Finite Group Databasebeta

← C7×#10388 #13187604

Subgroups — C7×#10388 #13187604

Order 1792.

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

normal subgroups only

This group has 594 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×#2608C211yesmaximal, characteristic, complemented
2C7×#1789C212yesmaximal
3C7×#1782C211yesmaximal, characteristic
4C7×#1306C211yesmaximal, characteristic, complemented
5C7×#1284C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#104—22no—
8C2×C7×#104—22no—
9C2×C7×#99—22no—
10C7×#407C2212yes—
11C7×#396—22no—
12C7×#389—22no—
13C7×#387—22no—
14C7×#381C2211yescharacteristic
15C7×#381C2211yescharacteristic
16C4×C7×C22⋊C4C2212yes—
17#10388C711yesmaximal, characteristic, direct product factor, Sylow
18C2×C7×#41—22no—
19C2×C7×#41—22no—
20C2×C7×#41—44no—
21C22×C7×Q8Q811yescharacteristic
22C22×C7×D4—21no—
23C22×C7×D4Q811yescharacteristic
24C23×C4×C7—21no—
25C23×C4×C7C2311yescharacteristic
26C2×C7×M16—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—21no—
32C2×C7×C22⋊C4D412yes—
33C2×C42×C7D412yes—
34C2×C42×C7D412yes—
35C7×#104—44no—
36C7×#104—44no—
37C7×#99—44no—
38C7×C22⋊C8—22no—
39C7×C22⋊C8—22no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44#2608C2×C711yescharacteristic, complemented
45#1789C2×C712yes—
46#1782C2×C711yescharacteristic
47#1306C2×C711yescharacteristic, complemented
48#1284C2×C712yescomplemented
49C24×C7—21no—
50C24×C7C2×D411yescharacteristic
51C7×#41—24no—
52C7×#41—24no—
53C7×#41—42no—
54C7×#41—42no—
55C7×#41—44no—
56C7×#41—44no—
57C2×C7×Q8—22no—
58C2×C7×Q8—42no—
59C2×C7×D4—24no—
60C2×C7×D4—22no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—41no—
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×Q811yescharacteristic
76C22×C4×C7#4112yes—
77C22×C4×C7#4111yescharacteristic
78C7×M16—42no—
79C7×M16—42no—
80C2×C7×C8—42no—
81C7×C4⋊C4—44no—
82C7×C4⋊C4—44no—
83C7×C4⋊C4—82no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C42×C7—42no—
88C42×C7—42no—
89C22×#41C22×C711yescharacteristic
90C2×#104—22no—
91C2×#104—22no—
92C2×#99—22no—
93#407C22×C712yes—
94#396—22no—
95#389—22no—
96#387—22no—
97#381C22×C711yescharacteristic
98#381C22×C711yescharacteristic
99C4×C22⋊C4C22×C712yes—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—42no—
103C23×C7—41no—
104C23×C7—41no—
105C23×C7—82no—
106C23×C7#12011yescharacteristic
107C23×C7C22⋊Q811yescharacteristic
108C23×C7#12411yescharacteristic
109C7×Q8—24no—
110C7×Q8—42no—
111C7×D4—24no—
112C7×D4—42no—
113C7×D4—44no—
114C7×D4—44no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—24no—
118C2×C4×C7—22no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—42no—
122C2×C4×C7—44no—
123C2×C4×C7—44no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—81no—
135C2×C4×C7—82no—
136C2×C4×C7C22⋊Q812yes—
137C2×C4×C7#12412yes—
138C7×C8—82no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—44no—
142C22×Q8C7×Q811yescharacteristic
143C22×D4—21no—
144C22×D4C7×Q811yescharacteristic
145C23×C4—21no—
146C23×C4C23×C711yescharacteristic
147C2×M16—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4C7×D412yes—
154C2×C42C7×D412yes—
155C2×C42C7×D412yes—
156#104—44no—
157#104—44no—
158#99—44no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165C22×C7—22no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—42no—
171C22×C7—82no—
172C22×C7—82no—
173C22×C7—82no—
174C22×C7—82no—
175C22×C7#39311yescharacteristic
176C22×C7B(4,𝔽2)12yes—
177C4×C7—22no—
178C4×C7—22no—
179C4×C7—44no—
180C4×C7—42no—
181C4×C7—82no—
182C4×C7—82no—
183C4×C7—161no—
184C24—21no—
185C24C2×C7×D411yescharacteristic
186#41—24no—
187#41—24no—
188#41—42no—
189#41—42no—
190#41—44no—
191#41—44no—
192C2×Q8—22no—
193C2×Q8—42no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—41no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4C2×C7×D412yes—
210C22×C4C2×C7×Q811yescharacteristic
211C22×C4C7×#4112yes—
212C22×C4C7×#4111yescharacteristic
213M16—42no—
214M16—42no—
215C2×C8—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—82no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—42no—
223C42—42no—
224C2×C7—22no—
225C2×C7—41no—
226C2×C7—41no—
227C2×C7—82no—
228C2×C7#202911yescharacteristic
229C2×C7#203112yes—
230C23—22no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—82no—
236C23C7×#12011yescharacteristic
237C23C7×C22⋊Q811yescharacteristic
238C23C7×#12411yescharacteristic
239Q8—24no—
240Q8—42no—
241D4—24no—
242D4—42no—
243D4—44no—
244D4—44no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4C7×C22⋊Q812yes—
267C2×C4C7×#12412yes—
268C8—82no—
269C7#1038811yescharacteristic, direct product factor, Sylow
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22C7×#39311yescharacteristic
281C22C7×B(4,𝔽2)12yes—
282C4—22no—
283C4—22no—
284C4—44no—
285C4—42no—
286C4—82no—
287C4—82no—
288C4—161no—
289C2—22no—
290C2—41no—
291C2—41no—
292C2—82no—
293C2C7×#202911yescharacteristic
294C2C7×#203112yes—
subgroupG/H
center, socleC22×C7#393
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2608C2semidirect
4. C7×#1306C2semidirect
5. C7×#1284C2semidirect
17. #10388C7direct (N × H)
44. #2608C2×C7semidirect
47. #1306C2×C7semidirect
48. #1284C2×C7semidirect
269. C7#10388direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26082
C7×#17892
C7×#17822
C7×#13062
C7×#12842
#103887