Finite Group Databasebeta

← #13052882

Subgroups — #13052882

Order 1344.

324 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 86 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×#66669C211yesmaximal, characteristic
2D3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#66558C211yesmaximal, characteristic, complemented
4#10620649C211yesmaximal, characteristic, complemented
5C3×#6383C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89738—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C4×D3×C7—21no—
11C2×C4×D3×C7C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13C2×#4245—21no—
14C2×Dic3×D7—21no—
15#66669—22no—
16#66669C414yescomplemented
17C3×C7×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C7×C4⋊Dic3C2211yescharacteristic
20C7×Dic3⋊C4—21no—
21C2×C3×C7⋊C8C2211yescharacteristic
22#66558—22no—
23C2×#1157—31no—
24C2×C7×C4⋊C4—31no—
25C2×#1136—31no—
26#6384—31no—
27#6384—31no—
28#6383—31no—
29#6383D311yescharacteristic, complemented
30#4771—71nomaximal, Hall
31C22×D3×C7D411yescharacteristic
32C22×C3×D7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36Dic42—21no—
37Dic42C2×C412yescomplemented
38C2×C7×Dic3—21no—
39C2×C7×Dic3D411yescharacteristic
40C4×D3×C7—42no—
41C4×D3×C7C2×C412yes—
42C4×D3×C7D412yes—
43C3×D28—21no—
44C3×D28C2×C412yescomplemented
45#4245—24no—
46Dic3×D7—24no—
47C3⋊(C7⋊C8)—21no—
48C3×C7⋊C8—21no—
49C22×C4×C7—31no—
50C22×C4×C7—61no—
51C2×C22⋊D7—61no—
52#1157—34no—
53#1157—62no—
54C2×D28C2×D311yescharacteristic
55C2×C4×D7—61no—
56C2×Dic14—31no—
57C7×C4⋊C4—31no—
58C7×C4⋊C4—61no—
59C7×C4⋊C4C2×D311yescharacteristic
60#1136—62no—
61C2×C7⋊C8—31no—
62C2×C7⋊C8C2×D311yescharacteristic
63C2×#237—71no—
64C2×ΓL(1,𝔽25)—71no—
65D3×C4⋊C4D711yescharacteristic, complemented
66C3×#102—71no—
67#832—71no—
68#816—71no—
69#810—71no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D3×C7—21no—
72C2×D3×C7—21no—
73C2×D3×C7C22⋊C412yes—
74C2×C3×D7—22no—
75C2×C3×D7—41no—
76C2×C3×D7—41no—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—22no—
81C7×Dic3—41no—
82C7×Dic3C22⋊C412yes—
83#417—211noSylow
84C23×C7—31no—
85C22×D7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—31no—
88C2×C4×C7—32no—
89C2×C4×C7—32no—
90C2×C4×C7—61no—
91C2×C4×C7—122no—
92C2×C4×C7C22×D311yescharacteristic
93C22⋊D7—64no—
94C2×Dic7—61no—
95D28—21no—
96D28C4×D312yescomplemented
97C4×D7—64no—
98Dic14—32no—
99Dic14—61no—
100C7⋊C8—21no—
101C7⋊C8—61no—
102C2×C3×D4—71no—
103C2×#67—141no—
104C22×Dic3—141no—
105#237—74no—
106#237—142no—
107C2×C4×D3—21no—
108C2×C4×D3C2×D711yescharacteristic
109C2×Dic6—71no—
110C2×C3×C8—71no—
111C3×C4⋊C4C2×D711yescharacteristic
112C4⋊Dic3C2×D711yescharacteristic
113Dic3⋊C4—21no—
114C2×C3⋊C8—71no—
115ΓL(1,𝔽25)—142no—
116C2×C3×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119C3×D7—42no—
120D3×C7—22no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—31no—
124#103—211no—
125#103—211no—
