Finite Group Databasebeta

← C7×#10299 #13187515

Subgroups — C7×#10299 #13187515

Order 1792.

576 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3229C212yesmaximal
2C7×#2868C211yesmaximal, characteristic
3C7×#1994C211yesmaximal, characteristic
4C7×#1546C212yesmaximal
5C7×#1276C211yesmaximal, characteristic
6C7×#533C2211yescharacteristic
7C2×C7×C4⋊Q8—22no—
8C2×C7×C4⋊Q8C2212yes—
9C7×#498—22no—
10C7×#487—22no—
11C7×#483—24no—
12C7×#475—24no—
13C7×#431C2212yes—
14C7×#423C2211yescharacteristic
15C7×#384—24no—
16C43×C7C2211yescharacteristic
17C7×#339—22no—
18C7×#332—24no—
19#10299C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—22no—
21C2×C7×M16—21no—
22C7×C4⋊Q8—22no—
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—22no—
26C7×C4⋊Q8—44no—
27C7×C4⋊Q8—44no—
28C7×C4⋊Q8D412yes—
29C7×#125—21no—
30C7×C22⋊Q8—22no—
31C4×C7×Q8—22no—
32C7×#117—21no—
33C2×C7×C4⋊C4—28no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4D414yes—
36C2×C42×C7—22no—
37C2×C42×C7D412yes—
38C2×C42×C7C2311yescharacteristic
39C7×#106—28no—
40C7×C4⋊C8—22no—
41C7×C4⋊C8D414yes—
42C7×#103—28no—
43C7×C22⋊C8—22no—
44C4×C7×C8—22no—
45#3229C2×C712yes—
46#2868C2×C711yescharacteristic
47#1994C2×C711yescharacteristic
48#1546C2×C712yes—
49#1276C2×C711yescharacteristic
50C2×C7×Q8—22no—
51C2×C7×Q8—42no—
52C2×C7×Q8—42no—
53C2×C7×Q8—44no—
54C2×C7×Q8—44no—
55C22×C4×C7—22no—
56C22×C4×C7—24no—
57C22×C4×C7C2×D412yes—
58C22×C4×C7C2×D411yescharacteristic
59C7×M16—42no—
60C2×C7×C8—24no—
61C2×C7×C8—41no—
62C7×C4⋊C4—24no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—24no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—416no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C22⋊C4—41no—
71C42×C7—28no—
72C42×C7—22no—
73C42×C7—41no—
74C42×C7SD1614yes—
75C42×C7C2×D411yescharacteristic
76C42×C7C2×D412yes—
77C42×C7#4111yescharacteristic
78#533C22×C711yescharacteristic
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C22×C712yes—
81#498—22no—
82#487—22no—
83#483—24no—
84#475—24no—
85#431C22×C712yes—
86#423C22×C711yescharacteristic
87#384—24no—
88C43C22×C711yescharacteristic
89#339—22no—
90#332—24no—
91C23×C7#12011yescharacteristic
92C7×Q8—44no—
93C7×Q8—84no—
94C7×Q8—82no—
95C7×Q8—82no—
96C2×C4×C7—28no—
97C2×C4×C7—24no—
98C2×C4×C7—24no—
99C2×C4×C7—42no—
100C2×C4×C7—44no—
101C2×C4×C7—48no—
102C2×C4×C7#12012yes—
103C2×C4×C7#12112yes—
104C2×C4×C7#12111yescharacteristic
105C2×C4×C7#12711yescharacteristic
106C2×C4×C7C2×SD1612yes—
107C2×C4×C7AGL(1,ℤ/8)12yes—
108C7×C8—44no—
109C7×C8—81no—
110C22×Q8—22no—
111C2×M16—21no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—44no—
117C4⋊Q8—44no—
118C4⋊Q8C7×D412yes—
119#125—21no—
120C22⋊Q8—22no—
121C4×Q8—22no—
122#117—21no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C7×D414yes—
126C2×C42—22no—
127C2×C42C7×D412yes—
128C2×C42C23×C711yescharacteristic
129#106—28no—
130C4⋊C8—22no—
131C4⋊C8C7×D414yes—
132#103—28no—
133C22⋊C8—22no—
134C4×C8—22no—
135C22×C7—22no—
136C22×C7#39211yescharacteristic
137C22×C7#45012yes—
138C4×C7—28no—
139C4×C7—22no—
140C4×C7—82no—
141C4×C7—84no—
142C4×C7#45614yes—
143C4×C7#46012yes—
144C4×C7#49211yescharacteristic
145C4×C7#49611yescharacteristic
146C2×Q8—22no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4C2×C7×D412yes—
154C22×C4C2×C7×D411yescharacteristic
155M16—42no—
156C2×C8—24no—
157C2×C8—41no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—416no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C22⋊C4—41no—
167C42—28no—
168C42—22no—
169C42—41no—
170C42C7×SD1614yes—
171C42C2×C7×D411yescharacteristic
172C42C2×C7×D412yes—
173C42C7×#4111yescharacteristic
174C2×C7—22no—
175C2×C7#200512yes—
176C2×C7#201411yescharacteristic
177C23C7×#12011yescharacteristic
178Q8—44no—
179Q8—82no—
180Q8—82no—
181Q8—84no—
182C2×C4—28no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—48no—
188C2×C4C7×#12012yes—
189C2×C4C7×#12112yes—
190C2×C4C7×#12111yescharacteristic
191C2×C4C7×#12711yescharacteristic
192C2×C4C2×C7×SD1612yes—
193C2×C4C7×AGL(1,ℤ/8)12yes—
194C8—44no—
195C8—81no—
196C7#1029911yescharacteristic, direct product factor, Sylow
197C22—22no—
198C22C7×#39211yescharacteristic
199C22C7×#45012yes—
200C4—28no—
201C4—22no—
202C4—82no—
203C4—84no—
204C4C7×#45614yes—
205C4C7×#46012yes—
206C4C7×#49211yescharacteristic
207C4C7×#49611yescharacteristic
208C2—22no—
209C2C7×#200512yes—
210C2C7×#201411yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#120
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
19. #10299C7direct (N × H)
196. C7#10299direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32292
C7×#28682
C7×#19942
C7×#15462
C7×#12762
#102997