Finite Group Databasebeta

← #13054665

Subgroups — #13054665

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×#66634C211yesmaximal, characteristic
2C7×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#1144C211yesmaximal, characteristic, complemented
4#10620936C211yesmaximal, characteristic
5Dic3⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic
8#89800—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66634—22no—
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×C4⋊Dic3—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C4×C7×Dic3—21no—
17C4×C7×Dic3C2211yescharacteristic
18C3×#1144—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66575—21no—
21#66575—21no—
22C4×C7×Q8—31no—
23C2×#1144D311yescharacteristic, complemented
24C2×#1143—31no—
25#6438—31no—
26#6384—31no—
27#6383—31no—
28C4⋊(C7⋊C8)—31no—
29#5185—71nomaximal, Hall
30C22×D21—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8D412yes—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35D84—21no—
36D84—21no—
37C2×C7×Dic3—21no—
38C2×C7×Dic3D411yescharacteristic
39C2×C7×Dic3D411yescharacteristic
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×D28—31no—
47C2×Dic14C2×D311yescharacteristic
48C7×C4⋊C4—31no—
49C7×C4⋊C4—61no—
50C42×C7—31no—
51C42×C7—61no—
52#1144—22no—
53#1143—62no—
54(C2×D7)⋊C4—61no—
55(C2×D7)⋊C4—61no—
56C2×C7⋊C8—31no—
57C2×C7⋊C8C2×D311yescharacteristic
58C2×C3×Q16—71no—
59#947—71no—
60Q8×Dic3D711yescharacteristic, complemented
61C2×#215—71no—
62#835—71no—
63#821—71no—
64Dic3⋊C8—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D21—41no—
67C2×D21—41no—
68C2×D21—41no—
69C3×C4×C7—22no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72C3×Dic7—41no—
73C7×Dic3—21no—
74C7×Dic3—21no—
75C7×Dic3—41no—
76#464—211noSylow
77C22×D7—61no—
78C7×Q8—21no—
79C7×Q8#6712yes—
80C2×C4×C7—21no—
81C2×C4×C7—31no—
82C2×C4×C7—31no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86D28—61no—
87D28—61no—
88Dic14—21no—
89Dic14—21no—
90C7⋊C8—21no—
91C7⋊C8—61no—
92C2×C3×Q8—71no—
93C2×C3×Q8C2×D711yescharacteristic
94C2×D12—71no—
95C3×Q16—142no—
96C2×C3×C8—71no—
97#215—142no—
98(C2×D3)⋊C4—141no—
99(C2×D3)⋊C4—141no—
100C4⋊Dic3—21no—
101C4⋊Dic3C2×D711yescharacteristic
102C4×Dic3—21no—
103C4×Dic3C2×D711yescharacteristic
104C2×C3⋊C8—71no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13511yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D21—81no—
109C2×Q16—71no—
110C2×SD16—211no—
111#124—211no—
112C4×Q8—31no—
113C4⋊C8—211no—
114#103—211no—
115#102—211no—
116C22×C7D3×D411yescharacteristic
117C2×D7—121no—
118C2×D7—121no—
119C2×D7—121no—
120C4×C7—22no—
121C4×C7—61no—
122C4×C7—61no—
123C4×C7—121no—
124C4×C7#23711yescharacteristic
125C4×C7C2×#6711yescharacteristic
126Dic7—41no—
127C22×D3—141no—
128C3×Q8—21no—
129C3×Q8—141no—
130C3×Q8—141no—
131C3×Q8D2812yes—
132C2×C3×C4—21no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—21no—
136C2×Dic3C22⋊D711yescharacteristic
137C2×Dic3C22⋊D711yescharacteristic
138D12—141no—
139D12—141no—
140C3×C8—141no—
141C3⋊C8—141no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8D3×D711yescharacteristic
145C2×D4—211no—
146Q16—142no—
147SD16—422no—
148C2×C8—71no—
149C2×C8—211no—
150C4⋊C4—31no—
151C4⋊C4—61no—
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—31no—
155C42—61no—
156C2×C7#91811yescharacteristic
157C2×C7#91911yescharacteristic
158C2×C7#93911yescharacteristic
159D7—241no—
160C22×C3C2×C22⋊D711yescharacteristic
161C2×D3—281no—
162C2×D3—281no—
163C2×D3—281no—
164C3×C4—22no—
165C3×C4—281no—
166C3×C4C2×D2811yescharacteristic
167C3×C4#115711yescharacteristic
168Dic3—21no—
169Dic3—21no—
170Dic3—41no—
171C23—421no—
172Q8—21no—
173Q8—141no—
174Q8—141no—
175Q8#424612yes—
176D4—421no—
177D4—421no—
178C2×C4—21no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—141no—
183C2×C4C2×D3×D711yescharacteristic
184C8—141no—
185C8—421no—
186C7#518511yescharacteristic, complemented, Sylow
187C2×C3#649311yescharacteristic
188C2×C3#650511yescharacteristic
189C2×C3#651311yescharacteristic
190D3—561no—
191C22—841no—
192C22—841no—
193C22—841no—
194C22D3×C22⋊D711yescharacteristic
195C4—22no—
196C4—61no—
197C4—61no—
198C4—121no—
199C4—281no—
200C4#6668311yescharacteristic
201C4C2×#424611yescharacteristic
202C3#8980011yescharacteristic, complemented, Sylow
203C2—1681no—
204C2#1062108111yescharacteristic
205C2#1062108211yescharacteristic
206C2#1062113111yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
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
2. C7×Q8×Dic3C2semidirect
3. C2×C3×#1144C2semidirect
5. Dic3⋊(C7⋊C8)C2semidirect
6. #10620505C2semidirect
23. C2×#1144D3semidirect
60. Q8×Dic3D7semidirect
142. C3×C7#464semidirect
186. C7#5185semidirect
202. C3#89800semidirect

Maximal subgroups

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

subgroupindex
C2×#666342
C7×Q8×Dic32
C2×C3×#11442
#106209362
Dic3⋊(C7⋊C8)2
#106205052
#106205042
#898003
#51857