Finite Group Databasebeta

← C7×#10327 #13187543

Subgroups — C7×#10327 #13187543

Order 1792.

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

normal subgroups only

This group has 806 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
1C7×#3192C211yesmaximal, characteristic
2C7×#2436C212yesmaximal, complemented
3C7×#1784C211yesmaximal, characteristic, complemented
4C7×#1507C212yesmaximal, complemented
5C7×#1297C211yesmaximal, characteristic, complemented
6C23×C7×Q8C2211yescharacteristic
7C2×C7×#124—22no—
8C2×C7×#121C2212yes—
9C7×#481—22no—
10C7×#469—22no—
11C7×#447—22no—
12C2×C7×#101—22no—
13C7×#407C2211yescharacteristic
14C7×#407C2211yescharacteristic
15C7×#393—24no—
16C7×#391—22no—
17C7×#386—24no—
18C4×C7×C22⋊C4C2212yes—
19C7×#327—24no—
20#10327C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—22no—
22C22×C7×Q8—21no—
23C22×C7×Q8—22no—
24C22×C7×Q8—21no—
25C22×C7×Q8—44no—
26C22×C7×D4—22no—
27C22×C7×D4D412yes—
28C23×C4×C7D412yes—
29C23×C4×C7C2311yescharacteristic
30C2×C7×D8—22no—
31C2×C7×M16—21no—
32C2×C7×M16D412yes—
33C7×#124—48no—
34C7×#121—22no—
35C7×#121—22no—
36C7×#121—22no—
37C7×#121—22no—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C22⋊C4—28no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C42×C7—22no—
43C7×#106—22no—
44C7×#102—24no—
45C7×#101—44no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#95—28no—
49C7×#95D414yes—
50#3192C2×C711yescharacteristic
51#2436C2×C712yescomplemented
52#1784C2×C711yescharacteristic, complemented
53#1507C2×C712yescomplemented
54#1297C2×C711yescharacteristic, complemented
55C24×C7—22no—
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—24no—
58C2×C7×Q8—28no—
59C2×C7×Q8—44no—
60C2×C7×Q8—42no—
61C2×C7×Q8—48no—
62C2×C7×Q8—44no—
63C2×C7×Q8—41no—
64C2×C7×Q8—42no—
65C2×C7×Q8—41no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—44no—
69C2×C7×D4—48no—
70C2×C7×D4—42no—
71C2×C7×D4—42no—
72C22×C4×C7—24no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—24no—
76C22×C4×C7—44no—
77C22×C4×C7—42no—
78C22×C4×C7C2×D412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D412yes—
81C22×C4×C7#4111yescharacteristic
82C7×D8—82no—
83C7×M16—42no—
84C7×M16—42no—
85C2×C7×C8—22no—
86C2×C7×C8—41no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—22no—
89C7×C22⋊C4—416no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C42×C7—44no—
94C23×Q8C22×C711yescharacteristic
95C2×#124—22no—
96C2×#121C22×C712yes—
97#481—22no—
98#469—22no—
99#447—22no—
100C2×#101—22no—
101#407C22×C711yescharacteristic
102#407C22×C711yescharacteristic
103#393—24no—
104#391—22no—
105#386—24no—
106C4×C22⋊C4C22×C712yes—
107#327—24no—
108C23×C7—24no—
109C23×C7—41no—
110C23×C7—44no—
111C23×C7—44no—
112C23×C7—42no—
113C23×C7—44no—
114C23×C7#12012yes—
115C23×C7#12111yescharacteristic
116C7×Q8—44no—
117C7×Q8—48no—
118C7×Q8—44no—
119C7×D4—44no—
120C7×D4—82no—
121C7×D4—84no—
122C7×D4—84no—
123C2×C4×C7—24no—
124C2×C4×C7—22no—
125C2×C4×C7—28no—
126C2×C4×C7—24no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—44no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—44no—
133C2×C4×C7—44no—
134C2×C4×C7—44no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7#12011yescharacteristic
138C2×C4×C7#12112yes—
139C2×C4×C7#12711yescharacteristic
140C7×C8—42no—
141C7×C8—81no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—44no—
147C22×D4—22no—
148C22×D4C7×D412yes—
149C23×C4C7×D412yes—
150C23×C4C23×C711yescharacteristic
151C2×D8—22no—
152C2×M16—21no—
153C2×M16C7×D412yes—
154#124—48no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121—22no—
159C2×C4⋊C4C7×D412yes—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—22no—
164#106—22no—
165#102—24no—
166#101—44no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169#95—28no—
170#95C7×D414yes—
171C22×C7—22no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—21no—
175C22×C7—44no—
176C22×C7—88no—
177C22×C7—82no—
178C22×C7—82no—
179C22×C7—82no—
180C22×C7#39211yescharacteristic
181C22×C7#45414yes—
182C4×C7—21no—
183C4×C7—22no—
184C4×C7—44no—
185C4×C7—84no—
186C4×C7—82no—
187C4×C7B(4,𝔽2)12yes—
188C24—22no—
189C24C2×C7×D411yescharacteristic
190C2×Q8—24no—
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—42no—
194C2×Q8—48no—
195C2×Q8—44no—
196C2×Q8—41no—
197C2×Q8—42no—
198C2×Q8—41no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—44no—
202C2×D4—48no—
203C2×D4—42no—
204C2×D4—42no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4C2×C7×D412yes—
212C22×C4C2×C7×D411yescharacteristic
213C22×C4C2×C7×D412yes—
214C22×C4C7×#4111yescharacteristic
215D8—82no—
216M16—42no—
217M16—42no—
218C2×C8—22no—
219C2×C8—41no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C22⋊C4—416no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—44no—
227C2×C7—24no—
228C2×C7—41no—
229C2×C7—162no—
230C2×C7#200312yes—
231C2×C7#202111yescharacteristic
232C23—24no—
233C23—41no—
234C23—44no—
235C23—44no—
236C23—42no—
237C23—44no—
238C23C7×#12012yes—
239C23C7×#12111yescharacteristic
240Q8—44no—
241Q8—48no—
242Q8—44no—
243D4—44no—
244D4—82no—
245D4—84no—
246D4—84no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C7×#12011yescharacteristic
262C2×C4C7×#12112yes—
263C2×C4C7×#12711yescharacteristic
264C8—42no—
265C8—81no—
266C7#1032711yescharacteristic, direct product factor, Sylow
267C22—22no—
268C22—22no—
269C22—22no—
270C22—21no—
271C22—44no—
272C22—88no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22C7×#39211yescharacteristic
277C22C7×#45414yes—
278C4—21no—
279C4—22no—
280C4—44no—
281C4—84no—
282C4—82no—
283C4C7×B(4,𝔽2)12yes—
284C2—24no—
285C2—41no—
286C2—162no—
287C2C7×#200312yes—
288C2C7×#202111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3192C2
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
2. C7×#2436C2semidirect
3. C7×#1784C2semidirect
4. C7×#1507C2semidirect
5. C7×#1297C2semidirect
20. #10327C7direct (N × H)
51. #2436C2×C7semidirect
52. #1784C2×C7semidirect
53. #1507C2×C7semidirect
54. #1297C2×C7semidirect
266. C7#10327direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31922
C7×#24362
C7×#17842
C7×#15072
C7×#12972
#103277