Finite Group Databasebeta

← #13054656

Subgroups — #13054656

Order 1344.

296 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 70 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×Q8×D21C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620952C211yesmaximal, characteristic
4C2×#66570C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic
8#89580—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10Q8×D21—22no—
11Q8×D21—22no—
12C2×C4×D21—21no—
13C2×C4×D21C2211yescharacteristic
14C2×Dic42—21no—
15C2×Dic42C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C7×C4⋊Dic3C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×#1144—22no—
20C2×C3×C7⋊C8C2211yescharacteristic
21Dic3⋊Dic7—21no—
22#66575—21no—
23#66570—22no—
24C2×Q8×D7—31no—
25C2×#1144D311yescharacteristic, complemented
26C2×Dic28—31no—
27#6460—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5186—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8D412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×Dic21D411yescharacteristic
39C4×D21—24no—
40C4×D21—21no—
41C4×D21—21no—
42Dic42—24no—
43Dic42—21no—
44Dic42D412yes—
45C2×C7×Dic3—21no—
46C2×C3×Dic7—21no—
47C3×Dic14—21no—
48C3×Dic14—21no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8C2×D311yescharacteristic
52Q8×D7—62no—
53Q8×D7—62no—
54C2×C4×D7—31no—
55C2×C4×D7—61no—
56C2×Dic14—31no—
57C2×Dic14—61no—
58C2×Dic14C2×D311yescharacteristic
59C2×C7×C8—31no—
60C7×C4⋊C4—31no—
61#1144—22no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8C2×D311yescharacteristic
65Dic28—62no—
66C2×D3×Q8—71no—
67C2×C3×Q16—71no—
68#946—71no—
69C2×#216—71no—
70#835D711yescharacteristic, complemented
71(C2×D3)⋊C8—71no—
72#816—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21Q1612yescomplemented
77C3×C4×C7—22no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—21no—
81Dic21—22no—
82C3×Dic7—41no—
83C7×Dic3—41no—
84#451—211noSylow
85C22×D7—31no—
86C7×Q8—21no—
87C7×Q8#6712yes—
88C2×C4×C7—21no—
89C2×C4×C7—61no—
90C2×C4×C7C22×D311yescharacteristic
91C2×Dic7—21no—
92C2×Dic7—31no—
93C2×Dic7—61no—
94C4×D7—64no—
95C4×D7—61no—
96C4×D7—61no—
97Dic14—21no—
98Dic14—21no—
99Dic14—32no—
100Dic14—64no—
101Dic14—61no—
102C7×C8—61no—
103C7⋊C8—21no—
104C2×C3×Q8—71no—
105C2×C3×Q8C2×D711yescharacteristic
106D3×Q8—142no—
107D3×Q8—142no—
108C2×C4×D3—71no—
109C2×C4×D3—141no—
110C2×Dic6—71no—
111C2×Dic6—141no—
112C3×Q16—142no—
113C2×C3×C8—71no—
114#216—142no—
115(C2×D3)⋊C4—141no—
116C4⋊Dic3C2×D711yescharacteristic
117Dic3⋊C4—141no—
118C2×C3⋊C8C2×D711yescharacteristic
119C2×C3×C7#12011yescharacteristic
120C2×C3×C7C2×Q1611yescharacteristic
121C2×C3×C7#13711yescharacteristic
122D21—22no—
123C22×Q8—211no—
124C2×Q16—71no—
125C2×Q16—211no—
126C22⋊Q8—211no—
127#103—31no—
128#103—211no—
129C22⋊C8—211no—
130C22×C7D3×D411yescharacteristic
131C2×D7—32no—
132C2×D7—61no—
133C2×D7—61no—
134C4×C7—22no—
135C4×C7—121no—
136C4×C7D3×D411yescharacteristic
137C4×C7C2×#6711yescharacteristic
138Dic7—41no—
139Dic7—62no—
140Dic7—61no—
141C22×D3—71no—
142C3×Q8—21no—
143C3×Q8—141no—
144C3×Q8—141no—
145C3×Q8D2812yes—
146C2×C3×C4—21no—
147C2×C3×C4—141no—
148C2×C3×C4C22×D711yescharacteristic
149C2×Dic3—21no—
150C2×Dic3—71no—
151C2×Dic3—141no—
152C4×D3—144no—
153C4×D3—141no—
154C4×D3—141no—
155Dic6—72no—
156Dic6—144no—
157Dic6—141no—
158C3×C8—141no—
159C3⋊C8—21no—
160C3×C7#45111yescharacteristic, complemented, Hall
161C2×Q8—71no—
162C2×Q8—211no—
163C2×Q8—421no—
164C2×Q8—422no—
165C2×Q8—422no—
166C2×Q8D3×D711yescharacteristic
167C22×C4—211no—
168C22×C4—421no—
169Q16—142no—
170Q16—422no—
171C2×C8—31no—
172C2×C8—71no—
173C4⋊C4—31no—
174C4⋊C4—421no—
175C22⋊C4—421no—
176C2×C7D3×Q1611yescharacteristic
177C2×C7#91811yescharacteristic
178C2×C7#93711yescharacteristic
179D7—62no—
180C22×C3D4×D711yescharacteristic
181C2×D3—72no—
182C2×D3—141no—
183C2×D3—141no—
184C3×C4—22no—
185C3×C4—281no—
186C3×C4C2×D2811yescharacteristic
187C3×C4D4×D711yescharacteristic
188Dic3—41no—
189Dic3—142no—
190Dic3—141no—
191C23—211no—
192Q8—21no—
193Q8—141no—
194Q8—141no—
195Q8—212no—
196Q8—424no—
197Q8—421no—
198Q8#424612yes—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—141no—
202C2×C4—211no—
203C2×C4—424no—
204C2×C4—421no—
205C2×C4—421no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—141no—
210C7#518611yescharacteristic, complemented, Sylow
211C2×C3#644511yescharacteristic
212C2×C3#647811yescharacteristic
213C2×C3D7×Q1611yescharacteristic
214D3—142no—
215C22—212no—
216C22—421no—
217C22—421no—
218C22#6669411yescharacteristic
219C4—22no—
220C4—121no—
221C4—281no—
222C4—421no—
223C4—422no—
224C4C2×#424611yescharacteristic
225C4#6669411yescharacteristic
226C3#8958011yescharacteristic, complemented, Sylow
227C2—422no—
228C2#1062107311yescharacteristic
229C2#1062108611yescharacteristic
230C2#1062113411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#451
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×C3×#1144C2semidirect
4. C2×#66570C2semidirect
5. C7×#835C2semidirect
6. #10620510C2semidirect
25. C2×#1144D3semidirect
70. #835D7semidirect
76. C2×D21Q16semidirect
160. C3×C7#451semidirect
210. C7#5186semidirect
226. C3#89580semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D212
C2×C3×#11442
#106209522
C2×#665702
C7×#8352
#106205102
#106204922
#895803
#51867