#13262
C7×#13172
#115477