Finite Group Databasebeta

← #13056676

Subgroups — #13056676

Order 1344.

240 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C3×C7×#121C211yesmaximal, characteristic
2C2×#66633C211yesmaximal, characteristic, complemented
3#10621407C211yesmaximal, characteristic, complemented
4C2×#66627C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89814—31nomaximal, Hall
9C2×C3×C7×D4—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C2×Dic42C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×C7×C22⋊C4—21no—
15#66636—21no—
16#66633—22no—
17C4⋊Dic21C2211yescharacteristic
18#66629—21no—
19#66627—22no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C7×#121D311yescharacteristic
23C2×#1142—31no—
24#6487—31no—
25C2×#1136—31no—
26#6406—31no—
27#6384—31no—
28#6381—31no—
29#5042—71nomaximal, Hall
30C23×C3×C7—21no—
31C23×C3×C7D411yescharacteristic
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C3×C7×D4—42no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7D411yescharacteristic
39C2×C3×C4×C7C2311yescharacteristic
40C2×Dic21—21no—
41C2×Dic21—21no—
42Dic42—21no—
43Dic42—21no—
44C3⋊(C7⋊C8)—21no—
45C3⋊(C7⋊C8)D412yescomplemented
46C2×C7×D4—21no—
47C2×C7×D4C2×D311yescharacteristic
48C22×C4×C7C2×D311yescharacteristic
49C2×Dic14—31no—
50C7×C4⋊C4C2×D311yescharacteristic
51C7×C22⋊C4—21no—
52C22⋊Dic7—61no—
53#1142—62no—
54C4⋊Dic7—31no—
55Dic7⋊C4—61no—
56#1136—62no—
57C2×C7⋊C8—31no—
58C2×C7⋊C8—31no—
59C3×#121D711yescharacteristic
60C2×#214—71no—
61#924—71no—
62C2×#208—71no—
63#832—71no—
64#810—71no—
65#807—71no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7—41no—
70C22×C3×C7—41no—
71C22×C3×C7—41no—
72C22×C3×C7C2×D411yescharacteristic
73C3×C4×C7—21no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7#4111yescharacteristic
77Dic21—41no—
78Dic21—41no—
79#468—211noSylow
80C23×C7—21no—
81C23×C7#6711yescharacteristic
82C7×D4—21no—
83C7×D4—21no—
84C7×D4—42no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7#6711yescharacteristic
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—61no—
91C2×Dic7—61no—
92Dic14—61no—
93Dic14—61no—
94C7⋊C8—32no—
95C7⋊C8—61no—
96C2×C3×D4—21no—
97C2×C3×D4C2×D711yescharacteristic
98C22×C3×C4C2×D711yescharacteristic
99C2×Dic6—71no—
100C3×C4⋊C4C2×D711yescharacteristic
101C3×C22⋊C4—21no—
102#217—141no—
103#214—142no—
104C4⋊Dic3—71no—
105Dic3⋊C4—141no—
106#208—142no—
107C2×C3⋊C8—71no—
108C2×C3⋊C8—71no—
109C2×C3×C7—41no—
110C2×C3×C7—81no—
111C2×C3×C7#12111yescharacteristic
112C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
113C2×C3×C7#13711yescharacteristic
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121D2111yescharacteristic
118#107—211no—
119#103—211no—
120#102—211no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7—21no—
124C22×C7—41no—
125C22×C7—41no—
126C22×C7—41no—
127C22×C7C2×#6711yescharacteristic
128C4×C7—21no—
129C4×C7—41no—
130C4×C7D3×D411yescharacteristic
131C4×C7#23711yescharacteristic
132Dic7—121no—
133Dic7—121no—
134C23×C3—21no—
135C23×C3C22⋊D711yescharacteristic
136C3×D4—21no—
137C3×D4—21no—
138C3×D4—42no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4C22⋊D711yescharacteristic
143C2×C3×C4C22×D711yescharacteristic
144C2×Dic3—141no—
145C2×Dic3—141no—
146Dic6—141no—
147Dic6—141no—
148C3⋊C8—72no—
149C3⋊C8—141no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—21no—
153C2×D4C2×D2111yescharacteristic
154C22×C4C2×D2111yescharacteristic
155SD16—422no—
156M16—422no—
157C2×C8—211no—
158C2×C8—211no—
159C4⋊C4—211no—
160C4⋊C4—421no—
161C4⋊C4C2×D2111yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—421no—
164C2×C7—41no—
165C2×C7—81no—
166C2×C7#93211yescharacteristic
167C2×C7#93911yescharacteristic
168C2×C7#95111yescharacteristic
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3C2×C22⋊D711yescharacteristic
176C3×C4—21no—
177C3×C4—41no—
178C3×C4D4×D711yescharacteristic
179C3×C4#115911yescharacteristic
180Dic3—281no—
181Dic3—281no—
182C23—21no—
183C23#426811yescharacteristic
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—42no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4#426811yescharacteristic
195C2×C4C22×D2111yescharacteristic
196C8—212no—
197C8—421no—
198C7#504211yescharacteristic, complemented, Sylow
199C2×C3—41no—
200C2×C3—81no—
201C2×C3#649511yescharacteristic
202C2×C3#650211yescharacteristic
203C2×C3#651411yescharacteristic
204C22—21no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22C2×#426811yescharacteristic
211C4—21no—
212C4—41no—
213C4—841no—
214C4—841no—
215C4D4×D2111yescharacteristic
216C4#6673011yescharacteristic
217C3#8981411yescharacteristic, complemented, Sylow
218C2—41no—
219C2—81no—
220C2#1062141511yescharacteristic
221C2#1062142211yescharacteristic
222C2#1062143411yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
Fitting, prime radicalC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#468
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×#66633C2semidirect
3. #10621407C2semidirect
4. C2×#66627C2semidirect
5. #10620671C2semidirect
6. #10620649C2semidirect
7. #10620646C2semidirect
45. C3⋊(C7⋊C8)D4semidirect
150. C3×C7#468semidirect
198. C7#5042semidirect
217. C3#89814semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
C2×#666332
#106214072
C2×#666272
#106206712
#106206492
#106206462
#898143
#50427