Finite Group Databasebeta

← #13056440

Subgroups — #13056440

Order 1344.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#66732C211yesmaximal, characteristic
2C2×C8×D21C211yesmaximal, characteristic, complemented
3#10621375C211yesmaximal, characteristic, complemented
4C3×C7×#103C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620660C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89576—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66732—22no—
11#66732C414yescomplemented
12C2×D84—21no—
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×C4×D21C2211yescharacteristic
16C2×C3×C7×C8C2211yescharacteristic
17C3×C7×C4⋊C4C2211yescharacteristic
18#66631—21no—
19C4⋊Dic21C2211yescharacteristic
20C4×Dic21—21no—
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C8×D21—22no—
23C2×#1161—31no—
24C2×C8×D7—31no—
25#6455—31no—
26C7×#103D311yescharacteristic, complemented
27#6409—31no—
28#6395—31no—
29#6383—31no—
30#4804—71nomaximal, Hall
31C22×D21—21no—
32C22×D21D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8C2×C412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40D84—24no—
41D84—21no—
42D84C2×C412yes—
43C4×D21—24no—
44C4×D21C2×C412yes—
45C4×D21D412yes—
46C3×C7×C8—21no—
47C3⋊(C7⋊C8)—21no—
48C2×C7×Q8C2×D311yescharacteristic
49#1161—34no—
50#1161—62no—
51C2×D28—31no—
52C2×D28—61no—
53C2×C4×D7—31no—
54C2×C4×D7—61no—
55C2×C7×C8C2×D311yescharacteristic
56C7×C4⋊C4C2×D311yescharacteristic
57(C2×D7)⋊C4—61no—
58C4⋊Dic7—31no—
59C4×Dic7—61no—
60C2×C7⋊C8—31no—
61C8×D7—62no—
62C2×#239—71no—
63C2×D3×C8—71no—
64#892—71no—
65C3×#103D711yescharacteristic, complemented
66#835—71no—
67#821—71no—
68#809—71no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—21no—
71C2×D21—21no—
72C2×D21—22no—
73C2×D21—41no—
74C2×D21—41no—
75C2×D21C22⋊C412yes—
76C3×C4×C7—22no—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—41no—
81Dic21C22⋊C412yes—
82#416—211noSylow
83C22×D7—31no—
84C22×D7—61no—
85C7×Q8—21no—
86C7×Q8C4×D312yes—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—31no—
91C2×Dic7—61no—
92D28—32no—
93D28—64no—
94D28—61no—
95C4×D7—32no—
96C4×D7—32no—
97C4×D7—64no—
98C7×C8—21no—
99C7⋊C8—61no—
100C2×C3×Q8C2×D711yescharacteristic
101#239—74no—
102#239—142no—
103C2×D12—71no—
104C2×D12—141no—
105C2×C4×D3—71no—
106C2×C4×D3—141no—
107C2×C3×C8C2×D711yescharacteristic
108C3×C4⋊C4C2×D711yescharacteristic
109(C2×D3)⋊C4—141no—
110C4⋊Dic3—71no—
111C4×Dic3—141no—
112C2×C3⋊C8—71no—
113D3×C8—142no—
114C2×C3×C7C2×C22⋊C411yescharacteristic
115C2×C3×C7#13511yescharacteristic
116C2×C3×C7#13511yescharacteristic
117D21—22no—
118D21—42no—
119C2×#41—211no—
120C22×C8—211no—
121#117—211no—
122#103—211no—
123#103D2111yescharacteristic, complemented
124#102—211no—
125#102—211no—
126C22×C7D3×D411yescharacteristic
127C2×D7—32no—
128C2×D7—61no—
129C2×D7—61no—
130C2×D7—62no—
131C2×D7—121no—
132C2×D7—121no—
133C4×C7—22no—
134C4×C7—41no—
135C4×C7C2×C4×D311yescharacteristic
136C4×C7D3×D411yescharacteristic
137Dic7—32no—
138Dic7—121no—
139C22×D3—71no—
140C22×D3—141no—
141C3×Q8—21no—
142C3×Q8C4×D712yes—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4C22×D711yescharacteristic
146C2×Dic3—71no—
147C2×Dic3—141no—
148D12—72no—
149D12—144no—
150D12—141no—
151C4×D3—72no—
152C4×D3—72no—
153C4×D3—144no—
154C3×C8—21no—
155C3⋊C8—141no—
156C3×C7#41611yescharacteristic, complemented, Hall
157#41—214no—
158#41—422no—
159C2×Q8C2×D2111yescharacteristic
160C2×D4—211no—
161C2×D4—421no—
162C22×C4—211no—
163C22×C4—421no—
164C2×C8—211no—
165C2×C8—422no—
166C2×C8C2×D2111yescharacteristic
167C4⋊C4—211no—
168C4⋊C4C2×D2111yescharacteristic
169C22⋊C4—421no—
170C42—421no—
171C2×C7D3×C22⋊C411yescharacteristic
172C2×C7#91611yescharacteristic
173C2×C7#91911yescharacteristic
174D7—62no—
175D7—122no—
176C22×C3D4×D711yescharacteristic
177C2×D3—72no—
178C2×D3—141no—
179C2×D3—141no—
180C2×D3—142no—
181C2×D3—281no—
182C2×D3—281no—
183C3×C4—22no—
184C3×C4—41no—
185C3×C4C2×C4×D711yescharacteristic
186C3×C4D4×D711yescharacteristic
187Dic3—72no—
188Dic3—281no—
189C23—211no—
190C23—421no—
191Q8—21no—
192Q8C4×D2112yes—
193D4—212no—
194D4—424no—
195D4—421no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—211no—
199C2×C4—212no—
200C2×C4—212no—
201C2×C4—424no—
202C2×C4—421no—
203C2×C4C22×D2111yescharacteristic
204C8—21no—
205C8—421no—
206C7#480411yescharacteristic, complemented, Sylow
207C2×C3D7×C22⋊C411yescharacteristic
208C2×C3#647911yescharacteristic
209C2×C3#648211yescharacteristic
210D3—142no—
211D3—282no—
212C22—212no—
213C22—421no—
214C22—421no—
215C22—422no—
216C22—841no—
217C22—841no—
218C22D4×D2111yescharacteristic
219C4—22no—
220C4—41no—
221C4—212no—
222C4—841no—
223C4C2×C4×D2111yescharacteristic
224C4D4×D2111yescharacteristic
225C3#8957611yescharacteristic, complemented, Sylow
226C2—422no—
227C2—842no—
228C2C22⋊C4×D2111yescharacteristic
229C2#1062139911yescharacteristic
230C2#1062140211yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×#66732C2
cosocleC2×C3×C4×C7C23
hypercenter#103D21
hypocenterC3×C7#416
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×C8×D21C2semidirect
3. #10621375C2semidirect
4. C3×C7×#103C2semidirect
5. #10620674C2semidirect
6. #10620660C2semidirect
7. #10620648C2semidirect
11. #66732C4semidirect
26. C7×#103D3semidirect
65. C3×#103D7semidirect
123. #103D21semidirect
156. C3×C7#416semidirect
206. C7#4804semidirect
225. C3#89576semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×C8×D212
#106213752
C3×C7×#1032
#106206742
#106206602
#106206482
#895763
#48047