Finite Group Databasebeta

← C7×#7481 #13184697

Subgroups — C7×#7481 #13184697

Order 1792.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×C7×#354C211yesmaximal, characteristic
2C7×#1888C211yesmaximal, characteristic
3C7×#1533C211yesmaximal, characteristic
4C2×C7×#124C2211yescharacteristic
5C7×#519C411yescharacteristic, complemented
6C7×#432—21no—
7C7×#418—21no—
8C2×C7×#101C411yescharacteristic, complemented
9C2×C7×#99—21no—
10C7×#354—21no—
11C7×#354C412yescomplemented
12C7×#339—21no—
13C7×#329—21no—
14C7×#329C412yescomplemented
15#7481C711yesmaximal, characteristic, direct product factor, Sylow
16C22×C7×Q8C2×C411yescharacteristic
17C22×C7×D4D411yescharacteristic
18C2×C7×M16—21no—
19C7×#124—41no—
20C7×#124C2×C411yescharacteristic
21C7×#124C2×C411yescharacteristic
22C7×#124D411yescharacteristic
23C7×#124Q811yescharacteristic
24C4×C7×Q8—42no—
25C7×#117—21no—
26C7×#117—41no—
27C2×C7×C22⋊C4—41no—
28C2×C7×C22⋊C4—41no—
29C2×C42×C7D411yescharacteristic
30C7×C4⋊C8—41no—
31C7×#102—42no—
32C7×#101—41no—
33C7×#99—22no—
34C7×#99—41no—
35C7×C22⋊C8—41no—
36C7×#97—22no—
37C2×#354C2×C711yescharacteristic
38#1888C2×C711yescharacteristic
39#1533C2×C711yescharacteristic
40C24×C7—21no—
41C2×C7×Q8—21no—
42C2×C7×Q8—41no—
43C2×C7×Q8—41no—
44C2×C7×Q8C4211yescharacteristic
45C2×C7×Q8C4⋊C411yescharacteristic
46C2×C7×D4—21no—
47C2×C7×D4—41no—
48C2×C7×D4—41no—
49C2×C7×D4C22⋊C411yescharacteristic
50C2×C7×D4C4⋊C411yescharacteristic
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7—81no—
54C22×C4×C7C22⋊C411yescharacteristic
55C7×M16—41no—
56C7×M16—41no—
57C2×C7×C8—42no—
58C2×C7×C8—41no—
59C7×C4⋊C4—22no—
60C7×C4⋊C4—42no—
61C7×C22⋊C4—41no—
62C7×C22⋊C4—41no—
63C7×C22⋊C4—41no—
64C7×C22⋊C4—41no—
65C7×C22⋊C4—41no—
66C7×C22⋊C4—42no—
67C7×C22⋊C4—81no—
68C42×C7—21no—
69C42×C7—41no—
70C42×C7—42no—
71C42×C7C22⋊C411yescharacteristic
72C42×C7C4⋊C411yescharacteristic
73C2×#124C22×C711yescharacteristic
74#519C4×C711yescharacteristic, complemented
75#432—21no—
76#418—21no—
77C2×#101C4×C711yescharacteristic, complemented
78C2×#99—21no—
79#354—21no—
80#354C4×C712yescomplemented
81#339—21no—
82#329—21no—
83#329C4×C712yescomplemented
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—41no—
87C23×C7—41no—
88C23×C7—82no—
89C23×C7#9911yescharacteristic
90C7×Q8—41no—
91C7×Q8—41no—
92C7×Q8—42no—
93C7×D4—41no—
94C7×D4—41no—
95C7×D4—81no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—81no—
106C2×C4×C7—82no—
107C2×C4×C7—162no—
108C2×C4×C7#9511yescharacteristic
109C2×C4×C7#9911yescharacteristic
110C7×C8—81no—
111C7×C8—82no—
112C22×Q8C2×C4×C711yescharacteristic
113C22×D4C7×D411yescharacteristic
114C2×M16—21no—
115#124—41no—
116#124C2×C4×C711yescharacteristic
117#124C2×C4×C711yescharacteristic
118#124C7×D411yescharacteristic
119#124C7×Q811yescharacteristic
120C4×Q8—42no—
121#117—21no—
122#117—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C42C7×D411yescharacteristic
126C4⋊C8—41no—
127#102—42no—
128#101—41no—
129#99—22no—
130#99—41no—
131C22⋊C8—41no—
132#97—22no—
133C22×C7—41no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7—81no—
137C22×C7—81no—
138C22×C7—82no—
139C22×C7—161no—
140C22×C7#34211yescharacteristic
141C22×C7AΣL(1,𝔽16)11yescharacteristic
142C22×C7#35211yescharacteristic
143C4×C7—21no—
144C4×C7—21no—
145C4×C7—41no—
146C4×C7—41no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7—82no—
150C4×C7—162no—
151C24—21no—
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8C42×C711yescharacteristic
156C2×Q8C7×C4⋊C411yescharacteristic
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C7×C22⋊C411yescharacteristic
161C2×D4C7×C4⋊C411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—81no—
165C22×C4C7×C22⋊C411yescharacteristic
166M16—41no—
167M16—41no—
168C2×C8—42no—
169C2×C8—41no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C22⋊C4—81no—
179C42—21no—
180C42—41no—
181C42—42no—
182C42C7×C22⋊C411yescharacteristic
183C42C7×C4⋊C411yescharacteristic
184C2×C7—21no—
185C2×C7—21no—
186C2×C7—21no—
187C2×C7—81no—
188C2×C7—81no—
189C2×C7#139011yescharacteristic
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23—82no—
195C23C7×#9911yescharacteristic
196Q8—41no—
197Q8—41no—
198Q8—42no—
199D4—41no—
200D4—41no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—162no—
214C2×C4C7×#9511yescharacteristic
215C2×C4C7×#9911yescharacteristic
216C8—81no—
217C8—82no—
218C7#748111yescharacteristic, direct product factor, Sylow
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—161no—
226C22C7×#34211yescharacteristic
227C22C7×AΣL(1,𝔽16)11yescharacteristic
228C22C7×#35211yescharacteristic
229C4—21no—
230C4—21no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—82no—
236C4—162no—
237C2—21no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—81no—
242C2C7×#139011yescharacteristic
subgroupG/H
center, socleC2×C7#1390
commutatorC2×Q8C42×C7
Frattini, cosocleC2×#124C22×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#124 ◁ G

Complements

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

NH ≅ G/Ntype
5. C7×#519C4semidirect
8. C2×C7×#101C4semidirect
11. C7×#354C4semidirect
14. C7×#329C4semidirect
15. #7481C7direct (N × H)
74. #519C4×C7semidirect
77. C2×#101C4×C7semidirect
80. #354C4×C7semidirect
83. #329C4×C7semidirect
218. C7#7481direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3542
C7×#18882
C7×#15332
#74817