Finite Group Databasebeta

← #13054201

Subgroups — #13054201

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×#1157C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic, complemented
3C2×#66569C211yesmaximal, characteristic, complemented
4C2×#66563C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89819—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C3×#1157—22no—
13C3×#1157—22no—
14C2×C3×D28C2211yescharacteristic
15C2×C3×C4×D7—21no—
16C2×C3×Dic14C2211yescharacteristic
17C2×C3⋊(C7⋊C8)C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C7×#217—21no—
20C7×C4⋊Dic3C2211yescharacteristic
21C7×Dic3⋊C4—21no—
22#66569—22no—
23#66563—22no—
24C2×#1157D311yescharacteristic
25C7×C22⋊Q8—31no—
26C2×#1144—31no—
27C2×#1143—31no—
28C22⋊(C7⋊C8)—31no—
29#6384—31no—
30#6383—31no—
31#5239—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×C3×D7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7C2311yescharacteristic
38C2×C7×Dic3—21no—
39C2×C7×Dic3—21no—
40C7×Dic6—21no—
41C7×Dic6—21no—
42C3×C22⋊D7—24no—
43C2×C3×Dic7—21no—
44C3×D28—21no—
45C3×D28D412yes—
46C3×C4×D7—24no—
47C3×Dic14—21no—
48C3×Dic14D412yes—
49C3⋊(C7⋊C8)—21no—
50C3⋊(C7⋊C8)—21no—
51C2×C7×Q8—31no—
52C22×C4×C7C2×D311yescharacteristic
53C2×C22⋊D7—21no—
54#1157—22no—
55#1157—22no—
56C2×D28C2×D311yescharacteristic
57C2×C4×D7—21no—
58C2×Dic14C2×D311yescharacteristic
59C7×C4⋊C4—31no—
60C7×C4⋊C4—61no—
61C7×C22⋊C4—61no—
62#1144—62no—
63#1143—62no—
64C2×C7⋊C8—31no—
65C2×C7⋊C8—31no—
66C2×C3×#41—71no—
67C2×#216—71no—
68C2×#214—71no—
69#924D711yescharacteristic, complemented
70#835—71no—
71#832—71no—
72#830—71no—
73C22×C3×C7—21no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7C2×D411yescharacteristic
77C2×C3×D7—22no—
78C2×C3×D7—41no—
79C2×C3×D7—41no—
80C3×C4×C7—21no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83C3×Dic7—22no—
84C7×Dic3—41no—
85C7×Dic3—41no—
86#452—211noSylow
87C23×C7#6711yescharacteristic
88C22×D7—21no—
89C7×Q8—61no—
90C7×Q8—61no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—61no—
94C2×C4×C7—61no—
95C2×C4×C7#6711yescharacteristic
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—24no—
98C2×Dic7—21no—
99D28—21no—
100D28#6712yes—
101C4×D7—24no—
102Dic14—21no—
103Dic14#6712yes—
104C7⋊C8—61no—
105C7⋊C8—61no—
106C3×#41—142no—
107C3×#41—142no—
108C2×C3×Q8—71no—
109C2×C3×D4—71no—
110C2×C3×D4—141no—
111C22×C3×C4—141no—
112C22×C3×C4C2×D711yescharacteristic
113C2×Dic6C2×D711yescharacteristic
114#217—21no—
115#216—142no—
116#214—142no—
117C4⋊Dic3C2×D711yescharacteristic
118Dic3⋊C4—21no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8—71no—
121C2×C3×C7—41no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125C3×D7—42no—
126C2×#41—71no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8—31no—
130#103—211no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7—21no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7C2×#6711yescharacteristic
137C2×D7—22no—
138C2×D7—41no—
139C2×D7—41no—
140C4×C7—21no—
141C4×C7—121no—
142C4×C7—121no—
143C4×C7C2×#6711yescharacteristic
144C4×C7C2×#6711yescharacteristic
145Dic7—22no—
146C23×C3—141no—
147C23×C3C22⋊D711yescharacteristic
148C3×Q8—72no—
149C3×Q8—141no—
150C3×D4—72no—
151C3×D4—144no—
152C3×D4—141no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—144no—
156C2×C3×C4—141no—
157C2×C3×C4C22⋊D711yescharacteristic
158C2×C3×C4C22×D711yescharacteristic
159C2×Dic3—21no—
160C2×Dic3—21no—
161Dic6—21no—
162Dic6—21no—
163C3⋊C8—141no—
164C3⋊C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—142no—
167#41—142no—
168C2×Q8—31no—
169C2×Q8—71no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—141no—
173C22×C4D3×D711yescharacteristic
174Q16—422no—
175SD16—422no—
176C2×C8—211no—
177C2×C8—211no—
178C4⋊C4—31no—
179C4⋊C4—61no—
180C22⋊C4—61no—
181C2×C7—41no—
182C2×C7#95011yescharacteristic
183C2×C7#95111yescharacteristic
184C2×C7#95311yescharacteristic
185D7—42no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—142no—
190C22×C3—281no—
191C22×C3—281no—
192C22×C3C2×C22⋊D711yescharacteristic
193C3×C4—21no—
194C3×C4—142no—
195C3×C4D4×D711yescharacteristic
196C3×C4D4×D711yescharacteristic
197Dic3—41no—
198Dic3—41no—
199C23—141no—
200C23#424511yescharacteristic
201Q8—61no—
202Q8—61no—
203Q8—72no—
204Q8—141no—
205D4—72no—
206D4—144no—
207D4—141no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—144no—
213C2×C4—141no—
214C2×C4#424511yescharacteristic
215C2×C4C2×D3×D711yescharacteristic
216C8—421no—
217C8—421no—
218C7#523911yescharacteristic, complemented, Sylow
219C2×C3—41no—
220C2×C3—282no—
221C2×C3#650011yescharacteristic
222C2×C3#650511yescharacteristic
223C2×C3#651311yescharacteristic
224C22—21no—
225C22—21no—
226C22—21no—
227C22—142no—
228C22—281no—
229C22—281no—
230C22C2×#424511yescharacteristic
231C4—21no—
232C4—121no—
233C4—121no—
234C4—142no—
235C4D7×#6711yescharacteristic
236C4D7×#6711yescharacteristic
237C3#8981911yescharacteristic, complemented, Sylow
238C2—41no—
239C2—282no—
240C2#1062085911yescharacteristic
241C2#1062086911yescharacteristic
242C2#1062112511yescharacteristic
subgroupG/H
centerC22C2×#4245
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1157C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C7×#924C2semidirect
3. C2×#66569C2semidirect
4. C2×#66563C2semidirect
5. #10620669C2semidirect
6. #10620505C2semidirect
7. #10620499C2semidirect
69. #924D7semidirect
165. C3×C7#452semidirect
218. C7#5239semidirect
237. C3#89819semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11572
C7×#9242
C2×#665692
C2×#665632
#106206692
#106205052
#106204992
#898193
#52397