Finite Group Databasebeta

← #13054658

Subgroups — #13054658

Order 1344.

224 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 68 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×#66635C211yesmaximal, characteristic, complemented
2C2×C7×#215C211yesmaximal, characteristic
3C3×Q8×Dic7C211yesmaximal, characteristic, complemented
4#10620935C211yesmaximal, characteristic
5Dic7⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620501C211yesmaximal, characteristic
8#89938—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12#66635—22no—
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×#215—22no—
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×C4×Dic7—21no—
19C3×C4×Dic7C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21(C2×D3)⋊Dic7—21no—
22C2×C7×SD16—31no—
23Q8×Dic7D311yescharacteristic, complemented
24C2×#1144—31no—
25#6499—31no—
26#6406—31no—
27#6390—31no—
28Dic7⋊C8—31no—
29#5028—71nomaximal, Hall
30C22×D3×C7—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8D412yes—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36Dic42—21no—
37Dic42—21no—
38C7×D12—21no—
39C7×D12—21no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C2×C3×Dic7D411yescharacteristic
43C3⋊(C7⋊C8)—21no—
44C7×C3⋊C8—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×C7×D4—31no—
47C2×Dic14—31no—
48C7×SD16—62no—
49C2×C7×C8—31no—
50C22⋊Dic7—61no—
51C22⋊Dic7—61no—
52#1144—62no—
53C4⋊Dic7—21no—
54C4⋊Dic7C2×D311yescharacteristic
55C4×Dic7—21no—
56C4×Dic7C2×D311yescharacteristic
57C2×C7⋊C8—31no—
58C3×C4×Q8—71no—
59C2×#216—71no—
60C2×#215D711yescharacteristic
61#875—71no—
62#810—71no—
63#809—71no—
64C4⋊(C3⋊C8)—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D3×C7—41no—
67C2×D3×C7—41no—
68C2×D3×C7—41no—
69C3×C4×C7—22no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73C3×Dic7—21no—
74C3×Dic7—21no—
75C3×Dic7—41no—
76#464—211noSylow
77C23×C7—61no—
78C7×Q8—21no—
79C7×Q8D1212yes—
80C7×D4—61no—
81C7×D4—61no—
82C2×C4×C7—21no—
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—21no—
85C2×Dic7—61no—
86C2×Dic7#6711yescharacteristic
87C2×Dic7#6711yescharacteristic
88Dic14—61no—
89Dic14—61no—
90C7×C8—61no—
91C7⋊C8—61no—
92C2×C3×Q8C2×D711yescharacteristic
93C2×D12C2×D711yescharacteristic
94C2×Dic6—71no—
95C3×C4⋊C4—71no—
96C3×C4⋊C4—141no—
97C3×C42—71no—
98C3×C42—141no—
99#216—142no—
100#215—22no—
101(C2×D3)⋊C4—141no—
102(C2×D3)⋊C4—141no—
103C2×C3⋊C8—71no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13511yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D3×C7—81no—
109C2×Q16—211no—
110C2×SD16—31no—
111#124—211no—
112C4×Q8—71no—
113C4⋊C8—211no—
114#103—211no—
115#102—211no—
116C22×C7—121no—
117C22×C7—121no—
118C22×C7—121no—
119C22×C7C2×#6711yescharacteristic
120C4×C7—22no—
121C4×C7C2×D1211yescharacteristic
122C4×C7#23511yescharacteristic
123Dic7—21no—
124Dic7—21no—
125Dic7—41no—
126Dic7—121no—
127C22×D3—21no—
128C3×Q8—21no—
129C3×Q8C22⋊D712yes—
130C2×C3×C4—21no—
131C2×C3×C4—71no—
132C2×C3×C4—71no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—141no—
136D12—21no—
137D12—21no—
138Dic6—141no—
139Dic6—141no—
140C3⋊C8—21no—
141C3⋊C8—141no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—211no—
144C2×Q8D3×D711yescharacteristic
145C2×D4—31no—
146Q16—422no—
147SD16—62no—
148C2×C8—31no—
149C2×C8—211no—
150C4⋊C4—71no—
151C4⋊C4—141no—
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—71no—
155C42—141no—
156C2×C7—241no—
157C2×C7#93011yescharacteristic
158C2×C7#94211yescharacteristic
159C2×C7#95011yescharacteristic
160C22×C3D4×D711yescharacteristic
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—41no—
164C3×C4—22no—
165C3×C4—141no—
166C3×C4—141no—
167C3×C4—281no—
168C3×C4#115911yescharacteristic
169C3×C4C2×C22⋊D711yescharacteristic
170Dic3—281no—
171C23—61no—
172Q8—21no—
173Q8—421no—
174Q8—421no—
175Q8#424712yes—
176D4—61no—
177D4—61no—
178C2×C4—21no—
179C2×C4—71no—
180C2×C4—71no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—421no—
186C7#502811yescharacteristic, complemented, Sylow
187C2×C3#647811yescharacteristic
188C2×C3#647911yescharacteristic
189C2×C3#650211yescharacteristic
190D3—81no—
191C22—121no—
192C22—121no—
193C22—121no—
194C22D7×#6711yescharacteristic
195C4—22no—
196C4—141no—
197C4—141no—
198C4—281no—
199C4—841no—
200C4#6668611yescharacteristic
201C4C2×#424711yescharacteristic
202C3#8993811yescharacteristic, complemented, Sylow
203C2—241no—
204C2#1062107511yescharacteristic
205C2#1062107611yescharacteristic
206C2#1062112811yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#464
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#66635C2semidirect
3. C3×Q8×Dic7C2semidirect
5. Dic7⋊(C3⋊C8)C2semidirect
6. #10620509C2semidirect
23. Q8×Dic7D3semidirect
142. C3×C7#464semidirect
186. C7#5028semidirect
202. C3#89938semidirect

Maximal subgroups

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

subgroupindex
C2×#666352
C2×C7×#2152
C3×Q8×Dic72
#106209352
Dic7⋊(C3⋊C8)2
#106205092
#106205012
#899383
#50287