Finite Group Databasebeta

← C7×#10387 #13187603

Subgroups — C7×#10387 #13187603

Order 1792.

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

normal subgroups only

This group has 414 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×#2607C211yesmaximal, characteristic
2C7×#1826C212yesmaximal
3C7×#1820C211yesmaximal, characteristic
4C7×#1306C211yesmaximal, characteristic
5C7×#1284C212yesmaximal
6C2×C7×#117C2211yescharacteristic
7C7×#407C2212yes—
8C7×#399—21no—
9C7×#393—21no—
10C7×#391—22no—
11C7×#387—22no—
12C7×#381C2211yescharacteristic
13C7×#381C2211yescharacteristic
14C4×C7×C22⋊C4C2212yes—
15C7×#342—21no—
16C7×#342—21no—
17C7×#339—22no—
18C7×#339—22no—
19#10387C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C23×C4×C7C2311yescharacteristic
22C2×C7×M16—22no—
23C7×#117—22no—
24C7×#117—22no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4Q812yescomplemented
28C2×C7×C22⋊C4—21no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—21no—
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4D411yescharacteristic
33C2×C42×C7—21no—
34C2×C42×C7D412yes—
35C2×C42×C7D412yes—
36C7×C22⋊C8—22no—
37C7×C22⋊C8—22no—
38C7×#95—21no—
39C7×#95—22no—
40C7×#95—21no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44#2607C2×C711yescharacteristic
45#1826C2×C712yes—
46#1820C2×C711yescharacteristic
47#1306C2×C711yescharacteristic
48#1284C2×C712yes—
49C24×C7C2×D411yescharacteristic
50C2×C7×Q8—44no—
51C2×C7×Q8—81no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7C2×D412yes—
61C22×C4×C7C2×Q811yescharacteristic
62C22×C4×C7#4112yes—
63C22×C4×C7#4111yescharacteristic
64C7×M16—42no—
65C7×M16—42no—
66C2×C7×C8—42no—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—44no—
69C7×C4⋊C4—81no—
70C7×C22⋊C4—44no—
71C7×C22⋊C4—41no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C7×C22⋊C4—41no—
75C7×C22⋊C4—41no—
76C7×C22⋊C4—41no—
77C42×C7—42no—
78C42×C7—42no—
79C42×C7—42no—
80C2×#117C22×C711yescharacteristic
81#407C22×C712yes—
82#399—21no—
83#393—21no—
84#391—22no—
85#387—22no—
86#381C22×C711yescharacteristic
87#381C22×C711yescharacteristic
88C4×C22⋊C4C22×C712yes—
89#342—21no—
90#342—21no—
91#339—22no—
92#339—22no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—42no—
96C23×C7#12011yescharacteristic
97C23×C7C22⋊Q811yescharacteristic
98C23×C7#12411yescharacteristic
99C7×Q8—84no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—24no—
103C2×C4×C7—41no—
104C2×C4×C7—41no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—82no—
117C2×C4×C7—81no—
118C2×C4×C7—82no—
119C2×C4×C7C22⋊Q812yes—
120C2×C4×C7#12412yes—
121C7×C8—82no—
122C22×Q8—21no—
123C23×C4C23×C711yescharacteristic
124C2×M16—22no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C7×Q812yescomplemented
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4C7×D411yescharacteristic
134C2×C22⋊C4C7×D411yescharacteristic
135C2×C42—21no—
136C2×C42C7×D412yes—
137C2×C42C7×D412yes—
138C22⋊C8—22no—
139C22⋊C8—22no—
140#95—21no—
141#95—22no—
142#95—21no—
143#95—22no—
144#95—22no—
145#95—22no—
146C22×C7—22no—
147C22×C7—21no—
148C22×C7—21no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7—81no—
152C22×C7—81no—
153C22×C7#39311yescharacteristic
154C22×C7#45811yescharacteristic
155C22×C7#45811yescharacteristic
156C4×C7—22no—
157C4×C7—22no—
158C4×C7—44no—
159C4×C7—82no—
160C4×C7—81no—
161C4×C7—81no—
162C4×C7—82no—
163C4×C7—161no—
164C24C2×C7×D411yescharacteristic
165C2×Q8—44no—
166C2×Q8—81no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4C2×C7×D412yes—
176C22×C4C2×C7×Q811yescharacteristic
177C22×C4C7×#4112yes—
178C22×C4C7×#4111yescharacteristic
179M16—42no—
180M16—42no—
181C2×C8—42no—
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—81no—
185C22⋊C4—44no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C42—42no—
193C42—42no—
194C42—42no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7—41no—
198C2×C7—41no—
199C2×C7#203011yescharacteristic
200C2×C7#203112yes—
201C23—21no—
202C23—21no—
203C23—42no—
204C23C7×#12011yescharacteristic
205C23C7×C22⋊Q811yescharacteristic
206C23C7×#12411yescharacteristic
207Q8—84no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4C7×C22⋊Q812yes—
228C2×C4C7×#12412yes—
229C8—82no—
230C7#1038711yescharacteristic, direct product factor, Sylow
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22C7×#39311yescharacteristic
239C22C7×#45811yescharacteristic
240C22C7×#45811yescharacteristic
241C4—22no—
242C4—22no—
243C4—44no—
244C4—81no—
245C4—82no—
246C4—81no—
247C4—82no—
248C4—161no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2C7×#203011yescharacteristic
254C2C7×#203112yes—
subgroupG/H
center, socleC22×C7#393
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
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
19. #10387C7direct (N × H)
27. C2×C7×C4⋊C4Q8semidirect
129. C2×C4⋊C4C7×Q8semidirect
230. C7#10387direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26072
C7×#18262
C7×#18202
C7×#13062
C7×#12842
#103877