Finite Group Databasebeta

← #13056423

Subgroups — #13056423

Order 1344.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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
1D4×Dic21C211yesmaximal, characteristic
2C2×#66633C211yesmaximal, characteristic
3#10621370C211yesmaximal, characteristic, complemented
4C3×C7×#102C211yesmaximal, characteristic
5#10620655C211yesmaximal, characteristic, complemented
6#10620654C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89529—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C22×Dic21—21no—
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66636—21no—
15#66633C414yescomplemented
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21D4×Dic7—31no—
22C2×#1142—31no—
23#6450—31no—
24C7×#102D311yescharacteristic
25#6390—31no—
26#6389—31no—
27#6381—31no—
28#4757—71nomaximal, Hall
29C23×C3×C7—21no—
30C3×C7×D4—21no—
31C3×C7×D4C2×C412yes—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21—21no—
36C2×Dic21—42no—
37C2×Dic21D411yescharacteristic
38C2×Dic21D411yescharacteristic
39Dic42—21no—
40Dic42C2×C412yescomplemented
41C3×C7×C8—21no—
42C3⋊(C7⋊C8)C2×C412yescomplemented
43C2×C7×D4C2×D311yescharacteristic
44C22×Dic7—61no—
45C2×Dic14—31no—
46C2×C7×C8C2×D311yescharacteristic
47C7×C4⋊C4C2×D311yescharacteristic
48C22⋊Dic7—61no—
49#1142—34no—
50C4⋊Dic7—31no—
51Dic7⋊C4—61no—
52C4×Dic7—31no—
53C4×Dic7—61no—
54C2×C7⋊C8—31no—
55D4×Dic3—71no—
56C2×#214—71no—
57#887—71no—
58C3×#102D711yescharacteristic
59#816—71no—
60#815—71no—
61#807—71no—
62C22×C3×C7—22no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7—41no—
67C3×C4×C7C22×C411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—22no—
70Dic21—21no—
71Dic21—21no—
72Dic21—41no—
73#440—211noSylow
74C23×C7—21no—
75C7×D4—21no—
76C7×D4C4×D312yes—
77C2×C4×C7—21no—
78C2×C4×C7C22×D311yescharacteristic
79C2×Dic7—31no—
80C2×Dic7—31no—
81C2×Dic7—61no—
82C2×Dic7—61no—
83C2×Dic7—122no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—21no—
87C7⋊C8—32no—
88C2×C3×D4C2×D711yescharacteristic
89C22×Dic3—141no—
90C2×Dic6—71no—
91C2×C3×C8C2×D711yescharacteristic
92C3×C4⋊C4C2×D711yescharacteristic
93#217—141no—
94#214—74no—
95C4⋊Dic3—71no—
96Dic3⋊C4—141no—
97C4×Dic3—71no—
98C4×Dic3—141no—
99C2×C3⋊C8—71no—
100C2×C3×C7—42no—
101C2×C3×C7C4×D411yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103C2×C3×C7#13711yescharacteristic
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—211no—
107#107—211no—
108#103—211no—
109#102D2111yescharacteristic
110#97—211no—
111C22×C7—22no—
112C22×C7—41no—
113C22×C7—41no—
114C22×C7D3×D411yescharacteristic
115C4×C7—41no—
116C4×C7C2×C4×D311yescharacteristic
117C4×C7#23711yescharacteristic
118Dic7—62no—
119Dic7—61no—
120Dic7—61no—
121Dic7—121no—
122C23×C3—21no—
123C3×D4—21no—
124C3×D4C4×D712yes—
125C2×C3×C4—21no—
126C2×C3×C4C22×D711yescharacteristic
127C2×Dic3—71no—
128C2×Dic3—71no—
129C2×Dic3—141no—
130C2×Dic3—141no—
131C2×Dic3—282no—
132Dic6—72no—
133Dic6—141no—
134C3×C8—21no—
135C3⋊C8—72no—
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4C2×D2111yescharacteristic
139C22×C4—421no—
140SD16—214no—
141C2×C8—211no—
142C2×C8C2×D2111yescharacteristic
143C4⋊C4—211no—
144C4⋊C4—421no—
145C4⋊C4C2×D2111yescharacteristic
146C22⋊C4—421no—
147C42—211no—
148C42—421no—
149C2×C7—42no—
150C2×C7#67⋊C411yescharacteristic
151C2×C7#91111yescharacteristic
152C2×C7#91511yescharacteristic
153C22×C3—22no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3D4×D711yescharacteristic
157C3×C4—41no—
158C3×C4C2×C4×D711yescharacteristic
159C3×C4#115911yescharacteristic
160Dic3—142no—
161Dic3—141no—
162Dic3—141no—
163Dic3—281no—
164C23—21no—
165Q8—212no—
166Q8—421no—
167D4—21no—
168D4C4×D2112yes—
169C2×C4—21no—
170C2×C4—211no—
171C2×C4—211no—
172C2×C4—421no—
173C2×C4—421no—
174C2×C4—842no—
175C2×C4C22×D2111yescharacteristic
176C8—21no—
177C8—212no—
178C7#475711yescharacteristic, complemented, Sylow
179C2×C3—42no—
180C2×C3(C22⋊D7)⋊C411yescharacteristic
181C2×C3#647411yescharacteristic
182C2×C3#647811yescharacteristic
183C22—22no—
184C22—41no—
185C22—41no—
186C22D4×D2111yescharacteristic
187C4—41no—
188C4—422no—
189C4—421no—
190C4—421no—
191C4—841no—
192C4C2×C4×D2111yescharacteristic
193C4#6673011yescharacteristic
194C3#8952911yescharacteristic, complemented, Sylow
195C2—42no—
196C2#1062136411yescharacteristic
197C2#1062139411yescharacteristic
198C2#1062139811yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×C7×D4C22
cosocleC2×C3×C4×C7C23
hypercenter#102D21
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
3. #10621370C2semidirect
5. #10620655C2semidirect
6. #10620654C2semidirect
7. #10620646C2semidirect
15. #66633C4semidirect
40. Dic42C2×C4semidirect
42. C3⋊(C7⋊C8)C2×C4semidirect
136. C3×C7#440semidirect
178. C7#4757semidirect
194. C3#89529semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
C2×#666332
#106213702
C3×C7×#1022
#106206552
#106206542
#106206462
#895293
#47577