Finite Group Databasebeta

← #13056394

Subgroups — #13056394

Order 1344.

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

normal subgroups only

This group has 280 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#10621423C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3C2×D168C211yesmaximal, characteristic, complemented
4#10621376C211yesmaximal, characteristic, complemented
5C3×C7×#102C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89558—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×#4268—21no—
12C2×D84C2211yescharacteristic
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×C3×C7×C8C2211yescharacteristic
16C3×C7×C4⋊C4C2211yescharacteristic
17#66632—22no—
18#66631—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21D168—22no—
22#6503—31no—
23C2×#1141—31no—
24C2×D56—31no—
25#6456—31no—
26C7×#102D311yescharacteristic, complemented
27Dic7⋊C8—31no—
28#6383—31no—
29#4786—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C22×D21—21no—
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—42no—
38#4268—42no—
39C2×Dic21D411yescharacteristic
40C2×Dic21D411yescharacteristic
41D84—21no—
42D84—21no—
43D84—21no—
44D84D412yescomplemented
45C4×D21—42no—
46C3×C7×C8—21no—
47C3⋊(C7⋊C8)—21no—
48C2×C7×D4C2×D311yescharacteristic
49C2×C22⋊D7—61no—
50C2×C22⋊D7—61no—
51C2×D28—31no—
52C2×D28—31no—
53C2×C4×D7—61no—
54C2×C7×C8C2×D311yescharacteristic
55C7×C4⋊C4C2×D311yescharacteristic
56#1141—62no—
57(C2×D7)⋊C4—61no—
58C4×Dic7—31no—
59C2×C7⋊C8—31no—
60D56—62no—
61#940—71no—
62C2×#213—71no—
63C2×D24—71no—
64#893—71no—
65C3×#102D711yescharacteristic, complemented
66Dic3⋊C8—71no—
67#809—71no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7—41no—
71C22×C3×C7C2×D411yescharacteristic
72C2×D21—22no—
73C2×D21—41no—
74C2×D21—41no—
75C2×D21—41no—
76C2×D21—41no—
77C2×D21—41no—
78C3×C4×C7—41no—
79C3×C4×C7C2×D411yescharacteristic
80C3×C4×C7#4111yescharacteristic
81Dic21—21no—
82Dic21D812yescomplemented
83#459—211noSylow
84C23×C7—21no—
85C22×D7—61no—
86C22×D7—61no—
87C7×D4—21no—
88C7×D4—21no—
89C2×C4×C7—21no—
90C2×C4×C7C22×D311yescharacteristic
91C22⋊D7—122no—
92C22⋊D7—122no—
93C2×Dic7—31no—
94C2×Dic7—31no—
95D28—32no—
96D28—61no—
97D28—61no—
98D28—61no—
99C4×D7—122no—
100C7×C8—21no—
101C7⋊C8—61no—
102C2×C3×D4C2×D711yescharacteristic
103C2×#67—141no—
104C2×#67—141no—
105C2×D12—71no—
106C2×D12—71no—
107C2×C4×D3—141no—
108C2×C3×C8C2×D711yescharacteristic
109C3×C4⋊C4C2×D711yescharacteristic
110#213—142no—
111(C2×D3)⋊C4—141no—
112C4×Dic3—71no—
113C2×C3⋊C8—71no—
114D24—142no—
115C2×C3×C7—81no—
116C2×C3×C7#12111yescharacteristic
117C2×C3×C7C2×D811yescharacteristic
118C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
119D21—42no—
120D21—81no—
121C2×D8—211no—
122C2×D8—211no—
123#127—211no—
124C4×D4—211no—
125C4⋊C8—211no—
126#102—211no—
127#102D2111yescharacteristic, complemented
128C22×C7—41no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7D3×D411yescharacteristic
132C2×D7—62no—
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C2×D7—121no—
137C2×D7—121no—
138C4×C7—41no—
139C4×C7#23511yescharacteristic
140C4×C7D3×D411yescharacteristic
141Dic7—32no—
142Dic7—61no—
143C23×C3—21no—
144C22×D3—141no—
145C22×D3—141no—
146C3×D4—21no—
147C3×D4—21no—
148C2×C3×C4—21no—
149C2×C3×C4C22×D711yescharacteristic
150#67—282no—
151#67—282no—
152C2×Dic3—71no—
153C2×Dic3—71no—
154D12—72no—
155D12—141no—
156D12—141no—
157D12—141no—
158C4×D3—282no—
159C3×C8—21no—
160C3⋊C8—141no—
161C3×C7#45911yescharacteristic, complemented, Hall
162C2×D4—211no—
163C2×D4—211no—
164C2×D4—421no—
165C2×D4—421no—
166C2×D4C2×D2111yescharacteristic
167C22×C4—421no—
168D8—422no—
169D8—422no—
170C2×C8—211no—
171C2×C8C2×D2111yescharacteristic
172C4⋊C4C2×D2111yescharacteristic
173C22⋊C4—421no—
174C42—211no—
175C2×C7—81no—
176C2×C7#88411yescharacteristic
177C2×C7D3×D811yescharacteristic
178C2×C7#91411yescharacteristic
179D7—122no—
180D7—241no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3D4×D711yescharacteristic
185C2×D3—142no—
186C2×D3—281no—
187C2×D3—281no—
188C2×D3—281no—
189C2×D3—281no—
190C2×D3—281no—
191C3×C4—41no—
192C3×C4#115711yescharacteristic
193C3×C4D4×D711yescharacteristic
194Dic3—72no—
195Dic3—141no—
196C23—21no—
197C23—421no—
198C23—421no—
199D4—21no—
200D4—21no—
201D4—212no—
202D4—421no—
203D4—421no—
204D4—421no—
205D4—842no—
206D4—842no—
207C2×C4—21no—
208C2×C4—211no—
209C2×C4—211no—
210C2×C4—842no—
211C2×C4C22×D2111yescharacteristic
212C8—21no—
213C8—421no—
214C7#478611yescharacteristic, complemented, Sylow
215C2×C3—81no—
216C2×C3#644711yescharacteristic
217C2×C3D7×D811yescharacteristic
218C2×C3#647711yescharacteristic
219D3—282no—
220D3—561no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—422no—
225C22—841no—
226C22—841no—
227C22—841no—
228C22—841no—
229C22—841no—
230C22D4×D2111yescharacteristic
231C4—41no—
232C4—212no—
233C4—421no—
234C4#6672811yescharacteristic
235C4D4×D2111yescharacteristic
236C3#8955811yescharacteristic, complemented, Sylow
237C2—81no—
238C2—842no—
239C2—1681no—
240C2#1062136711yescharacteristic
241C2D8×D2111yescharacteristic
242C2#1062139711yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
hypercenter#102D21
hypocenterC3×C7#459
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. #10621423C2semidirect
2. C2×#66632C2semidirect
3. C2×D168C2semidirect
4. #10621376C2semidirect
5. C3×C7×#102C2semidirect
6. #10620653C2semidirect
7. #10620648C2semidirect
26. C7×#102D3semidirect
44. D84D4semidirect
65. C3×#102D7semidirect
82. Dic21D8semidirect
127. #102D21semidirect
161. C3×C7#459semidirect
214. C7#4786semidirect
236. C3#89558semidirect

Maximal subgroups

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

subgroupindex
#106214232
C2×#666322
C2×D1682
#106213762
C3×C7×#1022
#106206532
#106206482
#895583
#47867