Finite Group Databasebeta

← #13054711

Subgroups — #13054711

Order 1344.

300 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 96 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×C3×Q8×D7C211yesmaximal, characteristic
2#10620920C211yesmaximal, characteristic
3C2×#66557C211yesmaximal, characteristic
4#10620674C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620506C211yesmaximal, characteristic, complemented
7#10620505C211yesmaximal, characteristic, complemented
8#89574—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×Q8×D7—22no—
11C3×Q8×D7C414yescomplemented
12C2×C3×C4×D7—21no—
13C2×C3×C4×D7C2211yescharacteristic
14C2×C3×Dic14—21no—
15C2×C3×Dic14C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17C2×C3⋊(C7⋊C8)C2211yescharacteristic
18C7×C4⋊Dic3C2211yescharacteristic
19C2×C7×C3⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21Dic3×Dic7—21no—
22#66557—22no—
23C2×Q8×D7D311yescharacteristic
24C2×#1131—31no—
25#6455—31no—
26C7×#103—31no—
27#6409—31no—
28#6390—31no—
29#6384—31no—
30#5225—71nomaximal, Hall
31C22×C3×D7D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8C2×C412yes—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C3×C4×D7—24no—
41C3×C4×D7C2×C412yes—
42C3×C4×D7D412yes—
43C3×Dic14—21no—
44C3×Dic14—24no—
45C3×Dic14C2×C412yes—
46C3⋊(C7⋊C8)—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8C2×D311yescharacteristic
49Q8×D7—22no—
50Q8×D7Dic314yescomplemented
51C2×C4×D7—21no—
52C2×C4×D7C2×D311yescharacteristic
53C2×Dic14—21no—
54C2×Dic14C2×D311yescharacteristic
55C2×C7×C8—31no—
56C7×C4⋊C4—31no—
57(C2×D7)⋊C4—61no—
58C4⋊Dic7—31no—
59C4×Dic7—61no—
60C2×C7⋊C8—31no—
61#1131—62no—
62C22×C3×Q8—71no—
63#926—71no—
64C2×#208—71no—
65#835—71no—
66#835—71no—
67#835—71no—
68#835D711yescharacteristic, complemented
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—21no—
71C2×C3×D7—21no—
72C2×C3×D7C22⋊C412yes—
73C3×C4×C7—22no—
74C3×C4×C7C22×C411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—41no—
77C3×Dic7—22no—
78C3×Dic7C22⋊C412yes—
79C7×Dic3—41no—
80#419—211noSylow
81C22×D7#6711yescharacteristic
82C7×Q8—21no—
83C7×Q8C2×Dic312yes—
84C2×C4×C7—21no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—21no—
88C2×Dic7—61no—
89C2×Dic7#6711yescharacteristic
90C4×D7—24no—
91C4×D7C2×Dic312yes—
92C4×D7#6712yes—
93Dic14—21no—
94Dic14—24no—
95Dic14C2×Dic312yes—
96C7×C8—61no—
97C7⋊C8—61no—
98C2×C3×Q8—74no—
99C2×C3×Q8—71no—
100C2×C3×Q8—141no—
101C2×C3×Q8—142no—
102C2×C3×Q8C2×D711yescharacteristic
103C22×C3×C4—71no—
104C22×C3×C4—141no—
105#217—141no—
106C4⋊Dic3—71no—
107C4⋊Dic3C2×D711yescharacteristic
108C4×Dic3—141no—
109#208—142no—
110C2×C3⋊C8—71no—
111C2×C3⋊C8C2×D711yescharacteristic
112C2×C3×C7C2×C22⋊C411yescharacteristic
113C2×C3×C7#13711yescharacteristic
114C2×C3×C7#13711yescharacteristic
115C3×D7—22no—
116C22×Q8—71no—
117C2×M16—211no—
118#117—211no—
119#103—31no—
120#103—211no—
121#103—211no—
122#103—211no—
123C22×C7C2×#6711yescharacteristic
124C2×D7—21no—
125C2×D7—21no—
126C2×D7#21712yes—
127C4×C7—22no—
128C4×C7—121no—
129C4×C7C22×Dic311yescharacteristic
130C4×C7C2×#6711yescharacteristic
131Dic7—22no—
132Dic7—121no—
133Dic7#21712yes—
134C23×C3—71no—
135C3×Q8—21no—
136C3×Q8—72no—
137C3×Q8—144no—
138C3×Q8—141no—
139C3×Q8C4×D712yes—
140C2×C3×C4—21no—
141C2×C3×C4—71no—
142C2×C3×C4—72no—
143C2×C3×C4—72no—
144C2×C3×C4—144no—
145C2×C3×C4—141no—
146C2×C3×C4C22×D711yescharacteristic
147C2×Dic3—21no—
148C2×Dic3—141no—
149C3⋊C8—21no—
150C3⋊C8—141no—
151C3×C7#41911yescharacteristic, complemented, Hall
152C2×Q8—74no—
153C2×Q8—71no—
154C2×Q8—141no—
155C2×Q8—142no—
156C2×Q8D3×D711yescharacteristic
157C22×C4—71no—
158C22×C4—141no—
159M16—422no—
160C2×C8—31no—
161C2×C8—211no—
162C4⋊C4—31no—
163C4⋊C4—211no—
164C22⋊C4—421no—
165C42—421no—
166C2×C7#94211yescharacteristic
167C2×C7#94211yescharacteristic
168C2×C7C2×#21711yescharacteristic
169D7—22no—
170C22×C3—72no—
171C22×C3—141no—
172C22×C3—141no—
173C22×C3D4×D711yescharacteristic
174C3×C4—22no—
175C3×C4—72no—
176C3×C4—142no—
177C3×C4C2×C4×D711yescharacteristic
178C3×C4D4×D711yescharacteristic
179Dic3—41no—
180Dic3—281no—
181C23—71no—
182Q8—21no—
183Q8—72no—
184Q8—144no—
185Q8—141no—
186Q8Dic3×D712yes—
187C2×C4—21no—
188C2×C4—61no—
189C2×C4—71no—
190C2×C4—72no—
191C2×C4—72no—
192C2×C4—144no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—421no—
198C7#522511yescharacteristic, complemented, Sylow
199C2×C3—142no—
200C2×C3D7×C22⋊C411yescharacteristic
201C2×C3#647811yescharacteristic
202C2×C3#648111yescharacteristic
203C22—72no—
204C22—141no—
205C22—141no—
206C22D7×#6711yescharacteristic
207C4—22no—
208C4—72no—
209C4—121no—
210C4—142no—
211C4—841no—
212C4C2×Dic3×D711yescharacteristic
213C4D7×#6711yescharacteristic
214C3#8957411yescharacteristic, complemented, Sylow
215C2—142no—
216C2#1062107511yescharacteristic
217C2#1062107711yescharacteristic
218C2D7×#21711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7#419
LCSG ▷ 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
4. #10620674C2semidirect
5. C7×#835C2semidirect
6. #10620506C2semidirect
7. #10620505C2semidirect
11. C3×Q8×D7C4semidirect
50. Q8×D7Dic3semidirect
68. #835D7semidirect
151. C3×C7#419semidirect
198. C7#5225semidirect
214. C3#89574semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D72
#106209202
C2×#665572
#106206742
C7×#8352
#106205062
#106205052
#895743
#52257