Finite Group Databasebeta

← C3×C7×#448 #13056959

Subgroups — C3×C7×#448 #13056959

Order 1344.

316 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C22×C3×C7×Q8C211yesmaximal, characteristic, complemented
2C2×C3×C7×SD16C212yesmaximal, complemented
3C3×C7×#121C211yesmaximal, characteristic
4C3×C7×#103C212yesmaximal, complemented
5C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
6C7×#448C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×Q8—24no—
8C2×C3×C7×Q8—21no—
9C2×C3×C7×Q8C2212yes—
10C2×C3×C7×D4—21no—
11C2×C3×C7×D4C2211yescharacteristic
12C22×C3×C4×C7—21no—
13C22×C3×C4×C7C2211yescharacteristic
14C3×C7×SD16—24no—
15C2×C3×C7×C8C2212yes—
16C3×C7×C4⋊C4C2211yescharacteristic
17C3×C7×C22⋊C4—21no—
18C22×C7×Q8C2×C311yescharacteristic, complemented
19C2×C7×SD16C2×C312yescomplemented
20C7×#121C2×C311yescharacteristic
21C7×#103C2×C312yescomplemented
22C7×C22⋊C8C2×C311yescharacteristic, complemented
23C3×#448C711yesmaximal, characteristic, direct product factor, Hall
24C23×C3×C7—21no—
25C23×C3×C7D411yescharacteristic
26C3×C7×Q8—24no—
27C3×C7×Q8—22no—
28C3×C7×Q8D414yescomplemented
29C3×C7×D4—22no—
30C3×C7×D4—42no—
31C2×C3×C4×C7—24no—
32C2×C3×C4×C7—22no—
33C2×C3×C4×C7—22no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7D411yescharacteristic
36C2×C3×C4×C7C2311yescharacteristic
37C3×C7×C8—22no—
38C2×C7×Q8—24no—
39C2×C7×Q8—21no—
40C2×C7×Q8C22×C312yes—
41C2×C7×D4—21no—
42C2×C7×D4C22×C311yescharacteristic
43C22×C4×C7—21no—
44C22×C4×C7C22×C311yescharacteristic
45C7×SD16—24no—
46C2×C7×C8C22×C312yes—
47C7×C4⋊C4C22×C311yescharacteristic
48C7×C22⋊C4—21no—
49C22×C3×Q8C2×C711yescharacteristic, complemented
50C2×C3×SD16C2×C712yescomplemented
51C3×#121C2×C711yescharacteristic
52C3×#103C2×C712yescomplemented
53C3×C22⋊C8C2×C711yescharacteristic, complemented
54C22×C3×C7—21no—
55C22×C3×C7—21no—
56C22×C3×C7—41no—
57C22×C3×C7—41no—
58C22×C3×C7—41no—
59C22×C3×C7SD1612yescomplemented
60C22×C3×C7C2×D411yescharacteristic
61C3×C4×C7—24no—
62C3×C4×C7—21no—
63C3×C4×C7—41no—
64C3×C4×C7C2×D412yes—
65#448C3×C711yescharacteristic, direct product factor, Sylow
66C23×C7—21no—
67C23×C7C3×D411yescharacteristic
68C7×Q8—24no—
69C7×Q8—22no—
70C7×Q8C3×D414yescomplemented
71C7×D4—22no—
72C7×D4—42no—
73C2×C4×C7—24no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—21no—
77C2×C4×C7C3×D411yescharacteristic
78C2×C4×C7C23×C311yescharacteristic
79C7×C8—22no—
80C2×C3×Q8—24no—
81C2×C3×Q8—21no—
82C2×C3×Q8C22×C712yes—
83C2×C3×D4—21no—
84C2×C3×D4C22×C711yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C22×C711yescharacteristic
87C3×SD16—24no—
88C2×C3×C8C22×C712yes—
89C3×C4⋊C4C22×C711yescharacteristic
90C3×C22⋊C4—21no—
91C2×C3×C7—22no—
92C2×C3×C7—81no—
93C2×C3×C7#12011yescharacteristic
94C2×C3×C7C2×SD1611yescharacteristic
95C2×C3×C7#13711yescharacteristic
96C22×Q8C2×C3×C711yescharacteristic, complemented
97C2×SD16C2×C3×C712yescomplemented
98#121C2×C3×C711yescharacteristic
99#103C2×C3×C712yescomplemented
100C22⋊C8C2×C3×C711yescharacteristic, complemented
101C22×C7—21no—
102C22×C7—21no—
103C22×C7—41no—
