Finite Group Databasebeta

← #13052608

Subgroups — #13052608

Order 1344.

228 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 64 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
1D3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2#10621018C211yesmaximal, characteristic
3#10620647C211yesmaximal, characteristic, complemented
4C3×#6383C211yesmaximal, characteristic, complemented
5#10620520C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#89740—31nomaximal, Hall
9C2×C4×D3×C7—21no—
10C2×C4×D3×C7C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×#4245—21no—
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C7×Dic3⋊C4—21no—
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D7)⋊Dic3—21no—
20C2×C7×C4⋊C4—31no—
21#6493—31no—
22C22⋊(C7⋊C8)—31no—
23#6383—31no—
24#6383D311yescharacteristic, complemented
25#6382—31no—
26#6382—31no—
27#4777—71nomaximal, Hall
28C22×D3×C7D411yescharacteristic
29C22×C3×D7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×C7×Dic3—21no—
34C2×C7×Dic3D411yescharacteristic
35C4×D3×C7—21no—
36C4×D3×C7—21no—
37C4×D3×C7—42no—
38C3×D28—21no—
39C3×D28—21no—
40#4245—42no—
41C3⋊(C7⋊C8)—21no—
42C3×C7⋊C8—21no—
43C22×C4×C7—31no—
44C22×C4×C7—61no—
45C2×C22⋊D7—61no—
46C2×D28C2×D311yescharacteristic
47C7×C4⋊C4—31no—
48C7×C4⋊C4—61no—
49C7×C4⋊C4C2×D311yescharacteristic
50(C2×D7)⋊C4—61no—
51C4⋊Dic7—31no—
52C2×C7⋊C8—31no—
53C2×C7⋊C8C2×D311yescharacteristic
54#938—71no—
55D3×C4⋊C4D711yescharacteristic, complemented
56C3×#102—71no—
57#832—71no—
58(C2×D3)⋊C8—71no—
59#817—71no—
60#808—71no—
61C22×C3×C7C2×D411yescharacteristic
62C2×D3×C7—21no—
63C2×D3×C7—21no—
64C2×D3×C7SD1612yes—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C2×C3×D7—41no—
68C3×C4×C7—41no—
69C3×C4×C7#4111yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—41no—
72C7×Dic3—21no—
73C7×Dic3—41no—
74#481—211noSylow
75C23×C7—31no—
76C22×D7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—31no—
79C2×C4×C7—61no—
80C2×C4×C7—61no—
81C2×C4×C7—61no—
82C2×C4×C7—122no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—122no—
85C2×Dic7—61no—
86D28—21no—
87D28—21no—
88C7⋊C8—21no—
89C7⋊C8—61no—
90C2×C3×D4—71no—
91C2×#67—141no—
92C2×C4×D3—21no—
93C2×C4×D3C2×D711yescharacteristic
94C2×C3×C8—71no—
95C3×C4⋊C4C2×D711yescharacteristic
96#217—141no—
97C4⋊Dic3—71no—
98C4⋊Dic3C2×D711yescharacteristic
99Dic3⋊C4—21no—
100C2×C3⋊C8—71no—
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7C2×SD1611yescharacteristic
103C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
104C3×D7—81no—
105D3×C7—22no—
106#121—211no—
107C2×C4⋊C4—31no—
108#106—211no—
109#106—211no—
110#102—71no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7—32no—
114C22×C7—61no—
115C22×C7—61no—
116C22×C7D3×D411yescharacteristic
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—41no—
121C4×C7—61no—
122C4×C7—121no—
123C4×C7#23511yescharacteristic
124C4×C7#23711yescharacteristic
125Dic7—121no—
126C23×C3—141no—
127C22×D3C22⋊D711yescharacteristic
128C3×D4—141no—
129C3×D4—141no—
130C2×C3×C4—21no—
131C2×C3×C4C22×D711yescharacteristic
132#67—282no—
133C2×Dic3—21no—
134C2×Dic3—141no—
135C2×Dic3C22⋊D711yescharacteristic
136C4×D3—21no—
137C4×D3—21no—
138C4×D3—42no—
139C3×C8—141no—
140C3⋊C8—141no—
141C3×C7#48111yescharacteristic, complemented, Hall
142C2×D4—71no—
143C2×D4—421no—
144C22×C4—31no—
145C22×C4—61no—
146C2×C8—71no—
147C2×C8—211no—
148C4⋊C4—31no—
149C4⋊C4—61no—
150C4⋊C4—211no—
151C4⋊C4D3×D711yescharacteristic
152C22⋊C4—421no—
153C2×C7—62no—
154C2×C7#88311yescharacteristic
155C2×C7#91111yescharacteristic
156C2×C7D3×SD1611yescharacteristic
157D7—81no—
158C22×C3—281no—
159C22×C3—281no—
160C22×C3—281no—
161C22×C3C2×C22⋊D711yescharacteristic
162C2×D3—21no—
163C2×D3—21no—
164C2×D3#114312yes—
165C3×C4—41no—
166C3×C4#115711yescharacteristic
167C3×C4#115711yescharacteristic
168Dic3—21no—
169Dic3—41no—
170Dic3—281no—
171C23—31no—
172C23—141no—
173D4—141no—
174D4—141no—
175D4—842no—
176C2×C4—21no—
177C2×C4—31no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—122no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—141no—
185C8—421no—
186C7#477711yescharacteristic, complemented, Sylow
187C2×C3—561no—
188C2×C3#649211yescharacteristic
189C2×C3#649511yescharacteristic
190C2×C3C2×#114311yescharacteristic
191D3—22no—
192C22—32no—
193C22—61no—
194C22—61no—
195C22—281no—
196C22—281no—
197C22—281no—
198C22D3×C22⋊D711yescharacteristic
199C4—41no—
200C4—61no—
201C4—121no—
202C4—841no—
203C4#6666911yescharacteristic
204C4#6667511yescharacteristic
205C3#8974011yescharacteristic, complemented, Sylow
206C2—62no—
207C2—561no—
208C2#1062101411yescharacteristic
209C2#1062105611yescharacteristic
210C2D3×#114311yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radical#10621018C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#481
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. D3×C7×C4⋊C4C2semidirect
3. #10620647C2semidirect
4. C3×#6383C2semidirect
5. #10620520C2semidirect
6. #10620499C2semidirect
7. (C2×D3)⋊(C7⋊C8)C2semidirect
24. #6383D3semidirect
55. D3×C4⋊C4D7semidirect
141. C3×C7#481semidirect
186. C7#4777semidirect
205. C3#89740semidirect

Maximal subgroups

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

subgroupindex
D3×C7×C4⋊C42
#106210182
#106206472
C3×#63832
#106205202
#106204992
(C2×D3)⋊(C7⋊C8)2
#897403
#47777