Finite Group Databasebeta

← C7×#11549 #13188765

Subgroups — C7×#11549 #13188765

Order 1792.

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

normal subgroups only

This group has 496 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×#2501C211yesmaximal, characteristic
2C7×#1463C211yesmaximal, characteristic, complemented
3C2×C7×#324C211yesmaximal, characteristic, complemented
4C7×#1325C212yesmaximal, complemented
5C7×#1318C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#123—21no—
8C2×C7×#121C411yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C7×#407—21no—
11C2×C7×C22⋊C8—21no—
12C7×#399—21no—
13C7×#399—21no—
14C7×#390C411yescharacteristic
15C7×#390C2211yescharacteristic
16C7×#384C411yescharacteristic
17C7×#384C2211yescharacteristic
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#324—21no—
20C7×#324—21no—
21C7×#324C2212yes—
22C7×#324C2212yes—
23#11549C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—41no—
30C22×C7×C8—41no—
31C7×#123—44no—
32C7×#123—44no—
33C7×#121—41no—
34C7×#121—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4—41no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4C2×C411yescharacteristic
43C2×C7×C4⋊C4D411yescharacteristic
44C2×C7×C4⋊C4D411yescharacteristic
45C2×C7×C4⋊C4C2311yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C42×C7C2×C411yescharacteristic
53C2×C42×C7C2×C411yescharacteristic
54C7×C22⋊C8—22no—
55C7×C22⋊C8—22no—
56C7×C22⋊C8—41no—
57C7×C22⋊C8—41no—
58C7×#95—21no—
59#2501C2×C711yescharacteristic
60#1463C2×C711yescharacteristic, complemented
61C2×#324C2×C711yescharacteristic, complemented
62#1325C2×C712yescomplemented
63#1318C2×C712yescomplemented
64C24×C7—21no—
65C24×C7C22⋊C411yescharacteristic
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4—42no—
70C2×C7×D4—42no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
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—22no—
82C22×C4×C7—21no—
83C22×C4×C7—41no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C7×M16—82no—
93C2×C7×C8—42no—
94C2×C7×C8—42no—
95C2×C7×C8—82no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—42no—
102C7×C4⋊C4—42no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—42no—
105C7×C22⋊C4—24no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C42×C7—22no—
112C42×C7—41no—
113C2×#123—21no—
114C2×#123—21no—
115C2×#121C4×C711yescharacteristic
116C22×C4⋊C4C22×C711yescharacteristic
117#407—21no—
118C2×C22⋊C8—21no—
119#399—21no—
120#399—21no—
121#390C4×C711yescharacteristic
122#390C22×C711yescharacteristic
123#384C4×C711yescharacteristic
124#384C22×C711yescharacteristic
125C4×C22⋊C4C4×C711yescharacteristic
126#324—21no—
127#324—21no—
128#324C22×C712yes—
129#324C22×C712yes—
130C23×C7—22no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7#9911yescharacteristic
142C23×C7#9911yescharacteristic
143C23×C7C2×C22⋊C411yescharacteristic
144C7×D4—82no—
145C7×D4—82no—
146C7×D4—82no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
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—81no—
174C2×C4×C7—81no—
175C2×C4×C7#13112yes—
176C7×C8—82no—
177C7×C8—82no—
178C22×D4—21no—
179C22×D4C2×C4×C711yescharacteristic
180C23×C4C2×C4×C711yescharacteristic
181C23×C4C7×D411yescharacteristic
182C23×C4C7×D411yescharacteristic
183C2×M16—41no—
184C22×C8—41no—
185#123—44no—
186#123—44no—
187#121—41no—
188#121—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C2×C4×C711yescharacteristic
196C2×C4⋊C4C2×C4×C711yescharacteristic
197C2×C4⋊C4C7×D411yescharacteristic
198C2×C4⋊C4C7×D411yescharacteristic
199C2×C4⋊C4C23×C711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42C2×C4×C711yescharacteristic
207C2×C42C2×C4×C711yescharacteristic
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#95—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—81no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7#35312yes—
230C22×C7#40811yescharacteristic
231C22×C7C2×#9911yescharacteristic
232C22×C7#41311yescharacteristic
233C4×C7—22no—
234C4×C7—42no—
235C4×C7—42no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241C4×C7—81no—
242C24—21no—
243C24C7×C22⋊C411yescharacteristic
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4C7×C22⋊C411yescharacteristic
265C22×C4C7×C22⋊C411yescharacteristic
266C22×C4C7×C22⋊C411yescharacteristic
267C22×C4C22×C4×C711yescharacteristic
268C22×C4C2×C7×D411yescharacteristic
269C22×C4C2×C7×D411yescharacteristic
270M16—82no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—22no—
290C42—41no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—22no—
294C2×C7—41no—
295C2×C7—161no—
296C2×C7#146811yescharacteristic
297C2×C7C2×#35311yescharacteristic
298C2×C7#213311yescharacteristic
299C23—22no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C7×#9911yescharacteristic
311C23C7×#9911yescharacteristic
312C23C2×C7×C22⋊C411yescharacteristic
313D4—82no—
314D4—82no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
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—81no—
343C2×C4—81no—
344C2×C4C7×#13112yes—
345C8—82no—
346C8—82no—
347C7#1154911yescharacteristic, direct product factor, Sylow
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C7×#35312yes—
365C22C7×#40811yescharacteristic
366C22C2×C7×#9911yescharacteristic
367C22C7×#41311yescharacteristic
368C4—22no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2C7×#146811yescharacteristic
383C2C2×C7×#35311yescharacteristic
384C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1463C2semidirect
3. C2×C7×#324C2semidirect
4. C7×#1325C2semidirect
5. C7×#1318C2semidirect
23. #11549C7direct (N × H)
60. #1463C2×C7semidirect
61. C2×#324C2×C7semidirect
62. #1325C2×C7semidirect
63. #1318C2×C7semidirect
347. C7#11549direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25012
C7×#14632
C2×C7×#3242
C7×#13252
C7×#13182
#115497