Finite Group Databasebeta

← C7×#11817 #13189033

Subgroups — C7×#11817 #13189033

Order 1792.

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

normal subgroups only

This group has 386 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×#2627C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3C7×#1463C211yesmaximal, characteristic
4C7×#1354C212yesmaximal, complemented
5C7×#1353C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C7×#407—21no—
10C7×#400C411yescharacteristic
11C7×#400C2211yescharacteristic
12C7×#398—21no—
13C7×#398—21no—
14C7×#395C411yescharacteristic
15C7×#395C2211yescharacteristic
16C7×#385—21no—
17C7×#381—21no—
18C7×#380C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11817C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—22no—
29C2×C7×M16—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—41no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4C2311yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—41no—
47C7×C22⋊C8—41no—
48C7×C22⋊C8—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2627C2×C711yescharacteristic
58#1821C2×C711yescharacteristic
59#1463C2×C711yescharacteristic
60#1354C2×C712yescomplemented
61#1353C2×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—22no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
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—
87C7×M16—82no—
88C2×C7×C8—42no—
89C2×C7×C8—42no—
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—81no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—81no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C2×C22⋊Q8C4×C711yescharacteristic
103C22×C4⋊C4C22×C711yescharacteristic
104#407—21no—
105#407—21no—
106#400C4×C711yescharacteristic
107#400C22×C711yescharacteristic
108#398—21no—
109#398—21no—
110#395C4×C711yescharacteristic
111#395C22×C711yescharacteristic
112#385—21no—
113#381—21no—
114#380C4×C711yescharacteristic
115#343—21no—
116#343C22×C712yes—
117#324—21no—
118#324C22×C712yes—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7#9911yescharacteristic
124C23×C7#9911yescharacteristic
125C23×C7C2×C22⋊C411yescharacteristic
126C7×Q8—82no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—81no—
155C2×C4×C7—81no—
156C2×C4×C7—81no—
157C2×C4×C7#13712yes—
158C7×C8—82no—
159C7×C8—82no—
160C22×Q8C2×C4×C711yescharacteristic
161C23×C4C2×C4×C711yescharacteristic
162C23×C4C7×D411yescharacteristic
163C23×C4C7×D411yescharacteristic
164C2×M16—22no—
165C2×M16—41no—
166C22⋊Q8—41no—
167C22⋊Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4C2×C4×C711yescharacteristic
175C2×C4⋊C4C2×C4×C711yescharacteristic
176C2×C4⋊C4C7×D411yescharacteristic
177C2×C4⋊C4C7×D411yescharacteristic
178C2×C4⋊C4C23×C711yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95C2×C4×C711yescharacteristic
192#95C2×C4×C711yescharacteristic
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—42no—
202C22×C7—42no—
203C22×C7C2×#9911yescharacteristic
204C22×C7#41911yescharacteristic
205C22×C7#42111yescharacteristic
206C4×C7—22no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—81no—
213C4×C7—81no—
214C4×C7—81no—
215C4×C7—81no—
216C24C7×C22⋊C411yescharacteristic
217C2×Q8—41no—
218C2×Q8—81no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4C7×C22⋊C411yescharacteristic
234C22×C4C7×C22⋊C411yescharacteristic
235C22×C4C7×C22⋊C411yescharacteristic
236C22×C4C22×C4×C711yescharacteristic
237C22×C4C2×C7×D411yescharacteristic
238C22×C4C2×C7×D411yescharacteristic
239M16—42no—
240M16—42no—
241M16—82no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C2×C7—21no—
257C2×C7—21no—
258C2×C7—41no—
259C2×C7—41no—
260C2×C7#151111yescharacteristic
261C2×C7#212111yescharacteristic
262C2×C7#213311yescharacteristic
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23C7×#9911yescharacteristic
268C23C7×#9911yescharacteristic
269C23C2×C7×C22⋊C411yescharacteristic
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C7×#13712yes—
302C8—82no—
303C8—82no—
304C7#1181711yescharacteristic, direct product factor, Sylow
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×C7×#9911yescharacteristic
316C22C7×#41911yescharacteristic
317C22C7×#42111yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2C7×#151111yescharacteristic
333C2C7×#212111yescharacteristic
334C2C7×#213311yescharacteristic
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×#1353C2semidirect
23. #11817C7direct (N × H)
60. #1354C2×C7semidirect
61. #1353C2×C7semidirect
304. C7#11817direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26272
C7×#18212
C7×#14632
C7×#13542
C7×#13532
#118177