Finite Group Databasebeta

← C7×#10216 #13187432

Subgroups — C7×#10216 #13187432

Order 1792.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3255C211yesmaximal, characteristic, complemented
2C7×#3235C212yesmaximal
3C7×#1963C211yesmaximal, characteristic
4C7×#1506C212yesmaximal, complemented
5C7×#1307C211yesmaximal, characteristic, complemented
6C7×#533C2212yes—
7C7×#532C2211yescharacteristic
8C7×#501—22no—
9C7×#490—22no—
10C7×#482—24no—
11C7×#476—22no—
12C7×#475—22no—
13C7×#474—22no—
14C7×#470—22no—
15C7×#462—22no—
16C7×#432C2212yes—
17C7×#423C2211yescharacteristic
18C2×C7×#104—22no—
19C7×#376C2211yescharacteristic
20C7×#328—24no—
21#10216C711yesmaximal, characteristic, direct product factor, Sylow
22C2×C7×#41—21no—
23C2×C7×Q16—22no—
24C2×C7×M16D412yes—
25C7×C4⋊Q8D412yes—
26C7×C4⋊Q8D412yes—
27C7×C4⋊Q8D411yescharacteristic
28C7×#127D411yescharacteristic
29C7×#125—22no—
30C7×#124—21no—
31C7×C22⋊Q8—22no—
32C7×C22⋊Q8—22no—
33C7×#121—22no—
34C4×C7×Q8—22no—
35C4×C7×Q8—22no—
36C4×C7×D4—22no—
37C7×#117—22no—
38C2×C42×C7—21no—
39C2×C42×C7C2311yescharacteristic
40C7×#107—24no—
41C7×#106—22no—
42C7×#106—24no—
43C7×C4⋊C8—22no—
44C7×C4⋊C8—22no—
45C7×#104—44no—
46C7×#103—24no—
47C7×#102—24no—
48C7×#102—24no—
49C7×C22⋊C8D414yes—
50C7×#97—22no—
51C7×#95—22no—
52#3255C2×C711yescharacteristic, complemented
53#3235C2×C712yes—
54#1963C2×C711yescharacteristic
55#1506C2×C712yescomplemented
56#1307C2×C711yescharacteristic, complemented
57C7×#41—42no—
58C7×#41—42no—
59C2×C7×Q8—22no—
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×D4—22no—
63C2×C7×D4—21no—
64C2×C7×D4—41no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D412yes—
69C7×Q16—82no—
70C7×M16—42no—
71C2×C7×C8—24no—
72C2×C7×C8—22no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C22⋊C4—42no—
81C7×C22⋊C4—42no—
82C42×C7—22no—
83C42×C7—41no—
84C42×C7—42no—
85C42×C7C2×D411yescharacteristic
86C42×C7C2×D412yes—
87C42×C7#4111yescharacteristic
88#533C22×C712yes—
89#532C22×C711yescharacteristic
90#501—22no—
91#490—22no—
92#482—24no—
93#476—22no—
94#475—22no—
95#474—22no—
96#470—22no—
97#462—22no—
98#432C22×C712yes—
99#423C22×C711yescharacteristic
100C2×#104—22no—
101#376C22×C711yescharacteristic
102#328—24no—
103C23×C7—41no—
104C23×C7#12711yescharacteristic
105C7×Q8—42no—
106C7×Q8—42no—
107C7×Q8—42no—
108C7×Q8—82no—
109C7×D4—44no—
110C7×D4—42no—
111C7×D4—82no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—82no—
120C2×C4×C7—81no—
121C2×C4×C7—81no—
122C2×C4×C7#12012yes—
123C2×C4×C7#12011yescharacteristic
124C2×C4×C7#12112yes—
125C2×C4×C7#12111yescharacteristic
126C7×C8—44no—
127C7×C8—42no—
128C2×#41—21no—
129C2×Q16—22no—
130C2×M16C7×D412yes—
131C4⋊Q8C7×D411yescharacteristic
132C4⋊Q8C7×D412yes—
133C4⋊Q8C7×D412yes—
134#127C7×D411yescharacteristic
135#125—22no—
136#124—21no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139#121—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×D4—22no—
143#117—22no—
144C2×C42—21no—
145C2×C42C23×C711yescharacteristic
146#107—24no—
147#106—22no—
148#106—24no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151#104—44no—
152#103—24no—
153#102—24no—
154#102—24no—
155C22⋊C8C7×D414yes—
156#97—22no—
157#95—22no—
158C22×C7—21no—
159C22×C7—22no—
160C22×C7—81no—
161C22×C7—82no—
162C22×C7#39211yescharacteristic
163C4×C7—21no—
164C4×C7—22no—
165C4×C7—22no—
166C4×C7—82no—
167C4×C7—81no—
168C4×C7—82no—
169C4×C7—82no—
170C4×C7#45311yescharacteristic
171C4×C7#45411yescharacteristic
172#41—42no—
173#41—42no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×D4—22no—
178C2×D4—21no—
179C2×D4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4C2×C7×D411yescharacteristic
183C22×C4C2×C7×D412yes—
184Q16—82no—
185M16—42no—
186C2×C8—24no—
187C2×C8—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—42no—
200C42C2×C7×D411yescharacteristic
201C42C2×C7×D412yes—
202C42C7×#4111yescharacteristic
203C2×C7—41no—
204C2×C7—161no—
205C2×C7#200611yescharacteristic
206C2×C7#200912yes—
207C23—41no—
208C23C7×#12711yescharacteristic
209Q8—42no—
210Q8—42no—
211Q8—42no—
212Q8—82no—
213D4—44no—
214D4—42no—
215D4—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4C7×#12012yes—
227C2×C4C7×#12011yescharacteristic
228C2×C4C7×#12112yes—
229C2×C4C7×#12111yescharacteristic
230C8—44no—
231C8—42no—
232C7#1021611yescharacteristic, direct product factor, Sylow
233C22—21no—
234C22—22no—
235C22—81no—
236C22—82no—
237C22C7×#39211yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—82no—
245C4C7×#45311yescharacteristic
246C4C7×#45411yescharacteristic
247C2—41no—
248C2—161no—
249C2C7×#200611yescharacteristic
250C2C7×#200912yes—
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3255C2semidirect
4. C7×#1506C2semidirect
5. C7×#1307C2semidirect
21. #10216C7direct (N × H)
52. #3255C2×C7semidirect
55. #1506C2×C7semidirect
56. #1307C2×C7semidirect
232. C7#10216direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32552
C7×#32352
C7×#19632
C7×#15062
C7×#13072
#102167