Finite Group Databasebeta

← C7×#11810 #13189026

Subgroups — C7×#11810 #13189026

Order 1792.

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

normal subgroups only

This group has 402 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×#1820C211yesmaximal, characteristic
3C7×#1464C211yesmaximal, characteristic
4C7×#1352C212yesmaximal
5C7×#1351C212yesmaximal
6C2×C4×C7×Q8C411yescharacteristic, complemented
7C2×C7×#117C2211yescharacteristic
8C7×#432—21no—
9C7×#400C411yescharacteristic, complemented
10C7×#399C2211yescharacteristic
11C7×#398—21no—
12C7×#396—21no—
13C7×#395C411yescharacteristic, complemented
14C7×#393C2211yescharacteristic
15C7×#381—21no—
16C4×C7×C4⋊C4—21no—
17C4×C7×C4⋊C4C411yescharacteristic, complemented
18C4×C7×C22⋊C4—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11810C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C4×C7×Q8—41no—
27C4×C7×Q8—41no—
28C7×#117—41no—
29C7×#117—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4C2311yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7—21no—
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×C22⋊C8—22no—
48C7×#97—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—41no—
54C7×#95—41no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2396C2×C711yescharacteristic
58#1820C2×C711yescharacteristic
59#1464C2×C711yescharacteristic
60#1352C2×C712yes—
61#1351C2×C712yes—
62C24×C7C2×D411yescharacteristic
63C2×C7×Q8—42no—
64C2×C7×Q8—41no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7—42no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C2×C7×C8—42no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—42no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
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, complemented
102C2×#117C22×C711yescharacteristic
103#432—21no—
104#400C4×C711yescharacteristic, complemented
105#399C22×C711yescharacteristic
106#398—21no—
107#396—21no—
108#395C4×C711yescharacteristic, complemented
109#393C22×C711yescharacteristic
110#381—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C4×C711yescharacteristic, complemented
113C4×C22⋊C4—21no—
114#342—21no—
115#342C22×C712yes—
116#324—21no—
117#323C22×C712yes—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7C2×C22⋊C411yescharacteristic
123C23×C7#13712yes—
124C7×Q8—42no—
125C7×Q8—42no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
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—21no—
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—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—82no—
154C2×C4×C7—82no—
155C2×C4×C7—82no—
156C2×C4×C7#9911yescharacteristic
157C2×C4×C7#9911yescharacteristic
158C7×C8—82no—
159C22×Q8C2×C4×C711yescharacteristic
160C23×C4C7×D411yescharacteristic
161C4×Q8—41no—
162C4×Q8—41no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×C4×C711yescharacteristic
170C2×C4⋊C4C2×C4×C711yescharacteristic
171C2×C4⋊C4C7×D411yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C7×D411yescharacteristic
176C2×C22⋊C4C23×C711yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×C4×C711yescharacteristic
180C2×C42C7×D411yescharacteristic
181C4⋊C8—41no—
182C22⋊C8—22no—
183#97—41no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—41no—
189#95—41no—
190#95C2×C4×C711yescharacteristic
191#95C2×C4×C711yescharacteristic
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—42no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—81no—
199C22×C7C2×#9911yescharacteristic
200C22×C7#41911yescharacteristic
201C22×C7#42111yescharacteristic
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—42no—
205C4×C7—41no—
206C4×C7—42no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—81no—
212C4×C7—81no—
213C4×C7—162no—
214C24C2×C7×D411yescharacteristic
215C2×Q8—42no—
216C2×Q8—41no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4C7×C22⋊C411yescharacteristic
227C22×C4C7×C22⋊C411yescharacteristic
228C22×C4C7×C22⋊C411yescharacteristic
229C22×C4C7×C22⋊C411yescharacteristic
230C22×C4C22×C4×C711yescharacteristic
231C22×C4C2×C7×D411yescharacteristic
232C2×C8—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C42—41no—
252C42—41no—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7—42no—
256C2×C7#151111yescharacteristic
257C2×C7#212011yescharacteristic
258C2×C7#212611yescharacteristic
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23C2×C7×C22⋊C411yescharacteristic
264C23C7×#13712yes—
265Q8—42no—
266Q8—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
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×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4C7×#9911yescharacteristic
298C2×C4C7×#9911yescharacteristic
299C8—82no—
300C7#1181011yescharacteristic, direct product factor, Sylow
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×C7×#9911yescharacteristic
309C22C7×#41911yescharacteristic
310C22C7×#42111yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—162no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2C7×#151111yescharacteristic
327C2C7×#212011yescharacteristic
328C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC24×C7C2×D4
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
6. C2×C4×C7×Q8C4semidirect
9. C7×#400C4semidirect
13. C7×#395C4semidirect
17. C4×C7×C4⋊C4C4semidirect
23. #11810C7direct (N × H)
101. C2×C4×Q8C4×C7semidirect
104. #400C4×C7semidirect
108. #395C4×C7semidirect
112. C4×C4⋊C4C4×C7semidirect
300. C7#11810direct (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×#18202
C7×#14642
C7×#13522
C7×#13512
#118107