Finite Group Databasebeta

← C7×#11852 #13189068

Subgroups — C7×#11852 #13189068

Order 1792.

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

normal subgroups only

This group has 438 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×#2591C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic, complemented
3C2×C7×#324C211yesmaximal, characteristic, complemented
4C7×#1353C212yesmaximal, complemented
5C7×#1345C212yesmaximal, complemented
6C2×C7×#121C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#400C411yescharacteristic
11C7×#400C2211yescharacteristic
12C7×#394C411yescharacteristic
13C7×#394C2211yescharacteristic
14C7×#388—21no—
15C7×#388—21no—
16C7×#386—21no—
17C7×#386—21no—
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11852C711yesmaximal, 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—
30C22×C7×C8—41no—
31C7×#121—41no—
32C7×#121—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C42×C7—21no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×C22⋊C8—41no—
50C7×C22⋊C8—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2591C2×C711yescharacteristic
58#1821C2×C711yescharacteristic, complemented
59C2×#324C2×C711yescharacteristic, complemented
60#1353C2×C712yescomplemented
61#1345C2×C712yescomplemented
62C24×C7—21no—
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—41no—
65C2×C7×D4—42no—
66C2×C7×D4—42no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4—81no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—41no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C7×M16—42no—
90C7×M16—42no—
91C2×C7×C8—42no—
92C2×C7×C8—42no—
93C2×C7×C8—82no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—81no—
104C7×C22⋊C4—81no—
105C42×C7—81no—
106C2×#121C4×C711yescharacteristic
107C22×C4⋊C4C22×C711yescharacteristic
108#407—21no—
109C2×C22⋊C8—21no—
110#400C4×C711yescharacteristic
111#400C22×C711yescharacteristic
112#394C4×C711yescharacteristic
113#394C22×C711yescharacteristic
114#388—21no—
115#388—21no—
116#386—21no—
117#386—21no—
118C4×C22⋊C4C4×C711yescharacteristic
119#343—21no—
120#343C22×C712yes—
121#324—21no—
122#324C22×C712yes—
123C23×C7—22no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7#9911yescharacteristic
135C23×C7#9911yescharacteristic
136C23×C7C2×C22⋊C411yescharacteristic
137C7×D4—82no—
138C7×D4—82no—
139C7×D4—82no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
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—21no—
149C2×C4×C7—42no—
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—41no—
165C2×C4×C7—81no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7#13512yes—
169C7×C8—82no—
170C7×C8—82no—
171C22×D4—21no—
172C22×D4C2×C4×C711yescharacteristic
173C23×C4C2×C4×C711yescharacteristic
174C23×C4C7×D411yescharacteristic
175C23×C4C7×D411yescharacteristic
176C2×M16—22no—
177C22×C8—41no—
178#121—41no—
179#121—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C2×C4×C711yescharacteristic
183C2×C4⋊C4C2×C4×C711yescharacteristic
184C2×C4⋊C4C7×D411yescharacteristic
185C2×C4⋊C4C7×D411yescharacteristic
186C2×C4⋊C4C23×C711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2×C4×C711yescharacteristic
203#95C2×C4×C711yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7#35512yes—
221C22×C7C2×#9911yescharacteristic
222C22×C7#41611yescharacteristic
223C22×C7#42111yescharacteristic
224C4×C7—22no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—81no—
233C24—21no—
234C24C7×C22⋊C411yescharacteristic
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—81no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C7×C22⋊C411yescharacteristic
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C22×C4×C711yescharacteristic
258C22×C4C2×C7×D411yescharacteristic
259C22×C4C2×C7×D411yescharacteristic
260M16—42no—
261M16—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—82no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—81no—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—22no—
280C2×C7—41no—
281C2×C7—161no—
282C2×C7#151611yescharacteristic
283C2×C7#212111yescharacteristic
284C2×C7C2×#35511yescharacteristic
285C23—22no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C7×#9911yescharacteristic
297C23C7×#9911yescharacteristic
298C23C2×C7×C22⋊C411yescharacteristic
299D4—82no—
300D4—82no—
301D4—82no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
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—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4C7×#13512yes—
331C8—82no—
332C8—82no—
333C7#1185211yescharacteristic, direct product factor, Sylow
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C7×#35512yes—
351C22C2×C7×#9911yescharacteristic
352C22C7×#41611yescharacteristic
353C22C7×#42111yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—161no—
368C2C7×#151611yescharacteristic
369C2C7×#212111yescharacteristic
370C2C2×C7×#35511yescharacteristic
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. C2×C7×#324C2semidirect
4. C7×#1353C2semidirect
5. C7×#1345C2semidirect
23. #11852C7direct (N × H)
58. #1821C2×C7semidirect
59. C2×#324C2×C7semidirect
60. #1353C2×C7semidirect
61. #1345C2×C7semidirect
333. C7#11852direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25912
C7×#18212
C2×C7×#3242
C7×#13532
C7×#13452
#118527