Finite Group Databasebeta

← C7×#11854 #13189070

Subgroups — C7×#11854 #13189070

Order 1792.

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

normal subgroups only

This group has 412 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×#2583C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3C2×C7×#324C211yesmaximal, characteristic
4C7×#1354C212yesmaximal, complemented
5C7×#1346C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#398C411yescharacteristic
11C7×#398C2211yescharacteristic
12C7×#395C411yescharacteristic
13C7×#395C2211yescharacteristic
14C7×#389—21no—
15C7×#389—21no—
16C7×#385—21no—
17C7×#385—21no—
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11854C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—22no—
29C22×C7×C8—41no—
30C7×C22⋊Q8—41no—
31C7×C22⋊Q8—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
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×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#2583C2×C711yescharacteristic
58#1821C2×C711yescharacteristic
59C2×#324C2×C711yescharacteristic
60#1354C2×C712yescomplemented
61#1346C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—41no—
64C2×C7×Q8—81no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—22no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—41no—
77C22×C4×C7—81no—
78C22×C4×C7—81no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C7×M16—42no—
86C7×M16—42no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C2×C7×C8—82no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—81no—
99C7×C4⋊C4—81no—
100C7×C22⋊C4—24no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C42×C7—81no—
104C2×C22⋊Q8C4×C711yescharacteristic
105C22×C4⋊C4C22×C711yescharacteristic
106#407—21no—
107C2×C22⋊C8—21no—
108#398C4×C711yescharacteristic
109#398C22×C711yescharacteristic
110#395C4×C711yescharacteristic
111#395C22×C711yescharacteristic
112#389—21no—
113#389—21no—
114#385—21no—
115#385—21no—
116C4×C22⋊C4C4×C711yescharacteristic
117#343—21no—
118#343C22×C712yes—
119#324—21no—
120#324C22×C712yes—
121C23×C7—22no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
125C23×C7#9911yescharacteristic
126C23×C7#9911yescharacteristic
127C23×C7C2×C22⋊C411yescharacteristic
128C7×Q8—82no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
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—41no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
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—81no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7#13512yes—
161C7×C8—82no—
162C7×C8—82no—
163C22×Q8C2×C4×C711yescharacteristic
164C23×C4C2×C4×C711yescharacteristic
165C23×C4C7×D411yescharacteristic
166C23×C4C7×D411yescharacteristic
167C2×M16—22no—
168C22×C8—41no—
169C22⋊Q8—41no—
170C22⋊Q8—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C2×C4×C711yescharacteristic
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C4⋊C4C7×D411yescharacteristic
181C2×C4⋊C4C7×D411yescharacteristic
182C2×C4⋊C4C23×C711yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42—21no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95C2×C4×C711yescharacteristic
195#95C2×C4×C711yescharacteristic
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—42no—
204C22×C7—42no—
205C22×C7#35612yes—
206C22×C7C2×#9911yescharacteristic
207C22×C7#41611yescharacteristic
208C22×C7#42111yescharacteristic
209C4×C7—22no—
210C4×C7—42no—
211C4×C7—42no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—81no—
218C4×C7—81no—
219C24C7×C22⋊C411yescharacteristic
220C2×Q8—41no—
221C2×Q8—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4C7×C22⋊C411yescharacteristic
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C22×C4×C711yescharacteristic
240C22×C4C2×C7×D411yescharacteristic
241C22×C4C2×C7×D411yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—82no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—81no—
261C2×C7—21no—
262C2×C7—21no—
263C2×C7—22no—
264C2×C7—41no—
265C2×C7#151611yescharacteristic
266C2×C7#212111yescharacteristic
267C2×C7C2×#35611yescharacteristic
268C23—22no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23C7×#9911yescharacteristic
273C23C7×#9911yescharacteristic
274C23C2×C7×C22⋊C411yescharacteristic
275Q8—82no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4C7×#13512yes—
308C8—82no—
309C8—82no—
310C7#1185411yescharacteristic, direct product factor, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22C7×#35612yes—
321C22C2×C7×#9911yescharacteristic
322C22C7×#41611yescharacteristic
323C22C7×#42111yescharacteristic
324C4—22no—
325C4—42no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2C7×#151611yescharacteristic
339C2C7×#212111yescharacteristic
340C2C2×C7×#35611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC24×C7C22⋊C4
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
4. C7×#1354C2semidirect
5. C7×#1346C2semidirect
23. #11854C7direct (N × H)
60. #1354C2×C7semidirect
61. #1346C2×C7semidirect
310. C7#11854direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25832
C7×#18212
C2×C7×#3242
C7×#13542
C7×#13462
#118547