104C22×C7—41no—
105C22×C7—41no—
106C22×C7C3×SD1612yescomplemented
107C22×C7C2×C3×D411yescharacteristic
108C4×C7—24no—
109C4×C7—21no—
110C4×C7—41no—
111C4×C7C2×C3×D412yes—
112C23×C3—21no—
113C23×C3C7×D411yescharacteristic
114C3×Q8—24no—
115C3×Q8—22no—
116C3×Q8C7×D414yescomplemented
117C3×D4—22no—
118C3×D4—42no—
119C2×C3×C4—24no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4C7×D411yescharacteristic
124C2×C3×C4C23×C711yescharacteristic
125C3×C8—22no—
126C3×C7#44811yescharacteristic, direct product factor, Hall
127C2×Q8—24no—
128C2×Q8—21no—
129C2×Q8C22×C3×C712yes—
130C2×D4—21no—
131C2×D4C22×C3×C711yescharacteristic
132C22×C4—21no—
133C22×C4C22×C3×C711yescharacteristic
134SD16—24no—
135C2×C8C22×C3×C712yes—
136C4⋊C4C22×C3×C711yescharacteristic
137C22⋊C4—21no—
138C2×C7—22no—
139C2×C7—81no—
140C2×C7C3×#12011yescharacteristic
141C2×C7C2×C3×SD1611yescharacteristic
142C2×C7C3×#13711yescharacteristic
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3C7×SD1612yescomplemented
149C22×C3C2×C7×D411yescharacteristic
150C3×C4—24no—
151C3×C4—21no—
152C3×C4—41no—
153C3×C4C2×C7×D412yes—
154C23—21no—
155C23C3×C7×D411yescharacteristic
156Q8—24no—
157Q8—22no—
158Q8C3×C7×D414yescomplemented
159D4—22no—
160D4—42no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4C3×C7×D411yescharacteristic
166C2×C4C23×C3×C711yescharacteristic
167C8—22no—
168C7C3×#44811yescharacteristic, direct product factor, Sylow
169C2×C3—22no—
170C2×C3—81no—
171C2×C3C7×#12011yescharacteristic
172C2×C3C2×C7×SD1611yescharacteristic
173C2×C3C7×#13711yescharacteristic
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22C3×C7×SD1612yescomplemented
180C22C2×C3×C7×D411yescharacteristic
181C4—24no—
182C4—21no—
183C4—41no—
184C4C2×C3×C7×D412yes—
185C3C7×#44811yescharacteristic, direct product factor, Sylow
186C2—22no—
187C2—81no—
188C2C3×C7×#12011yescharacteristic
189C2C2×C3×C7×SD1611yescharacteristic
190C2C3×C7×#13711yescharacteristic
subgroupG/H
center, socleC22×C3×C7C2×D4
commutator, Frattini, cosocleC2×C4C23×C3×C7
prime radicalC3×C7×#121C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×Q8C2semidirect
2. C2×C3×C7×SD16C2semidirect
4. C3×C7×#103C2semidirect
5. C3×C7×C22⋊C8C2semidirect
6. C7×#448C3direct (N × H)
18. C22×C7×Q8C2×C3semidirect
19. C2×C7×SD16C2×C3semidirect
21. C7×#103C2×C3semidirect
22. C7×C22⋊C8C2×C3semidirect
23. C3×#448C7direct (N × H)
28. C3×C7×Q8D4semidirect
49. C22×C3×Q8C2×C7semidirect
50. C2×C3×SD16C2×C7semidirect
52. C3×#103C2×C7semidirect
53. C3×C22⋊C8C2×C7semidirect
59. C22×C3×C7SD16semidirect
65. #448C3×C7direct (N × H)
70. C7×Q8C3×D4semidirect
96. C22×Q8C2×C3×C7semidirect
97. C2×SD16C2×C3×C7semidirect
99. #103C2×C3×C7semidirect
100. C22⋊C8C2×C3×C7semidirect
106. C22×C7C3×SD16semidirect
116. C3×Q8C7×D4semidirect
126. C3×C7#448direct (N × H)
148. C22×C3C7×SD16semidirect
158. Q8C3×C7×D4semidirect
168. C7C3×#448direct (N × H)
179. C22C3×C7×SD16semidirect
185. C3C7×#448direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×Q82
C2×C3×C7×SD162
C3×C7×#1212
C3×C7×#1032
C3×C7×C22⋊C82
C7×#4483
C3×#4487