Finite Group Databasebeta

← #13052627

Subgroups — #13052627

Order 1344.

248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 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×#6458C211yesmaximal, characteristic
2#10620922C211yesmaximal, characteristic, complemented
3C2×#66565C211yesmaximal, characteristic, complemented
4C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
5#10620649C211yesmaximal, characteristic, complemented
6C7×#808C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89636—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×D28—21no—
11C2×C3×D28C2211yescharacteristic
12C2×C3×C4×D7C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18Dic3⋊Dic7—21no—
19(C2×D7)⋊Dic3—21no—
20#66565—22no—
21D7×C3⋊C8—22no—
22C2×#1132—31no—
23C2×C8×D7—31no—
24#6460—31no—
25#6458D311yescharacteristic
26C7×#106—31no—
27#6384—31no—
28#6383—31no—
29#5041—71nomaximal, Hall
30C22×C3×D7—21no—
31C22×C3×D7D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35Dic42—21no—
36Dic42—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7D411yescharacteristic
39C3×D28—21no—
40C3×D28—21no—
41C3×D28—42no—
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44C3⋊(C7⋊C8)—21no—
45C7×C3⋊C8D412yescomplemented
46C2×D28—21no—
47C2×D28C2×D311yescharacteristic
48C2×C4×D7C2×D311yescharacteristic
49C2×Dic14—31no—
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52C7×C4⋊C4C2×D311yescharacteristic
53(C2×D7)⋊C4—21no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—61no—
56C2×C7⋊C8—31no—
57#1132—62no—
58C8×D7—62no—
59C3×#121—71no—
60C2×#214—71no—
61#924—71no—
62C22×C3⋊C8—71no—
63#832—71no—
64#810—71no—
65#808D711yescharacteristic, complemented
66C22×C3×C7C2×D411yescharacteristic
67C2×C3×D7—21no—
68C2×C3×D7—21no—
69C2×C3×D7—41no—
70C2×C3×D7—41no—
71C2×C3×D7—41no—
72C2×C3×D7SD1612yescomplemented
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—41no—
77C3×Dic7—21no—
78C7×Dic3—41no—
79#465—211noSylow
80C22×D7—21no—
81C22×D7#6711yescharacteristic
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—61no—
86C2×Dic7#6711yescharacteristic
87D28—21no—
88D28—21no—
89D28—42no—
90C4×D7—21no—
91C4×D7—21no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—32no—
95C7⋊C8—61no—
96C2×C3×D4—71no—
97C2×C3×D4—141no—
98C22×C3×C4—71no—
99C2×Dic6—71no—
100C3×C4⋊C4C2×D711yescharacteristic
101C3×C22⋊C4—141no—
102#217—141no—
103#214—142no—
104C4⋊Dic3C2×D711yescharacteristic
105Dic3⋊C4—141no—
106C2×C3⋊C8—71no—
107C2×C3⋊C8—142no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7C2×SD1611yescharacteristic
111C2×C3×C7#13511yescharacteristic
112C3×D7—22no—
113C3×D7—81no—
114C2×SD16—211no—
115C22×C8—211no—
116C22⋊Q8—211no—
117#121—71no—
118#106—31no—
119#103—211no—
120#102—211no—
121C22×C7C2×#6711yescharacteristic
122C2×D7—21no—
123C2×D7—21no—
124C2×D7—41no—
125C2×D7—41no—
126C2×D7—41no—
127C2×D7#21412yescomplemented
128C4×C7—41no—
129C4×C7—121no—
130C4×C7D3×D411yescharacteristic
131C4×C7#23711yescharacteristic
132Dic7—21no—
133Dic7—121no—
134C23×C3—71no—
135C23×C3—141no—
136C3×D4—141no—
137C3×D4—141no—
138C3×D4—282no—
139C2×C3×C4—21no—
140C2×C3×C4—71no—
141C2×C3×C4—141no—
142C2×C3×C4—141no—
143C2×C3×C4C22×D711yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—141no—
146Dic6—141no—
147Dic6—141no—
148C3⋊C8—141no—
149C3⋊C8D2812yescomplemented
150C3×C7#46511yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—71no—
153C2×D4—141no—
154C22×C4—71no—
155SD16—422no—
156C2×C8—31no—
157C2×C8—211no—
158C2×C8—422no—
159C4⋊C4—31no—
160C4⋊C4—421no—
161C4⋊C4D3×D711yescharacteristic
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7C2×#21411yescharacteristic
165C2×C7#93911yescharacteristic
166C2×C7#95011yescharacteristic
167D7—22no—
168D7—81no—
169C22×C3—72no—
170C22×C3—141no—
171C22×C3—141no—
172C22×C3—281no—
173C22×C3—281no—
174C22×C3—281no—
175C22×C3D4×D711yescharacteristic
176C3×C4—41no—
177C3×C4—141no—
178C3×C4C2×D2811yescharacteristic
179C3×C4#116111yescharacteristic
180Dic3—41no—
181Dic3—281no—
182C23—71no—
183C23—141no—
184Q8—421no—
185Q8—421no—
186D4—141no—
187D4—141no—
188D4—282no—
189C2×C4—21no—
190C2×C4—61no—
191C2×C4—71no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—32no—
197C8—421no—
198C7#504111yescharacteristic, complemented, Sylow
199C2×C3—142no—
200C2×C3—561no—
201C2×C3#645811yescharacteristic
202C2×C3D7×SD1611yescharacteristic
203C2×C3#647911yescharacteristic
204C22—72no—
205C22—141no—
206C22—141no—
207C22—281no—
208C22—281no—
209C22—281no—
210C22D7×#6711yescharacteristic
211C4—41no—
212C4—121no—
213C4—141no—
214C4—841no—
215C4#6667011yescharacteristic
216C4D3×D2811yescharacteristic
217C3#8963611yescharacteristic, complemented, Sylow
218C2—142no—
219C2—561no—
220C2#1062097111yescharacteristic
221C2D7×#21411yescharacteristic
222C2#1062107611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6458C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#465
LCSG ▷ C2×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
2. #10620922C2semidirect
3. C2×#66565C2semidirect
4. C2×D7×C3⋊C8C2semidirect
5. #10620649C2semidirect
6. C7×#808C2semidirect
7. #10620499C2semidirect
45. C7×C3⋊C8D4semidirect
65. #808D7semidirect
72. C2×C3×D7SD16semidirect
127. C2×D7#214semidirect
149. C3⋊C8D28semidirect
150. C3×C7#465semidirect
198. C7#5041semidirect
217. C3#89636semidirect

Maximal subgroups

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

subgroupindex
C3×#64582
#106209222
C2×#665652
C2×D7×C3⋊C82
#106206492
C7×#8082
#106204992
#896363
#50417