Finite Group Databasebeta

← C7×#11823 #13189039

Subgroups — C7×#11823 #13189039

Order 1792.

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

normal subgroups only

This group has 446 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×#2398C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1351C212yesmaximal, complemented
5C7×#1344C212yesmaximal, complemented
6C2×C4×C7×Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99—21no—
9C7×#407—21no—
10C7×#400—21no—
11C7×#397C411yescharacteristic, complemented
12C7×#397C2211yescharacteristic
13C7×#394—21no—
14C7×#393C411yescharacteristic, complemented
15C7×#393C2211yescharacteristic
16C7×#380—21no—
17C4×C7×C22⋊C4—21no—
18C4×C7×C22⋊C4—21no—
19C7×#376C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11823C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—41no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7—21no—
46C2×C42×C7C2×C411yescharacteristic
47C7×#99—42no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—41no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—41no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2398C2×C711yescharacteristic
59#1799C2×C711yescharacteristic
60#1465C2×C711yescharacteristic
61#1351C2×C712yescomplemented
62#1344C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—42no—
66C2×C7×Q8—41no—
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
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—42no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C7×M16—82no—
88C2×C7×C8—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—82no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—42no—
105C42×C7—41no—
106C42×C7—41no—
107C2×C4×Q8C4×C711yescharacteristic, complemented
108C2×C4×D4C22×C711yescharacteristic
109C2×#99—21no—
110#407—21no—
111#400—21no—
112#397C4×C711yescharacteristic, complemented
113#397C22×C711yescharacteristic
114#394—21no—
115#393C4×C711yescharacteristic, complemented
116#393C22×C711yescharacteristic
117#380—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C4×C711yescharacteristic, complemented
121#342C22×C712yes—
122#324—21no—
123#323C22×C712yes—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7C2×C22⋊C411yescharacteristic
134C23×C7AGL(1,ℤ/8)11yescharacteristic
135C23×C7#13711yescharacteristic
136C7×Q8—42no—
137C7×Q8—42no—
138C7×D4—42no—
139C7×D4—42no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
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—81no—
165C2×C4×C7—81no—
166C2×C4×C7—82no—
167C2×C4×C7—82no—
168C2×C4×C7—82no—
169C2×C4×C7#9911yescharacteristic
170C2×C4×C7#9911yescharacteristic
171C7×C8—82no—
172C22×Q8C2×C4×C711yescharacteristic
173C22×D4C7×D411yescharacteristic
174C23×C4C7×D411yescharacteristic
175C23×C4C7×D411yescharacteristic
176C2×M16—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C2×C4×C711yescharacteristic
184C2×C4⋊C4C7×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C2×C4×C711yescharacteristic
191C2×C22⋊C4C23×C711yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C2×C4×C711yescharacteristic
195#99—42no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—41no—
204#95C2×C4×C711yescharacteristic
205#95C2×C4×C711yescharacteristic
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—42no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7C2×#9911yescharacteristic
221C22×C7#41711yescharacteristic
222C22×C7#42111yescharacteristic
223C4×C7—21no—
224C4×C7—42no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—81no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—162no—
233C4×C7#35412yes—
234C24C7×C22⋊C411yescharacteristic
235C24C2×C7×D411yescharacteristic
236C2×Q8—42no—
237C2×Q8—41no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C7×C22⋊C411yescharacteristic
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C7×C22⋊C411yescharacteristic
256C22×C4C22×C4×C711yescharacteristic
257C22×C4C2×C7×D411yescharacteristic
258M16—82no—
259C2×C8—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42—41no—
277C42—41no—
278C2×C7—21no—
279C2×C7—21no—
280C2×C7—41no—
281C2×C7—41no—
282C2×C7—81no—
283C2×C7#151311yescharacteristic
284C2×C7#212011yescharacteristic
285C2×C7C2×#35411yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23C2×C7×C22⋊C411yescharacteristic
296C23C7×AGL(1,ℤ/8)11yescharacteristic
297C23C7×#13711yescharacteristic
298Q8—42no—
299Q8—42no—
300D4—42no—
301D4—42no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4C7×#9911yescharacteristic
332C2×C4C7×#9911yescharacteristic
333C8—82no—
334C7#1182311yescharacteristic, direct product factor, Sylow
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×C7×#9911yescharacteristic
350C22C7×#41711yescharacteristic
351C22C7×#42111yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—162no—
362C4C7×#35412yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2C7×#151311yescharacteristic
369C2C7×#212011yescharacteristic
370C2C2×C7×#35411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1351C2semidirect
5. C7×#1344C2semidirect
6. C2×C4×C7×Q8C4semidirect
11. C7×#397C4semidirect
14. C7×#393C4semidirect
19. C7×#376C4semidirect
23. #11823C7direct (N × H)
61. #1351C2×C7semidirect
62. #1344C2×C7semidirect
107. C2×C4×Q8C4×C7semidirect
112. #397C4×C7semidirect
115. #393C4×C7semidirect
120. #376C4×C7semidirect
334. C7#11823direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23982
C7×#17992
C7×#14652
C7×#13512
C7×#13442
#118237