Finite Group Databasebeta

← C2×#10621477 #13062069

Subgroups — C2×#10621477 #13062069

Order 1344.

330 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 53 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×A4×Dic7C211yesmaximal, characteristic
2C2×C7⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C7×SL(2,ℤ/4)C211yesmaximal, characteristic
4#10621477C214yesmaximal, direct product factor
5C2×#6441—31nomaximal, Hall
6C22×C7×A4C2211yescharacteristic
7C2×#4248—41nomaximal
8A4×Dic7C2212yes—
9C7⋊SL(2,ℤ/4)C2212yes—
10C7×SL(2,ℤ/4)C2212yes—
11C23×Dic7D311yescharacteristic
12C22×Dic14—31no—
13C2×C7×C22⋊C4—31no—
14C2×C22⋊Dic7—31no—
15C2×C4⋊Dic7—31no—
16C2×Dic7⋊C4—31no—
17C2×Dic7⋊C4—31no—
18#6441—34no—
19#6441—34no—
20C2×A4⋊Q8—71nomaximal, Hall
21C2×C7×A4Q812yescomplemented
22C2×C7×A4C2311yescharacteristic
23C2×Dic21—41no—
24C2×C7×Dic3—41no—
25C2×C3×Dic7—41no—
26#4248—44no—
27C24×C7C2×D311yescharacteristic
28C22×C4×C7—31no—
29C22×C4×C7—31no—
30C22×Dic7—31no—
31C22×Dic7—32no—
32C22×Dic7—31no—
33C22×Dic7—31no—
34C22×Dic7—31no—
35C22×Dic7C2×D312yes—
36C2×Dic14—34no—
37C2×Dic14—61no—
38C2×Dic14—62no—
39C2×Dic14—61no—
40C7×C22⋊C4—32no—
41C7×C22⋊C4—32no—
42C22⋊Dic7—32no—
43C22⋊Dic7—32no—
44C4⋊Dic7—34no—
45Dic7⋊C4—34no—
46Dic7⋊C4—34no—
47C2×C4×A4—71no—
48C2×SL(2,ℤ/4)—71no—
49C2×SL(2,ℤ/4)D711yescharacteristic
50A4⋊Q8—74no—
51C22×C3×C7—41no—
52C7×A4C2×Q811yescharacteristic, complemented
53Dic21—42no—
54C3×Dic7—42no—
55C7×Dic3—42no—
56C2×C22⋊Q8—211noSylow
57C23×C7—31no—
58C23×C7—31no—
59C23×C7—32no—
60C23×C7Dic612yescomplemented
61C23×C7C22×D311yescharacteristic
62C2×C4×C7—32no—
63C2×C4×C7—32no—
64C2×C4×C7—61no—
65C2×C4×C7—62no—
66C2×C4×C7—61no—
67C2×Dic7—31no—
68C2×Dic7—32no—
69C2×Dic7—32no—
70C2×Dic7—32no—
71C2×Dic7—32no—
72C2×Dic7—32no—
73C2×Dic7—61no—
74C2×Dic7—61no—
75C2×Dic7—62no—
76C2×Dic7—61no—
77C2×Dic7S411yescharacteristic
78Dic14—64no—
79Dic14—64no—
80C22×A4C2×D711yescharacteristic
81C2×Dic6—281no—
82C4×A4—72no—
83SL(2,ℤ/4)—72no—
84SL(2,ℤ/4)C2×D712yes—
85C2×C3×C7—42no—
86C2×C3×C7—41no—
87C22×Q8—211no—
88C23×C4—71no—
89C22⋊Q8—214no—
90C22⋊Q8—214no—
91C2×C4⋊C4—211no—
92C2×C4⋊C4—211no—
93C2×C4⋊C4—211no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—211no—
96C22×C7—32no—
97C22×C7—31no—
98C22×C7—32no—
99C22×C7—32no—
100C22×C7—31no—
101C22×C7—31no—
102C22×C7—61no—
103C22×C7C2×Dic611yescharacteristic, complemented
104C22×C7C2×S411yescharacteristic
105C4×C7—62no—
106C4×C7—62no—
107Dic7—32no—
108Dic7—62no—
109Dic7—62no—
110Dic7C2×S412yes—
111C2×A4Dic1412yescomplemented
112C2×A4C22×D711yescharacteristic
113C2×C3×C4—281no—
114C2×Dic3—41no—
115C2×Dic3—281no—
116Dic6—284no—
117C3×C7—41noHall
118C24D3×D711yescharacteristic
119C2×Q8—214no—
120C2×Q8—421no—
121C2×Q8—422no—
122C2×Q8—421no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—72no—
126C22×C4—211no—
127C22×C4—211no—
128C22×C4—212no—
129C22×C4—211no—
130C22×C4—211no—
131C4⋊C4—214no—
132C4⋊C4—214no—
133C4⋊C4—214no—
134C22⋊C4—32no—
135C22⋊C4—32no—
136C22⋊C4—212no—
137C22⋊C4—212no—
138C2×C7—31no—
139C2×C7—31no—
140C2×C7—32no—
141C2×C7A4⋊Q812yescomplemented
142C2×C7C22×S411yescharacteristic
143C22×C3—41no—
144A4C2×Dic1411yescharacteristic, complemented
145C3×C4—282no—
146Dic3—42no—
147Dic3—282no—
148C23—31no—
149C23—31no—
150C23—32no—
151C23#424812yescomplemented
152C23C2×D3×D711yescharacteristic
153Q8—424no—
154Q8—424no—
155C2×C4—32no—
156C2×C4—32no—
157C2×C4—62no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—71no—
161C2×C4—212no—
162C2×C4—212no—
163C2×C4—211no—
164C2×C4—212no—
165C2×C4—212no—
166C2×C4—212no—
167C2×C4—421no—
168C2×C4—422no—
169C2×C4—421no—
170C2×C4—421no—
171C7C2×A4⋊Q811yescharacteristic, complemented, Sylow
172C2×C3—42no—
173C2×C3—41no—
174C22—32no—
175C22—31no—
176C22—32no—
177C22—32no—
178C22—31no—
179C22—31no—
180C22—61no—
181C22C2×#424811yescharacteristic, complemented
182C22D7×S411yescharacteristic
183C4—62no—
184C4—62no—
185C4—72no—
186C4—212no—
187C4—422no—
188C4—422no—
189C3—41noSylow
190C2—31no—
191C2—31no—
192C2—32no—
193C2#1062147712yesdirect product factor
194C2C2×D7×S411yescharacteristic
subgroupG/H
center, hypercenterC22D7×S4
commutator, cosocleC2×C7×A4C23
FrattiniC2C2×D7×S4
Fitting, socleC24×C7C2×D3
prime radicalC22×C7×A4C22
hypocenterC7×A4C2×Q8
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #10621477C2direct (N × H)
21. C2×C7×A4Q8semidirect
52. C7×A4C2×Q8semidirect
60. C23×C7Dic6semidirect
103. C22×C7C2×Dic6semidirect
111. C2×A4Dic14semidirect
141. C2×C7A4⋊Q8semidirect
144. A4C2×Dic14semidirect
151. C23#4248semidirect
171. C7C2×A4⋊Q8semidirect
181. C22C2×#4248semidirect
193. C2#10621477direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×Dic72
C2×C7⋊SL(2,ℤ/4)2
C2×C7×SL(2,ℤ/4)2
#106214772
C2×#64413
C2×#42484
C2×A4⋊Q87