Finite Group Databasebeta

← #13054582

Subgroups — #13054582

Order 1344.

300 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 96 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×D3×C7×Q8C211yesmaximal, characteristic
2#10620900C211yesmaximal, characteristic
3C2×#66558C211yesmaximal, characteristic
4#10620674C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620507C211yesmaximal, characteristic, complemented
8#89997—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10D3×C7×Q8—22no—
11D3×C7×Q8C414yescomplemented
12C2×C4×D3×C7—21no—
13C2×C4×D3×C7C2211yescharacteristic
14C2×C7×Dic6—21no—
15C2×C7×Dic6C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17C2×C3⋊(C7⋊C8)C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21Dic3×Dic7—21no—
22#66558—22no—
23C22×C7×Q8—31no—
24#6489—31no—
25C2×#1136—31no—
26#6409—31no—
27#6409—31no—
28#6409—31no—
29#6409D311yescharacteristic, complemented
30#4802—71nomaximal, Hall
31C22×D3×C7D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8C2×C412yes—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×C7×Dic3—21no—
38C2×C7×Dic3D411yescharacteristic
39C4×D3×C7—24no—
40C4×D3×C7C2×C412yes—
41C4×D3×C7D412yes—
42C7×Dic6—21no—
43C7×Dic6—24no—
44C7×Dic6C2×C412yes—
45C2×C3×Dic7—21no—
46C3⋊(C7⋊C8)—21no—
47C3×C7⋊C8—21no—
48C2×C7×Q8—34no—
49C2×C7×Q8—31no—
50C2×C7×Q8—61no—
51C2×C7×Q8—62no—
52C2×C7×Q8C2×D311yescharacteristic
53C22×C4×C7—31no—
54C22×C4×C7—61no—
55C22⋊Dic7—61no—
56C4⋊Dic7—31no—
57C4⋊Dic7C2×D311yescharacteristic
58C4×Dic7—61no—
59#1136—62no—
60C2×C7⋊C8—31no—
61C2×C7⋊C8C2×D311yescharacteristic
62C2×D3×Q8D711yescharacteristic
63C2×ΓL(1,𝔽25)—71no—
64#892—71no—
65C3×#103—71no—
66#835—71no—
67#816—71no—
68#810—71no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D3×C7—21no—
71C2×D3×C7—21no—
72C2×D3×C7C22⋊C412yes—
73C3×C4×C7—22no—
74C3×C4×C7C22×C411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—41no—
77C3×Dic7—41no—
78C7×Dic3—22no—
79C7×Dic3C22⋊C412yes—
80#419—211noSylow
81C23×C7—31no—
82C7×Q8—21no—
83C7×Q8—32no—
84C7×Q8—64no—
85C7×Q8—61no—
86C7×Q8C4×D312yes—
87C2×C4×C7—21no—
88C2×C4×C7—31no—
89C2×C4×C7—32no—
90C2×C4×C7—32no—
91C2×C4×C7—64no—
92C2×C4×C7—61no—
93C2×C4×C7C22×D311yescharacteristic
94C2×Dic7—21no—
95C2×Dic7—61no—
96C7⋊C8—21no—
97C7⋊C8—61no—
98C2×C3×Q8C2×D711yescharacteristic
99D3×Q8—22no—
100D3×Q8Dic714yescomplemented
101C2×C4×D3—21no—
102C2×C4×D3C2×D711yescharacteristic
103C2×Dic6—21no—
104C2×Dic6C2×D711yescharacteristic
105C2×C3×C8—71no—
106C3×C4⋊C4—71no—
107(C2×D3)⋊C4—141no—
108C4⋊Dic3—71no—
109C4×Dic3—141no—
110C2×C3⋊C8—71no—
111ΓL(1,𝔽25)—142no—
112C2×C3×C7C2×C22⋊C411yescharacteristic
113C2×C3×C7#13711yescharacteristic
114C2×C3×C7#13711yescharacteristic
