Finite Group Databasebeta

← #13056697

Subgroups — #13056697

Order 1344.

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

normal subgroups only

This group has 224 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×#124C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic, complemented
3C2×#66633C211yesmaximal, characteristic
4C4×Dic42C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620671C211yesmaximal, characteristic
7C4⋊(C3⋊(C7⋊C8))C211yesmaximal, characteristic, complemented
8#89826—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×C7×D4C2211yescharacteristic
11C2×Dic42C2211yescharacteristic
12C3×C7×C22⋊C4—21no—
13C3×C7×C22⋊C4—21no—
14C3×C42×C7C2211yescharacteristic
15#66635—22no—
16#66633—22no—
17C4⋊Dic21C2211yescharacteristic
18#66629—21no—
19C4×Dic21—21no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C7×#124D311yescharacteristic
23C2×#1144—31no—
24C2×#1142—31no—
25C4×Dic14—31no—
26#6409—31no—
27#6406—31no—
28C4⋊(C7⋊C8)—31no—
29#5054—71nomaximal, Hall
30C23×C3×C7—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7D411yescharacteristic
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21—21no—
40C2×Dic21—21no—
41Dic42—21no—
42Dic42D412yes—
43C3⋊(C7⋊C8)—21no—
44C3⋊(C7⋊C8)—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×C7×D4C2×D311yescharacteristic
47C2×Dic14—31no—
48C7×C22⋊C4—21no—
49C7×C22⋊C4—21no—
50C42×C7C2×D311yescharacteristic
51#1144—62no—
52#1142—62no—
53C4⋊Dic7—31no—
54Dic7⋊C4—61no—
55C4×Dic7—61no—
56C2×C7⋊C8—31no—
57C2×C7⋊C8—31no—
58C3×#124D711yescharacteristic
59C2×#216—71no—
60C2×#214—71no—
61C4×Dic6—71no—
62#835—71no—
63#832—71no—
64C4⋊(C3⋊C8)—71no—
65C22×C3×C7—41no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7C2×D411yescharacteristic
69C3×C4×C7—21no—
70C3×C4×C7—21no—
71C3×C4×C7—41no—
72C3×C4×C7C2×D411yescharacteristic
73C3×C4×C7#4111yescharacteristic
74Dic21—22no—
75Dic21—41no—
76#464—211noSylow
77C23×C7—21no—
78C7×Q8—21no—
79C7×Q8—21no—
80C7×D4—21no—
81C7×D4—21no—
82C2×C4×C7—21no—
83C2×C4×C7#6711yescharacteristic
84C2×C4×C7#6711yescharacteristic
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—61no—
87C2×Dic7—61no—
88Dic14—32no—
89Dic14—61no—
90C7⋊C8—61no—
91C7⋊C8—61no—
92C2×C3×Q8C2×D711yescharacteristic
93C2×C3×D4C2×D711yescharacteristic
94C2×Dic6—71no—
95C3×C22⋊C4—21no—
96C3×C22⋊C4—21no—
97C3×C42C2×D711yescharacteristic
98#216—142no—
99#214—142no—
100C4⋊Dic3—71no—
101Dic3⋊C4—141no—
102C4×Dic3—141no—
103C2×C3⋊C8—71no—
104C2×C3⋊C8—71no—
105C2×C3×C7—81no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7#13511yescharacteristic
108C2×C3×C7#13711yescharacteristic
109C2×Q16—211no—
110C2×SD16—211no—
111#124D2111yescharacteristic
112C4×Q8—211no—
113C4⋊C8—211no—
114#103—211no—
115#102—211no—
116C22×C7—41no—
117C22×C7—41no—
118C22×C7—41no—
119C22×C7C2×#6711yescharacteristic
120C4×C7—21no—
121C4×C7—21no—
122C4×C7—41no—
123C4×C7D3×D411yescharacteristic
124C4×C7#23711yescharacteristic
125Dic7—62no—
126Dic7—121no—
127C23×C3—21no—
128C3×Q8—21no—
129C3×Q8—21no—
130C3×D4—21no—
131C3×D4—21no—
132C2×C3×C4—21no—
133C2×C3×C4C22⋊D711yescharacteristic
134C2×C3×C4C22⋊D711yescharacteristic
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—141no—
137C2×Dic3—141no—
138Dic6—72no—
139Dic6—141no—
140C3⋊C8—141no—
141C3⋊C8—141no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—211no—
144C2×Q8C2×D2111yescharacteristic
145C2×D4C2×D2111yescharacteristic
146Q16—422no—
147SD16—422no—
148C2×C8—211no—
149C2×C8—211no—
150C4⋊C4—211no—
151C4⋊C4—421no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C42—421no—
155C42C2×D2111yescharacteristic
156C2×C7—81no—
157C2×C7#93811yescharacteristic
158C2×C7#95011yescharacteristic
159C2×C7#95111yescharacteristic
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3C2×C22⋊D711yescharacteristic
164C3×C4—21no—
165C3×C4—21no—
166C3×C4—41no—
167C3×C4D4×D711yescharacteristic
168C3×C4#115911yescharacteristic
169Dic3—142no—
170Dic3—281no—
171C23—21no—
172Q8—21no—
173Q8—21no—
174Q8—212no—
175Q8—421no—
176D4—21no—
177D4—21no—
178C2×C4—21no—
179C2×C4—421no—
180C2×C4—421no—
181C2×C4#426811yescharacteristic
182C2×C4#426811yescharacteristic
183C2×C4C22×D2111yescharacteristic
184C8—421no—
185C8—421no—
186C7#505411yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3#650111yescharacteristic
189C2×C3#651311yescharacteristic
190C2×C3#651411yescharacteristic
191C22—41no—
192C22—41no—
193C22—41no—
194C22C2×#426811yescharacteristic
195C4—21no—
196C4—21no—
197C4—41no—
198C4—422no—
199C4—841no—
200C4D4×D2111yescharacteristic
201C4#6673011yescharacteristic
202C3#8982611yescharacteristic, complemented, Sylow
203C2—81no—
204C2#1062142111yescharacteristic
205C2#1062143311yescharacteristic
206C2#1062143411yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#124C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×C7×D4C22
hypercenter#124D21
hypocenterC3×C7#464
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. C2×#66635C2semidirect
4. C4×Dic42C2semidirect
5. #10620674C2semidirect
7. C4⋊(C3⋊(C7⋊C8))C2semidirect
142. C3×C7#464semidirect
186. C7#5054semidirect
202. C3#89826semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1242
C2×#666352
C2×#666332
C4×Dic422
#106206742
#106206712
C4⋊(C3⋊(C7⋊C8))2
#898263
#50547