Finite Group Databasebeta

← #13056411

Subgroups — #13056411

Order 1344.

268 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 65 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#10621421C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3C2×C8×D21C211yesmaximal, characteristic, complemented
4#10621378C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620660C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89547—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×D84—21no—
12C2×D84C2211yescharacteristic
13C2×C4×D21C2211yescharacteristic
14C2×C3×C7×C8C2211yescharacteristic
15C3×C7×C4⋊C4C2211yescharacteristic
16#66636—21no—
17#66632—22no—
18#66631—21no—
19C4⋊Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C8×D21—22no—
22#6501—31no—
23C2×#1141—31no—
24C2×C8×D7—31no—
25#6458—31no—
26C7×#102D311yescharacteristic, complemented
27#6395—31no—
28#6381—31no—
29#4775—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C22×D21D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—42no—
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40D84—21no—
41D84—21no—
42D84—42no—
43C4×D21—21no—
44C4×D21—21no—
45C3×C7×C8—21no—
46C3⋊(C7⋊C8)D412yescomplemented
47C2×C7×D4C2×D311yescharacteristic
48C2×C22⋊D7—61no—
49C2×D28—31no—
50C2×D28—61no—
51C2×C4×D7—31no—
52C2×C7×C8C2×D311yescharacteristic
53C7×C4⋊C4C2×D311yescharacteristic
54C22⋊Dic7—61no—
55#1141—62no—
56(C2×D7)⋊C4—61no—
57C4⋊Dic7—31no—
58C2×C7⋊C8—31no—
59C8×D7—62no—
60#938—71no—
61C2×#213—71no—
62C2×D3×C8—71no—
63#895—71no—
64C3×#102D711yescharacteristic, complemented
65#821—71no—
66#807—71no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—21no—
72C2×D21—21no—
73C2×D21—41no—
74C2×D21—41no—
75C2×D21—41no—
76C2×D21D812yescomplemented
77C3×C4×C7—41no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7#4111yescharacteristic
80Dic21—21no—
81Dic21—41no—
82#466—211noSylow
83C23×C7—21no—
84C22×D7—31no—
85C22×D7—61no—
86C7×D4—21no—
87C7×D4—21no—
88C2×C4×C7—21no—
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—122no—
91C2×Dic7—31no—
92C2×Dic7—61no—
93D28—61no—
94D28—61no—
95D28—122no—
96C4×D7—61no—
97C4×D7—61no—
98C7×C8—21no—
99C7⋊C8—32no—
100C2×C3×D4C2×D711yescharacteristic
101C2×#67—141no—
102C2×D12—71no—
103C2×D12—141no—
104C2×C4×D3—71no—
105C2×C3×C8C2×D711yescharacteristic
106C3×C4⋊C4C2×D711yescharacteristic
107#217—141no—
108#213—142no—
109(C2×D3)⋊C4—141no—
110C4⋊Dic3—71no—
111C2×C3⋊C8—71no—
112D3×C8—142no—
113C2×C3×C7—81no—
114C2×C3×C7#12111yescharacteristic
115C2×C3×C7C2×D811yescharacteristic
116C2×C3×C7#13511yescharacteristic
117D21—22no—
118D21—81no—
119C2×D8—211no—
120C22×C8—211no—
121#121—211no—
122#121—211no—
123#107—211no—
124#102—211no—
125#102D2111yescharacteristic, complemented
126C22×C7—41no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7D3×D411yescharacteristic
130C2×D7—32no—
131C2×D7—61no—
132C2×D7—61no—
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C4×C7—41no—
137C4×C7#23511yescharacteristic
138C4×C7D3×D411yescharacteristic
139Dic7—61no—
140Dic7—121no—
141C23×C3—21no—
142C22×D3—71no—
143C22×D3—141no—
144C3×D4—21no—
145C3×D4—21no—
146C2×C3×C4—21no—
147C2×C3×C4C22×D711yescharacteristic
148#67—282no—
149C2×Dic3—71no—
150C2×Dic3—141no—
151D12—141no—
152D12—141no—
153D12—282no—
154C4×D3—141no—
155C4×D3—141no—
156C3×C8—21no—
157C3⋊C8—72no—
158C3×C7#46611yescharacteristic, complemented, Hall
159C2×D4—211no—
160C2×D4—421no—
161C2×D4—421no—
162C2×D4C2×D2111yescharacteristic
163C22×C4—211no—
164D8—422no—
165C2×C8—211no—
166C2×C8—422no—
167C2×C8C2×D2111yescharacteristic
168C4⋊C4—211no—
169C4⋊C4C2×D2111yescharacteristic
170C22⋊C4—421no—
171C22⋊C4—421no—
172C2×C7—81no—
173C2×C7#88411yescharacteristic
174C2×C7D3×D811yescharacteristic
175C2×C7#91611yescharacteristic
176D7—62no—
177D7—241no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3D4×D711yescharacteristic
182C2×D3—72no—
183C2×D3—141no—
184C2×D3—141no—
185C2×D3—281no—
186C2×D3—281no—
187C2×D3—281no—
188C3×C4—41no—
189C3×C4#115711yescharacteristic
190C3×C4D4×D711yescharacteristic
191Dic3—141no—
192Dic3—281no—
193C23—21no—
194C23—211no—
195C23—421no—
196D4—21no—
197D4—21no—
198D4—421no—
199D4—421no—
200D4—842no—
201D4—842no—
202C2×C4—21no—
203C2×C4—211no—
204C2×C4—421no—
205C2×C4—421no—
206C2×C4—421no—
207C2×C4C22×D2111yescharacteristic
208C8—21no—
209C8—212no—
210C7#477511yescharacteristic, complemented, Sylow
211C2×C3—81no—
212C2×C3#644711yescharacteristic
213C2×C3D7×D811yescharacteristic
214C2×C3#647911yescharacteristic
215D3—142no—
216D3—561no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—212no—
221C22—421no—
222C22—421no—
223C22—841no—
224C22—841no—
225C22—841no—
226C22D4×D2111yescharacteristic
227C4—41no—
228C4—421no—
229C4—841no—
230C4#6672811yescharacteristic
231C4D4×D2111yescharacteristic
232C3#8954711yescharacteristic, complemented, Sylow
233C2—81no—
234C2—422no—
235C2—1681no—
236C2#1062136711yescharacteristic
237C2D8×D2111yescharacteristic
238C2#1062139911yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
hypercenter#102D21
hypocenterC3×C7#466
LCSG ▷ C2×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
1. #10621421C2semidirect
2. C2×#66632C2semidirect
3. C2×C8×D21C2semidirect
4. #10621378C2semidirect
5. C3×C7×#102C2semidirect
6. #10620660C2semidirect
7. #10620646C2semidirect
26. C7×#102D3semidirect
46. C3⋊(C7⋊C8)D4semidirect
64. C3×#102D7semidirect
76. C2×D21D8semidirect
125. #102D21semidirect
158. C3×C7#466semidirect
210. C7#4775semidirect
232. C3#89547semidirect

Maximal subgroups

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

subgroupindex
#106214212
C2×#666322
C2×C8×D212
#106213782
C3×C7×#1022
#106206602
#106206462
#895473
#47757