115D3×C7—22no—
116C22×Q8—31no—
117C2×M16—211no—
118#117—211no—
119#103—71no—
120#103—211no—
121#103—211no—
122#103—211no—
123C22×C7—32no—
124C22×C7—61no—
125C22×C7—61no—
126C22×C7D3×D411yescharacteristic
127C4×C7—22no—
128C4×C7—32no—
129C4×C7—62no—
130C4×C7C2×C4×D311yescharacteristic
131C4×C7D3×D411yescharacteristic
132Dic7—41no—
133Dic7—121no—
134C22×D3C22⋊D711yescharacteristic
135C3×Q8—21no—
136C3×Q8C2×Dic712yes—
137C2×C3×C4—21no—
138C2×C3×C4—141no—
139C2×C3×C4C22×D711yescharacteristic
140C2×Dic3—21no—
141C2×Dic3—141no—
142C2×Dic3C22⋊D711yescharacteristic
143C4×D3—24no—
144C4×D3C2×Dic712yes—
145C4×D3C22⋊D712yes—
146Dic6—21no—
147Dic6—24no—
148Dic6C2×Dic712yes—
149C3×C8—141no—
150C3⋊C8—141no—
151C3×C7#41911yescharacteristic, complemented, Hall
152C2×Q8—34no—
153C2×Q8—31no—
154C2×Q8—61no—
155C2×Q8—62no—
156C2×Q8D3×D711yescharacteristic
157C22×C4—31no—
158C22×C4—61no—
159M16—422no—
160C2×C8—71no—
161C2×C8—211no—
162C4⋊C4—71no—
163C4⋊C4—211no—
164C22⋊C4—421no—
165C42—421no—
166C2×C7—62no—
167C2×C7D3×C22⋊C411yescharacteristic
168C2×C7#91511yescharacteristic
169C2×C7#91811yescharacteristic
170C22×C3C2×C22⋊D711yescharacteristic
171C2×D3—21no—
172C2×D3—21no—
173C2×D3C22⋊Dic712yes—
174C3×C4—22no—
175C3×C4—281no—
176C3×C4C22×Dic711yescharacteristic
177C3×C4C2×C22⋊D711yescharacteristic
178Dic3—22no—
179Dic3—281no—
180Dic3C22⋊Dic712yes—
181C23—31no—
182Q8—21no—
183Q8—32no—
184Q8—64no—
185Q8—61no—
186Q8D3×Dic712yes—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—32no—
190C2×C4—32no—
191C2×C4—64no—
192C2×C4—61no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—141no—
197C8—421no—
198C7#480211yescharacteristic, complemented, Sylow
199C2×C3#650511yescharacteristic
200C2×C3#650511yescharacteristic
201C2×C3C2×C22⋊Dic711yescharacteristic
202D3—22no—
203C22—32no—
204C22—61no—
205C22—61no—
206C22D3×C22⋊D711yescharacteristic
207C4—22no—
208C4—32no—
209C4—62no—
210C4—281no—
211C4—841no—
212C4C2×D3×Dic711yescharacteristic
213C4D3×C22⋊D711yescharacteristic
214C3#8999711yescharacteristic, complemented, Sylow
215C2—62no—
216C2#1062108011yescharacteristic
217C2#1062108211yescharacteristic
218C2D3×C22⋊Dic711yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D3×C7D4
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7#419
LCSG ▷ 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
4. #10620674C2semidirect
5. C3×#6409C2semidirect
6. #10620508C2semidirect
7. #10620507C2semidirect
11. D3×C7×Q8C4semidirect
29. #6409D3semidirect
100. D3×Q8Dic7semidirect
151. C3×C7#419semidirect
198. C7#4802semidirect
214. C3#89997semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C7×Q82
#106209002
C2×#665582
#106206742
C3×#64092
#106205082
#106205072
#899973
#48027