Finite Group Databasebeta

← #13052870

Subgroups — #13052870

Order 1344.

300 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 78 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×D21⋊Q8C211yesmaximal, characteristic
2#10621375C211yesmaximal, characteristic
3C2×#66559C211yesmaximal, characteristic
4C7×#810C211yesmaximal, characteristic, complemented
5C3×#6384C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89574—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C2×#4248—21no—
13C2×D21⋊C4—21no—
14D21⋊Q8—22no—
15D21⋊Q8C414yescomplemented
16C3×C7×C4⋊C4C2211yescharacteristic
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19C4×Dic21—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66559—22no—
23C2×Q8×D7—31no—
24C2×#1131—31no—
25#6455—31no—
26C7×#103—31no—
27#6409—31no—
28#6390—31no—
29#6384D311yescharacteristic, complemented
30#4802—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C4×D21C2×C412yes—
37C4×D21D412yes—
38C2×C7×Dic3—21no—
39C7×Dic6—21no—
40C7×Dic6C2×C412yes—
41C2×C3×Dic7—21no—
42C3×Dic14—21no—
43C3×Dic14C2×C412yes—
44#4248—24no—
45D21⋊C4—24no—
46C3×C7⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8—31no—
49Q8×D7—34no—
50Q8×D7—62no—
51C2×C4×D7—31no—
52C2×C4×D7—61no—
53C2×Dic14—61no—
54C2×Dic14C2×D311yescharacteristic
55C2×C7×C8—31no—
56C7×C4⋊C4C2×D311yescharacteristic
57(C2×D7)⋊C4—61no—
58C4⋊Dic7—31no—
59C4×Dic7—61no—
60C2×C7⋊C8C2×D311yescharacteristic
61#1131—62no—
62C2×D3×Q8—71no—
63C2×ΓL(1,𝔽25)—71no—
64#892—71no—
65C3×#103—71no—
66#835—71no—
67#816—71no—
68#810D711yescharacteristic, complemented
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—21no—
71C2×D21—21no—
72C2×D21C22⋊C412yes—
73C3×C4×C7—41no—
74C3×C4×C7C22×C411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—41no—
77Dic21C22⋊C412yes—
78C3×Dic7—22no—
79C7×Dic3—22no—
80#419—211noSylow
81C22×D7—31no—
82C7×Q8—32no—
83C7×Q8—61no—
84C2×C4×C7—21no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—21no—
88C2×Dic7—31no—
89C2×Dic7—61no—
90C4×D7—32no—
91C4×D7—32no—
92C4×D7—64no—
93Dic14—21no—
94Dic14—64no—
95Dic14C4×D312yes—
96C7×C8—61no—
97C7⋊C8—21no—
98C2×C3×Q8—71no—
99D3×Q8—74no—
100D3×Q8—142no—
101C2×C4×D3—71no—
102C2×C4×D3—141no—
103C2×Dic6—141no—
104C2×Dic6C2×D711yescharacteristic
105C2×C3×C8—71no—
106C3×C4⋊C4C2×D711yescharacteristic
107(C2×D3)⋊C4—141no—
108C4⋊Dic3—71no—
109C4×Dic3—141no—
110C2×C3⋊C8C2×D711yescharacteristic
111ΓL(1,𝔽25)—142no—
112C2×C3×C7C2×C22⋊C411yescharacteristic
113C2×C3×C7#13711yescharacteristic
114C2×C3×C7#13711yescharacteristic
115D21—22no—
116C22×Q8—211no—
117C2×M16—211no—
118#117—211no—
119#103—31no—
120#103—71no—
121#103—211no—
122#103—211no—
123C22×C7D3×D411yescharacteristic
124C2×D7—32no—
125C2×D7—61no—
126C2×D7—61no—
127C4×C7—41no—
128C4×C7—62no—
129C4×C7C2×C4×D311yescharacteristic
130C4×C7D3×D411yescharacteristic
131Dic7—22no—
132Dic7—32no—
133Dic7—121no—
134C22×D3—71no—
135C3×Q8—72no—
136C3×Q8—141no—
137C2×C3×C4—21no—
138C2×C3×C4—141no—
139C2×C3×C4C22×D711yescharacteristic
140C2×Dic3—21no—
141C2×Dic3—71no—
142C2×Dic3—141no—
143C4×D3—72no—
144C4×D3—72no—
145C4×D3—144no—
146Dic6—21no—
147Dic6—144no—
148Dic6C4×D712yes—
149C3×C8—141no—
150C3⋊C8—21no—
151C3×C7#41911yescharacteristic, complemented, Hall
152C2×Q8—31no—
153C2×Q8—71no—
154C2×Q8—214no—
155C2×Q8—421no—
156C2×Q8—422no—
157C22×C4—211no—
158C22×C4—421no—
159M16—422no—
160C2×C8—31no—
161C2×C8—71no—
162C4⋊C4—211no—
163C4⋊C4D3×D711yescharacteristic
164C22⋊C4—421no—
165C42—421no—
166C2×C7D3×C22⋊C411yescharacteristic
167C2×C7#91511yescharacteristic
168C2×C7#91811yescharacteristic
169D7—62no—
170C22×C3D4×D711yescharacteristic
171C2×D3—72no—
172C2×D3—141no—
173C2×D3—141no—
174C3×C4—41no—
175C3×C4—142no—
176C3×C4C2×C4×D711yescharacteristic
177C3×C4D4×D711yescharacteristic
178Dic3—22no—
179Dic3—72no—
180Dic3—281no—
181C23—211no—
182Q8—32no—
183Q8—61no—
184Q8—72no—
185Q8—141no—
186Q8—424no—
187C2×C4—21no—
188C2×C4—61no—
189C2×C4—141no—
190C2×C4—211no—
191C2×C4—212no—
192C2×C4—212no—
193C2×C4—424no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—141no—
198C7#480211yescharacteristic, complemented, Sylow
199C2×C3D7×C22⋊C411yescharacteristic
200C2×C3#647811yescharacteristic
201C2×C3#648111yescharacteristic
202D3—142no—
203C22—212no—
204C22—421no—
205C22—421no—
206C22#6669411yescharacteristic
207C4—41no—
208C4—62no—
209C4—142no—
210C4—212no—
211C4—841no—
212C4C4×D3×D711yescharacteristic
213C4#6668111yescharacteristic
214C3#8957411yescharacteristic, complemented, Sylow
215C2—422no—
216C2#1062103511yescharacteristic
217C2#1062105011yescharacteristic
218C2#1062107411yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#419
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
4. C7×#810C2semidirect
5. C3×#6384C2semidirect
6. #10620508C2semidirect
7. #10620506C2semidirect
15. D21⋊Q8C4semidirect
29. #6384D3semidirect
68. #810D7semidirect
151. C3×C7#419semidirect
198. C7#4802semidirect
214. C3#89574semidirect

Maximal subgroups

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

subgroupindex
C2×D21⋊Q82
#106213752
C2×#665592
C7×#8102
C3×#63842
#106205082
#106205062
#895743
#48027