Finite Group Databasebeta

← C7×#12216 #13189432

Subgroups — C7×#12216 #13189432

Order 1792.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2030C211yesmaximal, characteristic
3C7×#1614C212yesmaximal
4C7×#1352C212yesmaximal
5C7×#1317C211yesmaximal, characteristic
6C7×#543C2211yescharacteristic
7C7×#484—22no—
8C7×#399C2212yes—
9C7×#393—21no—
10C7×#391—22no—
11C7×#387—22no—
12C7×#381—21no—
13C7×#381C2211yescharacteristic
14C4×C7×C22⋊C4—21no—
15C7×#342—21no—
16C7×#342C2211yescharacteristic
17C7×#328—22no—
18C7×#323C2212yes—
19#12216C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C23×C4×C7D411yescharacteristic
22C7×C22⋊Q8—22no—
23C7×C22⋊Q8—42no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4D412yes—
26C2×C7×C22⋊C4—21no—
27C2×C7×C22⋊C4—22no—
28C2×C7×C22⋊C4—21no—
29C2×C7×C22⋊C4D411yescharacteristic
30C2×C7×C22⋊C4C2311yescharacteristic
31C2×C42×C7—21no—
32C2×C42×C7—21no—
33C7×#107—42no—
34C7×#103—42no—
35C7×C22⋊C8—22no—
36C7×#95—22no—
37C7×#95—21no—
38C7×#95—22no—
39C7×#95—22no—
40C7×#95—42no—
41C7×#95D412yes—
42#2607C2×C711yescharacteristic
43#2030C2×C711yescharacteristic
44#1614C2×C712yes—
45#1352C2×C712yes—
46#1317C2×C711yescharacteristic
47C24×C7C2×D411yescharacteristic
48C2×C7×Q8—22no—
49C2×C7×Q8—44no—
50C2×C7×Q8—81no—
51C22×C4×C7—21no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7Q1612yes—
61C22×C4×C7C2×D412yes—
62C2×C7×C8—42no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—42no—
65C7×C4⋊C4—44no—
66C7×C4⋊C4—42no—
67C7×C22⋊C4—41no—
68C7×C22⋊C4—44no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—41no—
71C7×C22⋊C4—41no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C42×C7—81no—
75C42×C7—81no—
76#543C22×C711yescharacteristic
77#484—22no—
78#399C22×C712yes—
79#393—21no—
80#391—22no—
81#387—22no—
82#381—21no—
83#381C22×C711yescharacteristic
84C4×C22⋊C4—21no—
85#342—21no—
86#342C22×C711yescharacteristic
87#328—22no—
88#323C22×C712yes—
89C23×C7—21no—
90C23×C7—21no—
91C23×C7—42no—
92C23×C7#12011yescharacteristic
93C23×C7C2×Q1611yescharacteristic
94C23×C7#13711yescharacteristic
95C7×Q8—84no—
96C7×Q8—82no—
97C2×C4×C7—24no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—42no—
101C2×C4×C7—41no—
102C2×C4×C7—42no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—41no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—81no—
113C2×C4×C7—82no—
114C2×C4×C7—81no—
115C2×C4×C7—82no—
116C7×C8—82no—
117C22×Q8—21no—
118C23×C4C7×D411yescharacteristic
119C22⋊Q8—22no—
120C22⋊Q8—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C7×D412yes—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4C7×D411yescharacteristic
127C2×C22⋊C4C23×C711yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130#107—42no—
131#103—42no—
132C22⋊C8—22no—
133#95—22no—
134#95—21no—
135#95—22no—
136#95—22no—
137#95—42no—
138#95C7×D412yes—
139C22×C7—22no—
140C22×C7—21no—
141C22×C7—21no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7—42no—
145C22×C7—81no—
146C22×C7#45111yescharacteristic
147C22×C7#45511yescharacteristic
148C22×C7B(4,𝔽2)11yescharacteristic
149C4×C7—44no—
150C4×C7—42no—
151C4×C7—42no—
152C4×C7—81no—
153C4×C7—82no—
154C4×C7—82no—
155C4×C7—161no—
156C24C2×C7×D411yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8—81no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C7×Q1612yes—
170C22×C4C2×C7×D412yes—
171C2×C8—42no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C22⋊C4—41no—
177C22⋊C4—44no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C42—81no—
184C42—81no—
185C2×C7—21no—
186C2×C7—21no—
187C2×C7—41no—
188C2×C7—41no—
189C2×C7#160211yescharacteristic
190C2×C7#219812yes—
191C23—21no—
192C23—21no—
193C23—42no—
194C23C7×#12011yescharacteristic
195C23C2×C7×Q1611yescharacteristic
196C23C7×#13711yescharacteristic
197Q8—84no—
198Q8—82no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—82no—
218C8—82no—
219C7#1221611yescharacteristic, direct product factor, Sylow
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22C7×#45111yescharacteristic
228C22C7×#45511yescharacteristic
229C22C7×B(4,𝔽2)11yescharacteristic
230C4—44no—
231C4—42no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—82no—
236C4—161no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2C7×#160211yescharacteristic
242C2C7×#219812yes—
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12216C7direct (N × H)
219. C7#12216direct (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×#20302
C7×#16142
C7×#13522
C7×#13172
#122167