Finite Group Databasebeta

← #13054330

Subgroups — #13054330

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
1C7×#938C211yesmaximal, characteristic
2D3×C4⋊Dic7C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C3×#6406C211yesmaximal, characteristic, complemented
5#10620521C211yesmaximal, characteristic, complemented
6#10620520C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#89806—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×#67—21no—
11C2×C4×D3×C7C2211yescharacteristic
12C2×D3×Dic7—21no—
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×#217—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic3⋊Dic7—21no—
20C7×#121—31no—
21C2×C4⋊Dic7—31no—
22#6406—31no—
23#6406D311yescharacteristic, complemented
24C22⋊(C7⋊C8)—31no—
25#6382—31no—
26#6382—31no—
27#4777—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×D3×C7D411yescharacteristic
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C7×#67—42no—
35C2×C7×Dic3—21no—
36C2×C7×Dic3D411yescharacteristic
37C4×D3×C7—21no—
38C4×D3×C7—21no—
39C2×C3×Dic7—21no—
40D3×Dic7—42no—
41C3⋊(C7⋊C8)—21no—
42C3×C7⋊C8—21no—
43C2×C7×D4—61no—
44C2×C7×D4C2×D311yescharacteristic
45C22×C4×C7—31no—
46C22×Dic7—61no—
47C7×C4⋊C4—31no—
48C7×C22⋊C4—61no—
49C4⋊Dic7—31no—
50C4⋊Dic7—61no—
51C4⋊Dic7C2×D311yescharacteristic
52C2×C7⋊C8—31no—
53C2×C7⋊C8C2×D311yescharacteristic
54#938D711yescharacteristic
55D3×C4⋊C4—71no—
56C3×#102—71no—
57#832—71no—
58(C2×D3)⋊C8—71no—
59#817—71no—
60#808—71no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7—41no—
64C22×C3×C7C2×D411yescharacteristic
65C2×D3×C7—21no—
66C2×D3×C7—21no—
67C2×D3×C7SD1612yes—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—41no—
72C7×Dic3—21no—
73C7×Dic3—41no—
74#481—211noSylow
75C23×C7—21no—
76C23×C7—31no—
77C7×D4—21no—
78C7×D4—21no—
79C7×D4—122no—
80C2×C4×C7—31no—
81C2×C4×C7—61no—
82C2×C4×C7—61no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—61no—
87C2×Dic7—122no—
88C7⋊C8—21no—
89C7⋊C8—61no—
90C2×C3×D4C2×D711yescharacteristic
91C2×#67—21no—
92C2×C4×D3—141no—
93C2×C4×D3C2×D711yescharacteristic
94C2×C3×C8—71no—
95C3×C4⋊C4—71no—
96#217—21no—
97C4⋊Dic3—71no—
98C4⋊Dic3C2×D711yescharacteristic
99Dic3⋊C4—141no—
100C2×C3⋊C8—71no—
101C2×C3×C7—81no—
102C2×C3×C7#12311yescharacteristic
103C2×C3×C7C2×SD1611yescharacteristic
104C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
105D3×C7—22no—
106#121—31no—
107C2×C4⋊C4—211no—
108#106—211no—
109#106—211no—
110#102—71no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7—32no—
114C22×C7—41no—
115C22×C7—41no—
116C22×C7—41no—
117C22×C7—61no—
118C22×C7—61no—
119C22×C7D3×D411yescharacteristic
120C4×C7—61no—
121C4×C7—121no—
122C4×C7#23511yescharacteristic
123C4×C7#23711yescharacteristic
124Dic7—41no—
125Dic7—121no—
126C23×C3—21no—
127C22×D3C22⋊D711yescharacteristic
128C3×D4—21no—
129C3×D4—21no—
130C2×C3×C4—141no—
131C2×C3×C4C22×D711yescharacteristic
132#67—42no—
133C2×Dic3—21no—
134C2×Dic3—141no—
135C2×Dic3C22⋊D711yescharacteristic
136C4×D3—21no—
137C4×D3—21no—
138C4×D3—282no—
139C3×C8—141no—
140C3⋊C8—141no—
141C3×C7#48111yescharacteristic, complemented, Hall
142C2×D4—61no—
143C2×D4D3×D711yescharacteristic
144C22×C4—31no—
145C22×C4—421no—
146C2×C8—71no—
147C2×C8—211no—
148C4⋊C4—31no—
149C4⋊C4—71no—
150C4⋊C4—211no—
151C4⋊C4—421no—
152C22⋊C4—61no—
153C2×C7—62no—
154C2×C7—81no—
155C2×C7#88311yescharacteristic
156C2×C7#91111yescharacteristic
157C2×C7D3×SD1611yescharacteristic
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3C2×C22⋊D711yescharacteristic
162C2×D3—21no—
163C2×D3—21no—
164C2×D3#114212yes—
165C3×C4—281no—
166C3×C4#115911yescharacteristic
167C3×C4#115911yescharacteristic
168Dic3—21no—
169Dic3—41no—
170Dic3—281no—
171C23—21no—
172C23—31no—
173D4—21no—
174D4—21no—
175D4—122no—
176C2×C4—31no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—141no—
181C2×C4—421no—
182C2×C4—842no—
183C2×C4C2×D3×D711yescharacteristic
184C8—141no—
185C8—421no—
186C7#477711yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3C2×#114211yescharacteristic
189C2×C3#649811yescharacteristic
190C2×C3#651211yescharacteristic
191D3—22no—
192C22—32no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—61no—
197C22—61no—
198C22D3×C22⋊D711yescharacteristic
199C4—61no—
200C4—121no—
201C4—281no—
202C4—841no—
203C4#6668411yescharacteristic
204C4#6668611yescharacteristic
205C3#8980611yescharacteristic, complemented, Sylow
206C2—62no—
207C2—81no—
208C2#1062104211yescharacteristic
209C2D3×#114211yescharacteristic
210C2#1062111711yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC7×#938C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#481
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D3×C4⋊Dic7C2semidirect
3. #10620671C2semidirect
4. C3×#6406C2semidirect
5. #10620521C2semidirect
6. #10620520C2semidirect
7. (C2×D3)⋊(C7⋊C8)C2semidirect
23. #6406D3semidirect
141. C3×C7#481semidirect
186. C7#4777semidirect
205. C3#89806semidirect

Maximal subgroups

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

subgroupindex
C7×#9382
D3×C4⋊Dic72
#106206712
C3×#64062
#106205212
#106205202
(C2×D3)⋊(C7⋊C8)2
#898063
#47777