126#102—71no—
127#102—211no—
128C22×C7—32no—
129C22×C7—61no—
130C22×C7—61no—
131C22×C7D3×D411yescharacteristic
132C2×D7—22no—
133C2×D7—41no—
134C2×D7—41no—
135C4×C7—32no—
136C4×C7—41no—
137C4×C7—121no—
138C4×C7C2×C4×D311yescharacteristic
139C4×C7D3×D411yescharacteristic
140Dic7—62no—
141C23×C3—141no—
142C22×D3C22⋊D711yescharacteristic
143C3×D4—72no—
144C3×D4—141no—
145C2×C3×C4—21no—
146C2×C3×C4C22×D711yescharacteristic
147#67—144no—
148C2×Dic3—21no—
149C2×Dic3—141no—
150C2×Dic3—144no—
151C2×Dic3C22⋊D711yescharacteristic
152C4×D3—42no—
153C4×D3C4×D712yes—
154C4×D3D2812yes—
155Dic6—72no—
156Dic6—141no—
157C3×C8—141no—
158C3⋊C8—141no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8—211no—
163C2×D4—71no—
164C2×D4—421no—
165C22×C4—31no—
166C22×C4—61no—
167C22×C4—421no—
168M16—422no—
169C2×C8—71no—
170C2×C8—211no—
171C4⋊C4—31no—
172C4⋊C4—61no—
173C4⋊C4D3×D711yescharacteristic
174C2×C7—62no—
175C2×C7D3×C22⋊C411yescharacteristic
176C2×C7#91111yescharacteristic
177C2×C7#91511yescharacteristic
178D7—42no—
179C22×C3—142no—
180C22×C3—281no—
181C22×C3—281no—
182C22×C3C2×C22⋊D711yescharacteristic
183C2×D3—21no—
184C2×D3—21no—
185C2×D3(C2×D7)⋊C412yes—
186C3×C4—41no—
187C3×C4C2×C4×D711yescharacteristic
188C3×C4C2×D2811yescharacteristic
189Dic3—41no—
190Dic3—142no—
191Dic3(C2×D7)⋊C412yes—
192C23—31no—
193C23—141no—
194Q8—212no—
195Q8—421no—
196D4—72no—
197D4—141no—
198D4—424no—
199C2×C4—21no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—61no—
204C2×C4—122no—
205C2×C4—424no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—141no—
209C8—421no—
210C7#477111yescharacteristic, complemented, Sylow
211C2×C3—282no—
212C2×C3C2×(C2×D7)⋊C411yescharacteristic
213C2×C3#649511yescharacteristic
214C2×C3#650511yescharacteristic
215D3—22no—
216C22—32no—
217C22—61no—
218C22—61no—
219C22—142no—
220C22—281no—
221C22—281no—
222C22D3×C22⋊D711yescharacteristic
223C4—32no—
224C4—41no—
225C4—121no—
226C4—422no—
227C4C4×D3×D711yescharacteristic
228C4D3×D2811yescharacteristic
229C3#8973811yescharacteristic, complemented, Sylow
230C2—62no—
231C2—282no—
232C2D3×(C2×D7)⋊C411yescharacteristic
233C2#1062105611yescharacteristic
234C2#1062108011yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66669C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#417
LCSG ▷ 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
2. D3×C7×C4⋊C4C2semidirect
3. C2×#66558C2semidirect
4. #10620649C2semidirect
5. C3×#6383C2semidirect
6. #10620510C2semidirect
7. #10620499C2semidirect
16. #66669C4semidirect
29. #6383D3semidirect
37. Dic42C2×C4semidirect
44. C3×D28C2×C4semidirect
65. D3×C4⋊C4D7semidirect
96. D28C4×D3semidirect
159. C3×C7#417semidirect
210. C7#4771semidirect
229. C3#89738semidirect

Maximal subgroups

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

subgroupindex
C2×#666692
D3×C7×C4⋊C42
C2×#665582
#106206492
C3×#63832
#106205102
#106204992
#897383
#47717