Finite Group Databasebeta

← C7×#10333 #13187549

Subgroups — C7×#10333 #13187549

Order 1792.

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

normal subgroups only

This group has 914 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×#3195C211yesmaximal, characteristic, complemented
2C7×#2435C212yesmaximal
3C7×#1783C211yesmaximal, characteristic
4C7×#1508C212yesmaximal, complemented
5C7×#1297C211yesmaximal, characteristic, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#124—22no—
8C2×C7×C22⋊Q8C2212yes—
9C7×#483—22no—
10C7×#470—22no—
11C7×#451—22no—
12C2×C7×#100—22no—
13C7×#407C2211yescharacteristic
14C7×#407C2211yescharacteristic
15C7×#394—24no—
16C7×#390—22no—
17C7×#379—24no—
18C4×C7×C22⋊C4C2212yes—
19C7×#328—24no—
20#10333C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7D412yes—
22C22×C7×Q8D412yes—
23C22×C7×D4—22no—
24C22×C7×D4—21no—
25C22×C7×D4—22no—
26C22×C7×D4—21no—
27C22×C7×D4—44no—
28C23×C4×C7C2311yescharacteristic
29C2×C7×Q16—22no—
30C2×C7×M16—21no—
31C2×C7×M16D412yes—
32C7×#124—48no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—22no—
35C7×C22⋊Q8—22no—
36C7×C22⋊Q8—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C22⋊C4—28no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—28no—
43C2×C42×C7—22no—
44C7×#106—22no—
45C7×#103—24no—
46C7×#100—44no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×#95D414yes—
50#3195C2×C711yescharacteristic, complemented
51#2435C2×C712yes—
52#1783C2×C711yescharacteristic
53#1508C2×C712yescomplemented
54#1297C2×C711yescharacteristic, complemented
55C24×C7—24no—
56C24×C7—22no—
57C24×C7—44no—
58C24×C7—48no—
59C24×C7C2×D411yescharacteristic
60C2×C7×Q8—22no—
61C2×C7×Q8—22no—
62C2×C7×Q8—44no—
63C2×C7×D4—24no—
64C2×C7×D4—28no—
65C2×C7×D4—44no—
66C2×C7×D4—42no—
67C2×C7×D4—48no—
68C2×C7×D4—44no—
69C2×C7×D4—41no—
70C2×C7×D4—42no—
71C2×C7×D4—41no—
72C22×C4×C7—24no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—42no—
76C22×C4×C7C2×D412yes—
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7#4111yescharacteristic
80C7×Q16—82no—
81C7×M16—42no—
82C7×M16—42no—
83C2×C7×C8—22no—
84C2×C7×C8—41no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—42no—
89C7×C22⋊C4—44no—
90C7×C22⋊C4—416no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—88no—
94C42×C7—44no—
95C23×D4C22×C711yescharacteristic
96C2×#124—22no—
97C2×C22⋊Q8C22×C712yes—
98#483—22no—
99#470—22no—
100#451—22no—
101C2×#100—22no—
102#407C22×C711yescharacteristic
103#407C22×C711yescharacteristic
104#394—24no—
105#390—22no—
106#379—24no—
107C4×C22⋊C4C22×C712yes—
108#328—24no—
109C23×C7—24no—
110C23×C7—28no—
111C23×C7—24no—
112C23×C7—41no—
113C23×C7—44no—
114C23×C7—42no—
115C23×C7—48no—
116C23×C7—42no—
117C23×C7—48no—
118C23×C7—84no—
119C23×C7—816no—
120C23×C7#12012yes—
121C23×C7#12111yescharacteristic
122C7×Q8—44no—
123C7×Q8—82no—
124C7×D4—44no—
125C7×D4—48no—
126C7×D4—44no—
127C2×C4×C7—24no—
128C2×C4×C7—22no—
129C2×C4×C7—42no—
130C2×C4×C7—41no—
131C2×C4×C7—44no—
132C2×C4×C7—44no—
133C2×C4×C7—44no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7—44no—
138C2×C4×C7#12011yescharacteristic
139C2×C4×C7#12112yes—
140C2×C4×C7#12711yescharacteristic
141C7×C8—42no—
142C7×C8—81no—
143C25C7×D412yes—
144C22×Q8C7×D412yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—44no—
150C23×C4C23×C711yescharacteristic
151C2×Q16—22no—
152C2×M16—21no—
153C2×M16C7×D412yes—
154#124—48no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C7×D412yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—28no—
165C2×C42—22no—
166#106—22no—
167#103—24no—
168#100—44no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171#95C7×D414yes—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7—21no—
176C22×C7—44no—
177C22×C7—416no—
178C22×C7—44no—
179C22×C7—88no—
180C22×C7—84no—
181C22×C7—84no—
182C22×C7—84no—
183C22×C7#39211yescharacteristic
184C22×C7#45514yes—
185C4×C7—21no—
186C4×C7—22no—
187C4×C7—84no—
188C4×C7—82no—
189C4×C7—82no—
190C4×C7B(4,𝔽2)12yes—
191C24—24no—
192C24—22no—
193C24—44no—
194C24—48no—
195C24C2×C7×D411yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×D4—24no—
200C2×D4—28no—
201C2×D4—44no—
202C2×D4—42no—
203C2×D4—48no—
204C2×D4—44no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—41no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4C2×C7×D412yes—
213C22×C4C2×C7×D411yescharacteristic
214C22×C4C2×C7×D412yes—
215C22×C4C7×#4111yescharacteristic
216Q16—82no—
217M16—42no—
218M16—42no—
219C2×C8—22no—
220C2×C8—41no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—416no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—88no—
230C42—44no—
231C2×C7—24no—
232C2×C7—41no—
233C2×C7—84no—
234C2×C7#200412yes—
235C2×C7#202111yescharacteristic
236C23—24no—
237C23—28no—
238C23—24no—
239C23—41no—
240C23—44no—
241C23—42no—
242C23—48no—
243C23—42no—
244C23—48no—
245C23—84no—
246C23—816no—
247C23C7×#12012yes—
248C23C7×#12111yescharacteristic
249Q8—44no—
250Q8—82no—
251D4—44no—
252D4—48no—
253D4—44no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4C7×#12011yescharacteristic
266C2×C4C7×#12112yes—
267C2×C4C7×#12711yescharacteristic
268C8—42no—
269C8—81no—
270C7#1033311yescharacteristic, direct product factor, Sylow
271C22—22no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—44no—
276C22—416no—
277C22—44no—
278C22—88no—
279C22—84no—
280C22—84no—
281C22—84no—
282C22C7×#39211yescharacteristic
283C22C7×#45514yes—
284C4—21no—
285C4—22no—
286C4—84no—
287C4—82no—
288C4—82no—
289C4C7×B(4,𝔽2)12yes—
290C2—24no—
291C2—41no—
292C2—84no—
293C2C7×#200412yes—
294C2C7×#202111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC23×C7×D4C22
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×#3195C2semidirect
4. C7×#1508C2semidirect
5. C7×#1297C2semidirect
20. #10333C7direct (N × H)
50. #3195C2×C7semidirect
53. #1508C2×C7semidirect
54. #1297C2×C7semidirect
270. C7#10333direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C7×#24352
C7×#17832
C7×#15082
C7×#12972
#103337