Finite Group Databasebeta

← C7×#11805 #13189021

Subgroups — C7×#11805 #13189021

Order 1792.

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

normal subgroups only

This group has 448 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×#2618C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic, complemented
3C7×#1463C211yesmaximal, characteristic, complemented
4C7×#1346C212yesmaximal, complemented
5C7×#1345C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C411yescharacteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#407—21no—
10C7×#407—21no—
11C7×#398C411yescharacteristic
12C7×#398C2211yescharacteristic
13C7×#396—21no—
14C7×#396—21no—
15C7×#394C411yescharacteristic
16C7×#394C2211yescharacteristic
17C7×#386—21no—
18C7×#380C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11805C711yesmaximal, 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—22no—
30C2×C7×M16—41no—
31C7×#123—44no—
32C7×#121—41no—
33C7×#121—41no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4—41no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4C2311yescharacteristic
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—41no—
52C7×C22⋊C8—41no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2618C2×C711yescharacteristic
59#1821C2×C711yescharacteristic, complemented
60#1463C2×C711yescharacteristic, complemented
61#1346C2×C712yescomplemented
62#1345C2×C712yescomplemented
63C24×C7—21no—
64C24×C7C22⋊C411yescharacteristic
65C2×C7×D4—41no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—81no—
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—22no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C7×M16—42no—
91C7×M16—42no—
92C7×M16—82no—
93C2×C7×C8—42no—
94C2×C7×C8—42no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—42no—
102C7×C4⋊C4—81no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—81no—
109C2×#123—21no—
110C2×#121C4×C711yescharacteristic
111C22×C4⋊C4C22×C711yescharacteristic
112#407—21no—
113#407—21no—
114#398C4×C711yescharacteristic
115#398C22×C711yescharacteristic
116#396—21no—
117#396—21no—
118#394C4×C711yescharacteristic
119#394C22×C711yescharacteristic
120#386—21no—
121#380C4×C711yescharacteristic
122#343—21no—
123#343C22×C712yes—
124#324—21no—
125#324C22×C712yes—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7#9911yescharacteristic
138C23×C7#9911yescharacteristic
139C23×C7C2×C22⋊C411yescharacteristic
140C7×D4—82no—
141C7×D4—82no—
142C7×D4—82no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—42no—
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—81no—
169C2×C4×C7—81no—
170C2×C4×C7—81no—
171C2×C4×C7AGL(1,ℤ/8)12yes—
172C7×C8—82no—
173C7×C8—82no—
174C22×D4—21no—
175C22×D4C2×C4×C711yescharacteristic
176C23×C4C2×C4×C711yescharacteristic
177C23×C4C7×D411yescharacteristic
178C23×C4C7×D411yescharacteristic
179C2×M16—22no—
180C2×M16—41no—
181#123—44no—
182#121—41no—
183#121—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×C4×C711yescharacteristic
190C2×C4⋊C4C2×C4×C711yescharacteristic
191C2×C4⋊C4C7×D411yescharacteristic
192C2×C4⋊C4C7×D411yescharacteristic
193C2×C4⋊C4C23×C711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#95—21no—
205#95—21no—
206#95C2×C4×C711yescharacteristic
207#95C2×C4×C711yescharacteristic
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—42no—
217C22×C7—42no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7—81no—
222C22×C7—81no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7C2×#9911yescharacteristic
226C22×C7#41811yescharacteristic
227C22×C7#42111yescharacteristic
228C4×C7—22no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—81no—
235C4×C7—81no—
236C4×C7—81no—
237C24—21no—
238C24C7×C22⋊C411yescharacteristic
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4C7×C22⋊C411yescharacteristic
259C22×C4C7×C22⋊C411yescharacteristic
260C22×C4C7×C22⋊C411yescharacteristic
261C22×C4C22×C4×C711yescharacteristic
262C22×C4C2×C7×D411yescharacteristic
263C22×C4C2×C7×D411yescharacteristic
264M16—42no—
265M16—42no—
266M16—82no—
267C2×C8—42no—
268C2×C8—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—81no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C2×C7—21no—
284C2×C7—21no—
285C2×C7—41no—
286C2×C7—41no—
287C2×C7—161no—
288C2×C7#151011yescharacteristic
289C2×C7#212111yescharacteristic
290C2×C7#213311yescharacteristic
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23C7×#9911yescharacteristic
303C23C7×#9911yescharacteristic
304C23C2×C7×C22⋊C411yescharacteristic
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4C7×AGL(1,ℤ/8)12yes—
337C8—82no—
338C8—82no—
339C7#1180511yescharacteristic, direct product factor, Sylow
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×C7×#9911yescharacteristic
358C22C7×#41811yescharacteristic
359C22C7×#42111yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—161no—
374C2C7×#151011yescharacteristic
375C2C7×#212111yescharacteristic
376C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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×#1821C2semidirect
3. C7×#1463C2semidirect
4. C7×#1346C2semidirect
5. C7×#1345C2semidirect
23. #11805C7direct (N × H)
59. #1821C2×C7semidirect
60. #1463C2×C7semidirect
61. #1346C2×C7semidirect
62. #1345C2×C7semidirect
339. C7#11805direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26182
C7×#18212
C7×#14632
C7×#13462
C7×#13452
#118057