Finite Group Databasebeta

← #13054648

Subgroups — #13054648

Order 1344.

304 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 72 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×#1161C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic, complemented
3#10620922C211yesmaximal, characteristic, complemented
4C2×#66565C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89582—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C3×#1161—22no—
12C3×#1161—22no—
13C2×C3×D28—21no—
14C2×C3×D28C2211yescharacteristic
15C2×C3×C4×D7—21no—
16C2×C3×C4×D7C2211yescharacteristic
17#66635—22no—
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C7×C4⋊Dic3C2211yescharacteristic
20C2×C7×C3⋊C8C2211yescharacteristic
21Dic3⋊Dic7—21no—
22(C2×D7)⋊Dic3—21no—
23#66565—22no—
24C2×#1161D311yescharacteristic
25C2×#1144—31no—
26C2×#1132—31no—
27#6460—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6383—31no—
31#5239—71nomaximal, Hall
32C22×C3×D7—21no—
33C22×C3×D7D411yescharacteristic
34C3×C7×Q8—21no—
35C3×C7×Q8D412yes—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39Dic42—21no—
40Dic42—21no—
41C2×C7×Dic3—21no—
42C2×C3×Dic7D411yescharacteristic
43C3×D28—24no—
44C3×D28—21no—
45C3×D28D412yes—
46C3×C4×D7—24no—
47C3×C4×D7—21no—
48C3×C4×D7—21no—
49C3⋊(C7⋊C8)—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8C2×D311yescharacteristic
52#1161—22no—
53#1161—22no—
54C2×D28—21no—
55C2×D28C2×D311yescharacteristic
56C2×C4×D7—21no—
57C2×C4×D7C2×D311yescharacteristic
58C2×Dic14—31no—
59C2×C7×C8—31no—
60C7×C4⋊C4—31no—
61#1144—62no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8—31no—
65#1132—62no—
66C2×C3×#41—71no—
67C2×#216—71no—
68C2×#214—71no—
69#924—71no—
70#835D711yescharacteristic, complemented
71#832—71no—
72#830—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×C3×D7—21no—
75C2×C3×D7—22no—
76C2×C3×D7—21no—
77C2×C3×D7—21no—
78C2×C3×D7—41no—
79C2×C3×D7—41no—
80C3×C4×C7—22no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—41no—
84C3×Dic7—21no—
85C7×Dic3—41no—
86#452—211noSylow
87C22×D7—21no—
88C22×D7#6711yescharacteristic
89C7×Q8—21no—
90C7×Q8#6712yes—
91C2×C4×C7—21no—
92C2×C4×C7—61no—
93C2×C4×C7C22×D311yescharacteristic
94C2×Dic7—61no—
95C2×Dic7#6711yescharacteristic
96D28—24no—
97D28—21no—
98D28#6712yes—
99C4×D7—24no—
100C4×D7—21no—
101C4×D7—21no—
102Dic14—61no—
103Dic14—61no—
104C7×C8—61no—
105C7⋊C8—61no—
106C3×#41—142no—
107C3×#41—142no—
108C2×C3×Q8C2×D711yescharacteristic
109C2×C3×D4—71no—
110C2×C3×D4—141no—
111C22×C3×C4—71no—
112C22×C3×C4—141no—
113C2×Dic6—71no—
114#217—141no—
115#216—142no—
116#214—142no—
117C4⋊Dic3C2×D711yescharacteristic
118Dic3⋊C4—141no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124C3×D7—41no—
125C3×D7—42no—
126C2×#41—71no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8—211no—
130#103—31no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7C2×#6711yescharacteristic
134C2×D7—21no—
135C2×D7—21no—
136C2×D7—21no—
137C2×D7—22no—
138C2×D7—41no—
139C2×D7—41no—
140C4×C7—22no—
141C4×C7—121no—
142C4×C7C2×#6711yescharacteristic
143C4×C7C2×#6711yescharacteristic
144Dic7—21no—
145Dic7—121no—
146C23×C3—71no—
147C23×C3—141no—
148C3×Q8—21no—
149C3×Q8D2812yes—
150C3×D4—72no—
151C3×D4—144no—
152C3×D4—141no—
153C2×C3×C4—21no—
154C2×C3×C4—71no—
155C2×C3×C4—144no—
156C2×C3×C4—141no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159C2×Dic3—21no—
160C2×Dic3—141no—
161Dic6—141no—
162Dic6—141no—
163C3⋊C8—21no—
164C3⋊C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—142no—
167#41—142no—
168C2×Q8—211no—
169C2×Q8D3×D711yescharacteristic
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—71no—
173C22×C4—141no—
174Q16—422no—
175SD16—422no—
176C2×C8—31no—
177C2×C8—211no—
178C4⋊C4—31no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7#95011yescharacteristic
182C2×C7#95111yescharacteristic
183C2×C7#95311yescharacteristic
184D7—41no—
185D7—42no—
186C22×C3—142no—
187C22×C3—141no—
188C22×C3—141no—
189C22×C3—141no—
190C22×C3—281no—
191C22×C3—281no—
192C22×C3D4×D711yescharacteristic
193C3×C4—22no—
194C3×C4—141no—
195C3×C4C2×D2811yescharacteristic
196C3×C4D4×D711yescharacteristic
197Dic3—41no—
198Dic3—281no—
199C23—71no—
200C23—141no—
201Q8—21no—
202Q8—421no—
203Q8—421no—
204Q8#424612yes—
205D4—72no—
206D4—144no—
207D4—141no—
208C2×C4—21no—
209C2×C4—61no—
210C2×C4—71no—
211C2×C4—144no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—421no—
218C7#523911yescharacteristic, complemented, Sylow
219C2×C3—281no—
220C2×C3—282no—
221C2×C3#644511yescharacteristic
222C2×C3#647911yescharacteristic
223C2×C3#648111yescharacteristic
224C22—141no—
225C22—141no—
226C22—142no—
227C22—141no—
228C22—281no—
229C22—281no—
230C22D7×#6711yescharacteristic
231C4—22no—
232C4—121no—
233C4—141no—
234C4—841no—
235C4C2×#424611yescharacteristic
236C4D7×#6711yescharacteristic
237C3#8958211yescharacteristic, complemented, Sylow
238C2—281no—
239C2—282no—
240C2#1062107011yescharacteristic
241C2#1062107611yescharacteristic
242C2#1062112611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1161C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#452
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. C2×#66635C2semidirect
3. #10620922C2semidirect
4. C2×#66565C2semidirect
5. C7×#835C2semidirect
6. #10620499C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
70. #835D7semidirect
165. C3×C7#452semidirect
218. C7#5239semidirect
237. C3#89582semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11612
C2×#666352
#106209222
C2×#665652
C7×#8352
#106204992
(C2×D7)⋊(C3⋊C8)2
#895823
#52397