Finite Group Databasebeta

← C7×#11771 #13188987

Subgroups — C7×#11771 #13188987

Order 1792.

292 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3177C211yesmaximal, characteristic
2C7×#2166C211yesmaximal, characteristic
3C7×#1747C211yesmaximal, characteristic
4C7×#1342C212yesmaximal
5C7×#1340C212yesmaximal
6C7×#541—21no—
7C7×#499C411yescharacteristic, complemented
8C7×#494C411yescharacteristic, complemented
9C7×#491—21no—
10C7×#470C2212yes—
11C7×#467—21no—
12C7×#441C412yescomplemented
13C4×C7×Q16—21no—
14C7×#405—21no—
15C7×#405C2211yescharacteristic
16C7×#364—21no—
17C7×#349C2212yes—
18C7×#347—21no—
19C7×#338C2211yescharacteristic
20C7×H(3,ℤ/4)—21no—
21C7×H(3,ℤ/4)C2211yescharacteristic
22#11771C711yesmaximal, characteristic, direct product factor, Sylow
23C2×C7×Q16—21no—
24C2×C7×Q16C2×C412yes—
25C2×C7×M16—21no—
26C2×C7×M16C2311yescharacteristic
27C22×C7×C8D411yescharacteristic
28C7×C4⋊Q8—21no—
29C7×#126—42no—
30C7×#125—21no—
31C7×#125—42no—
32C7×C22⋊Q8—22no—
33C4×C7×Q8—22no—
34C7×#117—21no—
35C7×#117D411yescharacteristic
36C7×M32—22no—
37C7×#107—21no—
38C7×#106C2×C412yes—
39C7×#103—22no—
40C7×#103—22no—
41C7×#97—22no—
42C7×#97C2×C411yescharacteristic
43C7×#97D411yescharacteristic
44C4×C7×C8—22no—
45C4×C7×C8C2×C411yescharacteristic
46C4×C7×C8D411yescharacteristic
47#3177C2×C711yescharacteristic
48#2166C2×C711yescharacteristic
49#1747C2×C711yescharacteristic
50#1342C2×C712yes—
51#1340C2×C712yes—
52C2×C7×Q8—22no—
53C22×C4×C7C2×D411yescharacteristic
54C7×Q16—42no—
55C7×Q16—41no—
56C7×Q16—41no—
57C7×M16—41no—
58C7×M16—41no—
59C7×M16—41no—
60C2×C7×C8—21no—
61C2×C7×C8—21no—
62C2×C7×C8—22no—
63C2×C7×C8C22⋊C411yescharacteristic
64C2×C7×C8C22⋊C411yescharacteristic
65C2×C7×C8C22×C411yescharacteristic
66C2×C7×C8C2×D411yescharacteristic
67C7×C4⋊C4—21no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C22⋊C4—21no—
75C7×C22⋊C4—41no—
76C7×C22⋊C4—42no—
77C42×C7—22no—
78C42×C7—42no—
79C42×C7C22⋊C411yescharacteristic
80C42×C7C22⋊C411yescharacteristic
81C7×C16—42no—
82#541—21no—
83#499C4×C711yescharacteristic, complemented
84#494C4×C711yescharacteristic, complemented
85#491—21no—
86#470C22×C712yes—
87#467—21no—
88#441C4×C712yescomplemented
89C4×Q16—21no—
90#405—21no—
91#405C22×C711yescharacteristic
92#364—21no—
93#349C22×C712yes—
94#347—21no—
95#338C22×C711yescharacteristic
96H(3,ℤ/4)—21no—
97H(3,ℤ/4)C22×C711yescharacteristic
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C7×Q8—42no—
100C7×Q8—42no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7C2×C22⋊C411yescharacteristic
109C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
110C7×C8—21no—
111C7×C8—21no—
112C7×C8—21no—
113C7×C8—42no—
114C7×C8#10411yescharacteristic
115C7×C8#10411yescharacteristic
116C2×Q16—21no—
117C2×Q16C2×C4×C712yes—
118C2×M16—21no—
119C2×M16C23×C711yescharacteristic
120C22×C8C7×D411yescharacteristic
121C4⋊Q8—21no—
122#126—42no—
123#125—21no—
124#125—42no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117—21no—
128#117C7×D411yescharacteristic
129M32—22no—
130#107—21no—
131#106C2×C4×C712yes—
132#103—22no—
133#103—22no—
134#97—22no—
135#97C2×C4×C711yescharacteristic
136#97C7×D411yescharacteristic
137C4×C8—22no—
138C4×C8C2×C4×C711yescharacteristic
139C4×C8C7×D411yescharacteristic
140C22×C7—21no—
141C22×C7—41no—
142C22×C7#41811yescharacteristic
143C4×C7—21no—
144C4×C7—41no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7—82no—
148C4×C7—82no—
149C4×C7#41011yescharacteristic
150C4×C7C2×#10411yescharacteristic
151C2×Q8—22no—
152C22×C4C2×C7×D411yescharacteristic
153Q16—42no—
154Q16—41no—
155Q16—41no—
156M16—41no—
157M16—41no—
158M16—41no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8C7×C22⋊C411yescharacteristic
163C2×C8C7×C22⋊C411yescharacteristic
164C2×C8C22×C4×C711yescharacteristic
165C2×C8C2×C7×D411yescharacteristic
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C22⋊C4—21no—
174C22⋊C4—41no—
175C22⋊C4—42no—
176C42—22no—
177C42—42no—
178C42C7×C22⋊C411yescharacteristic
179C42C7×C22⋊C411yescharacteristic
180C16—42no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7#150511yescharacteristic
184C23C7×AGL(1,ℤ/8)11yescharacteristic
185Q8—42no—
186Q8—42no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4C2×C7×C22⋊C411yescharacteristic
195C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—42no—
200C8C7×#10411yescharacteristic
201C8C7×#10411yescharacteristic
202C7#1177111yescharacteristic, direct product factor, Sylow
203C22—21no—
204C22—41no—
205C22C7×#41811yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—82no—
212C4C7×#41011yescharacteristic
213C4C2×C7×#10411yescharacteristic
214C2—21no—
215C2—41no—
216C2C7×#150511yescharacteristic
subgroupG/H
center, socleC2×C7#1505
commutatorC2×C8C22×C4×C7
Frattini, cosocleC2×M16C23×C7
prime radicalC23×C7AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#405 ◁ G

Complements

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

NH ≅ G/Ntype
7. C7×#499C4semidirect
8. C7×#494C4semidirect
12. C7×#441C4semidirect
22. #11771C7direct (N × H)
83. #499C4×C7semidirect
84. #494C4×C7semidirect
88. #441C4×C7semidirect
202. C7#11771direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31772
C7×#21662
C7×#17472
C7×#13422
C7×#13402
#117717