Finite Group Databasebeta

← #13052838

Subgroups — #13052838

Order 1344.

248 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#10621378C211yesmaximal, characteristic
2#10620952C211yesmaximal, characteristic, complemented
3C2×#66566C211yesmaximal, characteristic, complemented
4C2×D21⋊C8C211yesmaximal, characteristic, complemented
5C7×#808C211yesmaximal, characteristic, complemented
6C3×#6384C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic, complemented
8#89636—31nomaximal, Hall
9C2×D84—21no—
10C2×D84C2211yescharacteristic
11C2×C4×D21C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66631—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic3⋊Dic7—21no—
19#66575—21no—
20#66566—22no—
21D21⋊C8—22no—
22C2×#1132—31no—
23C2×C8×D7—31no—
24#6460—31no—
25#6458—31no—
26C7×#106—31no—
27#6384D311yescharacteristic, complemented
28#6383—31no—
29#4807—71nomaximal, Hall
30C22×D21—21no—
31C22×D21D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21D411yescharacteristic
35D84—21no—
36D84—21no—
37D84—42no—
38C4×D21—21no—
39C4×D21—21no—
40C2×C7×Dic3—21no—
41C2×C3×Dic7—21no—
42C3×Dic14—21no—
43C3×Dic14—21no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8D412yescomplemented
46C2×D28—31no—
47C2×D28—61no—
48C2×C4×D7—31no—
49C2×Dic14C2×D311yescharacteristic
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52C7×C4⋊C4C2×D311yescharacteristic
53(C2×D7)⋊C4—61no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—61no—
56C2×C7⋊C8C2×D311yescharacteristic
57#1132—62no—
58C8×D7—62no—
59#946—71no—
60C2×#215—71no—
61C2×D3×C8—71no—
62#895—71no—
63C3×#103—71no—
64#821—71no—
65#808D711yescharacteristic, complemented
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—21no—
68C2×D21—21no—
69C2×D21—41no—
70C2×D21—41no—
71C2×D21—41no—
72C2×D21SD1612yescomplemented
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—21no—
77C3×Dic7—41no—
78C7×Dic3—41no—
79#465—211noSylow
80C22×D7—31no—
81C22×D7—61no—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—31no—
87D28—61no—
88D28—61no—
89D28—122no—
90C4×D7—61no—
91C4×D7—61no—
92Dic14—21no—
93Dic14—21no—
94C7×C8—32no—
95C7⋊C8—21no—
96C2×C3×Q8—71no—
97C2×D12—71no—
98C2×D12—141no—
99C2×C4×D3—71no—
100C2×C3×C8—71no—
101C3×C4⋊C4C2×D711yescharacteristic
102#215—142no—
103(C2×D3)⋊C4—141no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3C2×D711yescharacteristic
106Dic3⋊C4—141no—
107C2×C3⋊C8C2×D711yescharacteristic
108D3×C8—142no—
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7C2×SD1611yescharacteristic
111C2×C3×C7#13511yescharacteristic
112D21—22no—
113D21—81no—
114C2×SD16—211no—
115C22×C8—211no—
116C22⋊Q8—211no—
117#121—211no—
118#106—31no—
119#103—71no—
120#102—211no—
121C22×C7D3×D411yescharacteristic
122C2×D7—32no—
123C2×D7—61no—
124C2×D7—61no—
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7—121no—
130C4×C7#23511yescharacteristic
131C4×C7D3×D411yescharacteristic
132Dic7—41no—
133Dic7—61no—
134C22×D3—71no—
135C22×D3—141no—
136C3×Q8—141no—
137C3×Q8—141no—
138C2×C3×C4—21no—
139C2×C3×C4—141no—
140C2×C3×C4C22×D711yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—71no—
143D12—141no—
144D12—141no—
145D12—282no—
146C4×D3—141no—
147C4×D3—141no—
148C3×C8—141no—
149C3⋊C8D2812yescomplemented
150C3×C7#46511yescharacteristic, complemented, Hall
151C2×Q8—71no—
152C2×D4—211no—
153C2×D4—421no—
154C22×C4—211no—
155SD16—422no—
156C2×C8—31no—
157C2×C8—71no—
158C2×C8—422no—
159C4⋊C4—31no—
160C4⋊C4—421no—
161C4⋊C4D3×D711yescharacteristic
162C22⋊C4—421no—
163C22⋊C4—421no—
164C2×C7#88411yescharacteristic
165C2×C7D3×SD1611yescharacteristic
166C2×C7#91911yescharacteristic
167D7—62no—
168D7—241no—
169C22×C3D4×D711yescharacteristic
170C2×D3—72no—
171C2×D3—141no—
172C2×D3—141no—
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4—281no—
178C3×C4C2×D2811yescharacteristic
179C3×C4#116111yescharacteristic
180Dic3—41no—
181Dic3—141no—
182C23—211no—
183C23—421no—
184Q8—141no—
185Q8—141no—
186D4—421no—
187D4—421no—
188D4—842no—
189C2×C4—21no—
190C2×C4—61no—
191C2×C4—141no—
192C2×C4—211no—
193C2×C4—421no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—32no—
197C8—141no—
198C7#480711yescharacteristic, complemented, Sylow
199C2×C3#645811yescharacteristic
200C2×C3D7×SD1611yescharacteristic
201C2×C3#647911yescharacteristic
202D3—142no—
203D3—561no—
204C22—212no—
205C22—421no—
206C22—421no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22#6669411yescharacteristic
211C4—41no—
212C4—121no—
213C4—281no—
214C4—421no—
215C4#6667711yescharacteristic
216C4D3×D2811yescharacteristic
217C3#8963611yescharacteristic, complemented, Sylow
218C2—422no—
219C2—1681no—
220C2#1062102111yescharacteristic
221C2#1062104911yescharacteristic
222C2#1062108511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radical#10621378C2
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. #10620952C2semidirect
3. C2×#66566C2semidirect
4. C2×D21⋊C8C2semidirect
5. C7×#808C2semidirect
6. C3×#6384C2semidirect
7. #10620504C2semidirect
27. #6384D3semidirect
45. C7×C3⋊C8D4semidirect
65. #808D7semidirect
72. C2×D21SD16semidirect
149. C3⋊C8D28semidirect
150. C3×C7#465semidirect
198. C7#4807semidirect
217. C3#89636semidirect

Maximal subgroups

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

subgroupindex
#106213782
#106209522
C2×#665662
C2×D21⋊C82
C7×#8082
C3×#63842
#106205042
#896363
#48077