Finite Group Databasebeta

← C7×#11815 #13189031

Subgroups — C7×#11815 #13189031

Order 1792.

586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×C7×#395C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3C7×#1463C211yesmaximal, characteristic
4C7×#1354C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic
6C22×C7×C4⋊C4—21no—
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C7×#407—21no—
10C7×#395—21no—
11C7×#395—21no—
12C7×#395C412yes—
13C7×#395C2212yes—
14C2×C7×#95—21no—
15C2×C7×#95C411yescharacteristic
16C7×#343—21no—
17C7×#343C2212yes—
18C7×#324—21no—
19C7×#324C2212yes—
20#11815C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—44no—
23C22×C7×Q8—41no—
24C22×C7×Q8—41no—
25C22×C7×Q8C2×C412yes—
26C23×C4×C7—21no—
27C23×C4×C7—21no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—22no—
32C2×C7×M16—41no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—41no—
36C2×C7×C4⋊C4—41no—
37C2×C7×C4⋊C4—44no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4C2311yescharacteristic
42C7×C22⋊C8—22no—
43C7×C22⋊C8—41no—
44C7×C22⋊C8—41no—
45C7×C22⋊C8—41no—
46C7×#95—22no—
47C7×#95—21no—
48C7×#95—22no—
49C7×#95C2×C412yes—
50C2×#395C2×C711yescharacteristic
51#1821C2×C711yescharacteristic
52#1463C2×C711yescharacteristic
53#1354C2×C714yescomplemented
54C24×C7C22⋊C411yescharacteristic
55C2×C7×Q8—44no—
56C2×C7×Q8—42no—
57C2×C7×Q8—44no—
58C2×C7×Q8—42no—
59C2×C7×Q8—48no—
60C2×C7×Q8—44no—
61C2×C7×Q8—44no—
62C22×C4×C7—24no—
63C22×C4×C7—24no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—22no—
73C22×C4×C7—41no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7—42no—
80C22×C4×C7—42no—
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22×C411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C7×M16—42no—
88C7×M16—42no—
89C7×M16—82no—
90C2×C7×C8—42no—
91C2×C7×C8—42no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—44no—
97C7×C4⋊C4—44no—
98C23×Q8C4×C711yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4C22×C711yescharacteristic
101#407—21no—
102#407—21no—
103#395—21no—
104#395—21no—
105#395C4×C712yes—
106#395C22×C712yes—
107C2×#95—21no—
108C2×#95C4×C711yescharacteristic
109#343—21no—
110#343C22×C712yes—
111#324—21no—
112#324C22×C712yes—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7#9911yescharacteristic
118C23×C7#9911yescharacteristic
119C23×C7C2×C22⋊C411yescharacteristic
120C7×Q8—44no—
121C7×Q8—48no—
122C7×Q8—44no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—24no—
126C2×C4×C7—24no—
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—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—48no—
138C2×C4×C7—42no—
139C2×C4×C7—44no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—81no—
151C2×C4×C7#13712yes—
152C7×C8—82no—
153C7×C8—82no—
154C22×Q8—21no—
155C22×Q8—44no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8C2×C4×C712yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×C4×C711yescharacteristic
162C23×C4C7×D411yescharacteristic
163C23×C4C7×D411yescharacteristic
164C2×M16—22no—
165C2×M16—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C2×C4×C711yescharacteristic
172C2×C4⋊C4C7×D411yescharacteristic
173C2×C4⋊C4C7×D411yescharacteristic
174C2×C4⋊C4C23×C711yescharacteristic
175C22⋊C8—22no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#95—22no—
180#95—21no—
181#95—22no—
182#95C2×C4×C712yes—
183C22×C7—24no—
184C22×C7—24no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7AΣL(1,𝔽16)12yes—
192C22×C7#35612yes—
193C22×C7C2×#9911yescharacteristic
194C22×C7#41911yescharacteristic
195C22×C7#42111yescharacteristic
196C4×C7—22no—
197C4×C7—44no—
198C4×C7—41no—
199C4×C7—44no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C24C7×C22⋊C411yescharacteristic
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—48no—
210C2×Q8—44no—
211C2×Q8—44no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C7×C22⋊C411yescharacteristic
232C22×C4C7×C22⋊C411yescharacteristic
233C22×C4C7×C22⋊C411yescharacteristic
234C22×C4C22×C4×C711yescharacteristic
235C22×C4C2×C7×D411yescharacteristic
236C22×C4C2×C7×D411yescharacteristic
237M16—42no—
238M16—42no—
239M16—82no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C2×C7—21no—
249C2×C7—22no—
250C2×C7—22no—
251C2×C7—21no—
252C2×C7#151111yescharacteristic
253C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
254C2×C7C2×#35611yescharacteristic
255C23—22no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23C7×#9911yescharacteristic
260C23C7×#9911yescharacteristic
261C23C2×C7×C22⋊C411yescharacteristic
262Q8—44no—
263Q8—44no—
264Q8—48no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—48no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4C7×#13712yes—
294C8—82no—
295C8—82no—
296C7#1181511yescharacteristic, direct product factor, Sylow
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C7×AΣL(1,𝔽16)12yes—
306C22C7×#35612yes—
307C22C2×C7×#9911yescharacteristic
308C22C7×#41911yescharacteristic
309C22C7×#42111yescharacteristic
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—44no—
315C4—41no—
316C4—41no—
317C4—41no—
318C2—21no—
319C2—22no—
320C2—22no—
321C2—21no—
322C2C7×#151111yescharacteristic
323C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
324C2C2×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
20. #11815C7direct (N × H)
53. #1354C2×C7semidirect
296. C7#11815direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3952
C7×#18212
C7×#14632
C7×#13542
#118157