Finite Group Databasebeta

← C7×#10392 #13187608

Subgroups — C7×#10392 #13187608

Order 1792.

746 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2842C211yesmaximal, characteristic, complemented
2C7×#1781C211yesmaximal, characteristic
3C7×#1781C212yesmaximal
4C7×#1304C211yesmaximal, characteristic, complemented
5C7×#1304C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#99—24no—
8C2×C7×#99—22no—
9C7×#399—22no—
10C7×#396—24no—
11C7×#379C2212yes—
12C7×#379C2211yescharacteristic
13C7×#379C2211yescharacteristic
14C7×#379C2212yes—
15#10392C711yesmaximal, characteristic, direct product factor, Sylow
16C25×C7Q811yescharacteristic
17C25×C7C2311yescharacteristic
18C22×C7×D4—22no—
19C22×C7×D4—21no—
20C22×C7×D4—22no—
21C22×C7×D4—21no—
22C22×C7×D4—44no—
23C23×C4×C7Q811yescharacteristic
24C2×C7×C4⋊C4—24no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C22⋊C4—22no—
27C2×C7×C22⋊C4—21no—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4D414yes—
35C2×C7×C22⋊C4D412yes—
36C7×#99—48no—
37C7×#99—44no—
38C7×#95—22no—
39#2842C2×C711yescharacteristic, complemented
40#1781C2×C711yescharacteristic
41#1781C2×C712yes—
42#1304C2×C711yescharacteristic, complemented
43#1304C2×C712yescomplemented
44C24×C7—22no—
45C24×C7—21no—
46C24×C7—42no—
47C24×C7—42no—
48C24×C7—82no—
49C24×C7—82no—
50C24×C7C2×D412yes—
51C24×C7C2×D411yescharacteristic
52C24×C7C2×Q811yescharacteristic
53C24×C7#4112yes—
54C24×C7#4111yescharacteristic
55C2×C7×D4—28no—
56C2×C7×D4—24no—
57C2×C7×D4—44no—
58C2×C7×D4—42no—
59C2×C7×D4—44no—
60C2×C7×D4—42no—
61C2×C7×D4—44no—
62C2×C7×D4—42no—
63C2×C7×D4—41no—
64C2×C7×D4—42no—
65C2×C7×D4—41no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—24no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—41no—
73C7×C4⋊C4—84no—
74C7×C4⋊C4—82no—
75C7×C22⋊C4—44no—
76C7×C22⋊C4—42no—
77C7×C22⋊C4—44no—
78C7×C22⋊C4—42no—
79C7×C22⋊C4—82no—
80C7×C22⋊C4—82no—
81C7×C22⋊C4—82no—
82C7×C22⋊C4—82no—
83C7×C22⋊C4—82no—
84C7×C22⋊C4—82no—
85C23×D4C22×C711yescharacteristic
86C2×#99—24no—
87C2×#99—22no—
88#399—22no—
89#396—24no—
90#379C22×C712yes—
91#379C22×C711yescharacteristic
92#379C22×C711yescharacteristic
93#379C22×C712yes—
94C23×C7—24no—
95C23×C7—22no—
96C23×C7—24no—
97C23×C7—22no—
98C23×C7—41no—
99C23×C7—42no—
100C23×C7—42no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—41no—
108C23×C7—42no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7—82no—
112C23×C7—82no—
113C23×C7—82no—
114C23×C7—88no—
115C23×C7—88no—
116C23×C7#12011yescharacteristic
117C23×C7C22⋊Q812yes—
118C23×C7C22⋊Q811yescharacteristic
119C23×C7#12412yes—
120C23×C7#12411yescharacteristic
121C7×D4—48no—
122C7×D4—44no—
123C7×D4—44no—
124C2×C4×C7—24no—
125C2×C4×C7—22no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—44no—
130C2×C4×C7—44no—
131C2×C4×C7—44no—
132C2×C4×C7—44no—
133C2×C4×C7—42no—
134C2×C4×C7—82no—
135C2×C4×C7—81no—
136C2×C4×C7—82no—
137C2×C4×C7—82no—
138C2×C4×C7—82no—
139C25C7×Q811yescharacteristic
140C25C23×C711yescharacteristic
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—44no—
146C23×C4C7×Q811yescharacteristic
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C7×D414yes—
158C2×C22⋊C4C7×D412yes—
159#99—48no—
160#99—44no—
161#95—22no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—21no—
165C22×C7—22no—
166C22×C7—21no—
167C22×C7—22no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—22no—
172C22×C7—22no—
173C22×C7—44no—
174C22×C7—42no—
175C22×C7—84no—
176C22×C7—82no—
177C22×C7—82no—
178C22×C7—84no—
179C22×C7—84no—
180C22×C7—82no—
181C22×C7—84no—
182C22×C7—82no—
183C22×C7—82no—
184C22×C7#39311yescharacteristic
185C22×C7B(4,𝔽2)12yes—
186C4×C7—42no—
187C4×C7—82no—
188C4×C7—84no—
189C4×C7—162no—
190C4×C7—161no—
191C24—22no—
192C24—21no—
193C24—42no—
194C24—42no—
195C24—82no—
196C24—82no—
197C24C2×C7×D412yes—
198C24C2×C7×D411yescharacteristic
199C24C2×C7×Q811yescharacteristic
200C24C7×#4112yes—
201C24C7×#4111yescharacteristic
202C2×D4—28no—
203C2×D4—24no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—42no—
212C2×D4—41no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C2×C7—22no—
233C2×C7—42no—
234C2×C7—41no—
235C2×C7—42no—
236C2×C7—41no—
237C2×C7—82no—
238C2×C7#202911yescharacteristic
239C2×C7#203212yes—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—22no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23—88no—
261C23—88no—
262C23C7×#12011yescharacteristic
263C23C7×C22⋊Q812yes—
264C23C7×C22⋊Q811yescharacteristic
265C23C7×#12412yes—
266C23C7×#12411yescharacteristic
267D4—48no—
268D4—44no—
269D4—44no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C7#1039211yescharacteristic, direct product factor, Sylow
286C22—22no—
287C22—22no—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—44no—
298C22—42no—
299C22—84no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22—82no—
305C22—84no—
306C22—82no—
307C22—82no—
308C22C7×#39311yescharacteristic
309C22C7×B(4,𝔽2)12yes—
310C4—42no—
311C4—82no—
312C4—84no—
313C4—162no—
314C4—161no—
315C2—22no—
316C2—42no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2—82no—
321C2C7×#202911yescharacteristic
322C2C7×#203212yes—
subgroupG/H
center, socleC22×C7#393
commutator, Frattini, cosocleC25C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2842C2semidirect
4. C7×#1304C2semidirect
5. C7×#1304C2semidirect
15. #10392C7direct (N × H)
39. #2842C2×C7semidirect
42. #1304C2×C7semidirect
43. #1304C2×C7semidirect
285. C7#10392direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28422
C7×#17812
C7×#13042
#103927