Finite Group Databasebeta

← C7×#11816 #13189032

Subgroups — C7×#11816 #13189032

Order 1792.

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

normal subgroups only

This group has 388 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×#2396C211yesmaximal, characteristic
2C7×#1822C211yesmaximal, characteristic
3C7×#1464C211yesmaximal, characteristic
4C7×#1354C212yesmaximal, complemented
5C7×#1353C212yesmaximal, complemented
6C2×C4×C7×Q8C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#432—21no—
9C7×#407—21no—
10C7×#400C2211yescharacteristic
11C7×#399C411yescharacteristic
12C7×#398—21no—
13C7×#396—21no—
14C7×#395C2211yescharacteristic
15C7×#393C411yescharacteristic
16C4×C7×C4⋊C4—21no—
17C4×C7×C4⋊C4C411yescharacteristic
18C4×C7×C22⋊C4—21no—
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324C2212yes—
22C7×#323—21no—
23#11816C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C2×C7×M16—22no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4C2311yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7—21no—
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×C22⋊C8—41no—
50C7×#97—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2396C2×C711yescharacteristic
58#1822C2×C711yescharacteristic
59#1464C2×C711yescharacteristic
60#1354C2×C712yescomplemented
61#1353C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—42no—
64C2×C7×Q8—41no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C7×M16—42no—
80C7×M16—42no—
81C2×C7×C8—42no—
82C2×C7×C8—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—42no—
99C42×C7—41no—
100C42×C7—41no—
101C2×C4×Q8C4×C711yescharacteristic
102C2×#117C22×C711yescharacteristic
103#432—21no—
104#407—21no—
105#400C22×C711yescharacteristic
106#399C4×C711yescharacteristic
107#398—21no—
108#396—21no—
109#395C22×C711yescharacteristic
110#393C4×C711yescharacteristic
111C4×C4⋊C4—21no—
112C4×C4⋊C4C4×C711yescharacteristic
113C4×C22⋊C4—21no—
114#343—21no—
115#343C22×C712yes—
116#324C22×C712yes—
117#323—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—42no—
125C7×Q8—42no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
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—41no—
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—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7#9911yescharacteristic
154C2×C4×C7#13712yes—
155C7×C8—82no—
156C7×C8—82no—
157C22×Q8C2×C4×C711yescharacteristic
158C23×C4C7×D411yescharacteristic
159C2×M16—22no—
160C4×Q8—41no—
161C4×Q8—41no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C2×C4×C711yescharacteristic
169C2×C4⋊C4C7×D411yescharacteristic
170C2×C4⋊C4C23×C711yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C2×C4×C711yescharacteristic
174C2×C22⋊C4C7×D411yescharacteristic
175C2×C42—21no—
176C2×C42—21no—
177C2×C42C2×C4×C711yescharacteristic
178C2×C42C7×D411yescharacteristic
179C4⋊C8—41no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#97—41no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95C2×C4×C711yescharacteristic
189#95C2×C4×C711yescharacteristic
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—42no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—81no—
197C22×C7C2×#9911yescharacteristic
198C22×C7#41911yescharacteristic
199C22×C7#42111yescharacteristic
200C4×C7—22no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—42no—
204C4×C7—41no—
205C4×C7—42no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—81no—
210C4×C7—81no—
211C24C7×C22⋊C411yescharacteristic
212C2×Q8—42no—
213C2×Q8—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4C7×C22⋊C411yescharacteristic
223C22×C4C7×C22⋊C411yescharacteristic
224C22×C4C7×C22⋊C411yescharacteristic
225C22×C4C22×C4×C711yescharacteristic
226C22×C4C2×C7×D411yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228M16—42no—
229M16—42no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42—41no—
249C42—41no—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7—81no—
253C2×C7#151111yescharacteristic
254C2×C7#212011yescharacteristic
255C2×C7#213211yescharacteristic
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23C2×C7×C22⋊C411yescharacteristic
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4C7×#9911yescharacteristic
291C2×C4C7×#9911yescharacteristic
292C2×C4C7×#13712yes—
293C8—82no—
294C8—82no—
295C7#1181611yescharacteristic, direct product factor, Sylow
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×C7×#9911yescharacteristic
304C22C7×#41911yescharacteristic
305C22C7×#42111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2C7×#151111yescharacteristic
321C2C7×#212011yescharacteristic
322C2C7×#213211yescharacteristic
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 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C7×#1354C2semidirect
5. C7×#1353C2semidirect
23. #11816C7direct (N × H)
60. #1354C2×C7semidirect
61. #1353C2×C7semidirect
295. C7#11816direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23962
C7×#18222
C7×#14642
C7×#13542
C7×#13532
#118167