Finite Group Databasebeta

← C7×#11826 #13189042

Subgroups — C7×#11826 #13189042

Order 1792.

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

normal subgroups only

This group has 414 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×#2638C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1466C211yesmaximal, characteristic
4C7×#1351C212yesmaximal, complemented
5C7×#1344C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#432—21no—
9C2×C7×#99—21no—
10C7×#398C411yescharacteristic, complemented
11C7×#397—21no—
12C7×#397C2211yescharacteristic
13C7×#395C411yescharacteristic, complemented
14C7×#394—21no—
15C7×#393C2211yescharacteristic
16C7×#387—21no—
17C7×#385—21no—
18C7×#381C411yescharacteristic, complemented
19C7×#380—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11826C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×C22⋊Q8—41no—
29C7×C22⋊Q8—41no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—41no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C7×C22⋊C4C2311yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×#99—42no—
48C7×C22⋊C8—22no—
49C7×#97—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#2638C2×C711yescharacteristic
59#1799C2×C711yescharacteristic
60#1466C2×C711yescharacteristic
61#1351C2×C712yescomplemented
62#1344C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—41no—
66C2×C7×Q8—81no—
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—81no—
81C22×C4×C7—81no—
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C2×C7×C8—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—81no—
94C7×C4⋊C4—82no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—81no—
103C7×C22⋊C4—81no—
104C42×C7—21no—
105C42×C7—21no—
106C2×C22⋊Q8C4×C711yescharacteristic, complemented
107C2×C4×D4C22×C711yescharacteristic
108#432—21no—
109C2×#99—21no—
110#398C4×C711yescharacteristic, complemented
111#397—21no—
112#397C22×C711yescharacteristic
113#395C4×C711yescharacteristic, complemented
114#394—21no—
115#393C22×C711yescharacteristic
116#387—21no—
117#385—21no—
118#381C4×C711yescharacteristic, complemented
119#380—21no—
120#342C22×C712yes—
121#323—21no—
122#323C22×C712yes—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7C2×C22⋊C411yescharacteristic
133C23×C7AGL(1,ℤ/8)11yescharacteristic
134C23×C7#13711yescharacteristic
135C7×Q8—82no—
136C7×D4—42no—
137C7×D4—42no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
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—81no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7—82no—
165C2×C4×C7—82no—
166C2×C4×C7—82no—
167C2×C4×C7#9911yescharacteristic
168C2×C4×C7#9911yescharacteristic
169C7×C8—82no—
170C22×Q8C2×C4×C711yescharacteristic
171C22×D4C7×D411yescharacteristic
172C23×C4C2×C4×C711yescharacteristic
173C23×C4C7×D411yescharacteristic
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C4×D4—41no—
177C4×D4—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C2×C4×C711yescharacteristic
183C2×C4⋊C4C2×C4×C711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4C7×D411yescharacteristic
190C2×C22⋊C4C23×C711yescharacteristic
191C2×C42C7×D411yescharacteristic
192C4⋊C8—41no—
193#99—42no—
194C22⋊C8—22no—
195#97—41no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—41no—
202#95C2×C4×C711yescharacteristic
203#95C2×C4×C711yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7C2×#9911yescharacteristic
219C22×C7#41711yescharacteristic
220C22×C7#42111yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7—81no—
231C4×C7—162no—
232C24C7×C22⋊C411yescharacteristic
233C24C2×C7×D411yescharacteristic
234C2×Q8—41no—
235C2×Q8—81no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C7×C22⋊C411yescharacteristic
252C22×C4C7×C22⋊C411yescharacteristic
253C22×C4C7×C22⋊C411yescharacteristic
254C22×C4C22×C4×C711yescharacteristic
255C22×C4C2×C7×D411yescharacteristic
256C2×C8—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—82no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—21no—
274C42—21no—
275C2×C7—21no—
276C2×C7—21no—
277C2×C7—41no—
278C2×C7—41no—
279C2×C7—81no—
280C2×C7#151311yescharacteristic
281C2×C7#212111yescharacteristic
282C2×C7#212911yescharacteristic
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C7×C22⋊C411yescharacteristic
293C23C7×AGL(1,ℤ/8)11yescharacteristic
294C23C7×#13711yescharacteristic
295Q8—82no—
296D4—42no—
297D4—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4C7×#9911yescharacteristic
328C2×C4C7×#9911yescharacteristic
329C8—82no—
330C7#1182611yescharacteristic, direct product factor, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×C7×#9911yescharacteristic
346C22C7×#41711yescharacteristic
347C22C7×#42111yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—162no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2C7×#151311yescharacteristic
365C2C7×#212111yescharacteristic
366C2C7×#212911yescharacteristic
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×C7×C22⋊Q8C4semidirect
10. C7×#398C4semidirect
13. C7×#395C4semidirect
18. C7×#381C4semidirect
23. #11826C7direct (N × H)
61. #1351C2×C7semidirect
62. #1344C2×C7semidirect
106. C2×C22⋊Q8C4×C7semidirect
110. #398C4×C7semidirect
113. #395C4×C7semidirect
118. #381C4×C7semidirect
330. C7#11826direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26382
C7×#17992
C7×#14662
C7×#13512
C7×#13442
#118267