Finite Group Databasebeta

← C7×#11797 #13189013

Subgroups — C7×#11797 #13189013

Order 1792.

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

normal subgroups only

This group has 572 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×#397C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic
3C7×#1462C211yesmaximal, characteristic
4C7×#1344C214yesmaximal, complemented
5C22×C7×C4⋊C4C411yescharacteristic, complemented
6C22×C7×C22⋊C4—21no—
7C22×C7×C22⋊C4C2211yescharacteristic
8C7×#407—21no—
9C7×#397—21no—
10C7×#397—21no—
11C7×#397C412yescomplemented
12C7×#397C2212yes—
13C7×#379—21no—
14C2×C7×#95—21no—
15C2×C7×#95C411yescharacteristic, complemented
16C7×#342—21no—
17C7×#342C2212yes—
18C7×#323—21no—
19C7×#323C2212yes—
20#11797C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7—21no—
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C2×C7×M16—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—41no—
29C2×C7×C4⋊C4—41no—
30C2×C7×C4⋊C4C2×C412yes—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—41no—
35C2×C7×C22⋊C4—41no—
36C2×C7×C22⋊C4—44no—
37C2×C7×C22⋊C4—41no—
38C2×C7×C22⋊C4—41no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4C2311yescharacteristic
43C7×C22⋊C8—22no—
44C7×C22⋊C8—41no—
45C7×#95—22no—
46C7×#95—21no—
47C7×#95—22no—
48C7×#95C2×C412yes—
49C2×#397C2×C711yescharacteristic
50#1819C2×C711yescharacteristic
51#1462C2×C711yescharacteristic
52#1344C2×C714yescomplemented
53C24×C7—22no—
54C24×C7—21no—
55C24×C7—21no—
56C24×C7—41no—
57C24×C7—42no—
58C24×C7—42no—
59C24×C7C22⋊C411yescharacteristic
60C24×C7C22⋊C411yescharacteristic
61C24×C7C2×D411yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—41no—
76C22×C4×C7—41no—
77C22×C4×C7—42no—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×M16—82no—
83C2×C7×C8—42no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—42no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—44no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—82no—
95C7×C22⋊C4—82no—
96C22×C4⋊C4C4×C711yescharacteristic, complemented
97C22×C22⋊C4—21no—
98C22×C22⋊C4C22×C711yescharacteristic
99#407—21no—
100#397—21no—
101#397—21no—
102#397C4×C712yescomplemented
103#397C22×C712yes—
104#379—21no—
105C2×#95—21no—
106C2×#95C4×C711yescharacteristic, complemented
107#342—21no—
108#342C22×C712yes—
109#323—21no—
110#323C22×C712yes—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—48no—
127C23×C7—42no—
128C23×C7—42no—
129C23×C7—81no—
130C23×C7—81no—
131C23×C7—84no—
132C23×C7#9911yescharacteristic
133C23×C7#9911yescharacteristic
134C23×C7C2×C22⋊C411yescharacteristic
135C23×C7AGL(1,ℤ/8)12yes—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—81no—
154C2×C4×C7—84no—
155C2×C4×C7—82no—
156C2×C4×C7—82no—
157C2×C4×C7—82no—
158C2×C4×C7—82no—
159C2×C4×C7—82no—
160C7×C8—82no—
161C25C7×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C2×C4×C711yescharacteristic
165C23×C4C7×D411yescharacteristic
166C2×M16—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×C4×C712yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4C2×C4×C711yescharacteristic
180C2×C22⋊C4C7×D411yescharacteristic
181C2×C22⋊C4C7×D411yescharacteristic
182C2×C22⋊C4C23×C711yescharacteristic
183C22⋊C8—22no—
184C22⋊C8—41no—
185#95—22no—
186#95—21no—
187#95—22no—
188#95C2×C4×C712yes—
189C22×C7—24no—
190C22×C7—24no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—22no—
198C22×C7—42no—
199C22×C7—42no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—81no—
206C22×C7—84no—
207C22×C7—84no—
208C22×C7—81no—
209C22×C7#35212yes—
210C22×C7#35412yes—
211C22×C7C2×#9911yescharacteristic
212C22×C7#41811yescharacteristic
213C22×C7#42111yescharacteristic
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—82no—
219C4×C7—82no—
220C4×C7—162no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—41no—
225C24—42no—
226C24—42no—
227C24C7×C22⋊C411yescharacteristic
228C24C7×C22⋊C411yescharacteristic
229C24C2×C7×D411yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4C7×C22⋊C411yescharacteristic
247C22×C4C7×C22⋊C411yescharacteristic
248C22×C4C22×C4×C711yescharacteristic
249C22×C4C2×C7×D411yescharacteristic
250M16—82no—
251C2×C8—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C2×C7—21no—
265C2×C7—22no—
266C2×C7—22no—
267C2×C7—21no—
268C2×C7—42no—
269C2×C7—81no—
270C2×C7#151011yescharacteristic
271C2×C7C2×#35211yescharacteristic
272C2×C7C2×#35411yescharacteristic
273C23—22no—
274C23—22no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—42no—
283C23—42no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—48no—
289C23—42no—
290C23—42no—
291C23—81no—
292C23—81no—
293C23—84no—
294C23C7×#9911yescharacteristic
295C23C7×#9911yescharacteristic
296C23C2×C7×C22⋊C411yescharacteristic
297C23C7×AGL(1,ℤ/8)12yes—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C8—82no—
323C7#1179711yescharacteristic, direct product factor, Sylow
324C22—24no—
325C22—24no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—84no—
342C22—84no—
343C22—81no—
344C22C7×#35212yes—
345C22C7×#35412yes—
346C22C2×C7×#9911yescharacteristic
347C22C7×#41811yescharacteristic
348C22C7×#42111yescharacteristic
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—82no—
354C4—82no—
355C4—162no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2C7×#151011yescharacteristic
363C2C2×C7×#35211yescharacteristic
364C2C2×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×#1344C2semidirect
5. C22×C7×C4⋊C4C4semidirect
11. C7×#397C4semidirect
15. C2×C7×#95C4semidirect
20. #11797C7direct (N × H)
52. #1344C2×C7semidirect
96. C22×C4⋊C4C4×C7semidirect
102. #397C4×C7semidirect
106. C2×#95C4×C7semidirect
323. C7#11797direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3972
C7×#18192
C7×#14622
C7×#13442
#117977