Finite Group Databasebeta

← #13056523

Subgroups — #13056523

Order 1344.

240 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 69 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×#66623C211yesmaximal, characteristic, complemented
2C2×#66622C211yesmaximal, characteristic, complemented
3#10621380C211yesmaximal, characteristic, complemented
4#10621378C211yesmaximal, characteristic
5C3×C7×#107C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89658—31nomaximal, Hall
9C2×D84—21no—
10C2×D84C2211yescharacteristic
11C2×C4×D21C2211yescharacteristic
12C2×Dic42C2211yescharacteristic
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C4⋊C4C2211yescharacteristic
16#66631—21no—
17#66631—21no—
18C4⋊Dic21—21no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#66623—22no—
21#66622—22no—
22C2×#1132—31no—
23C2×#1131—31no—
24#6460—31no—
25#6458—31no—
26C7×#107D311yescharacteristic, complemented
27#6384—31no—
28#6383—31no—
29#4886—71nomaximal, Hall
30C22×D21—21no—
31C22×D21D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37D84—21no—
38D84—21no—
39D84—42no—
40C4×D21—21no—
41C4×D21—21no—
42Dic42—21no—
43Dic42—21no—
44C3×C7×C8D412yescomplemented
45C3⋊(C7⋊C8)—21no—
46C2×D28—31no—
47C2×D28—61no—
48C2×C4×D7—31no—
49C2×Dic14—31no—
50C2×C7×C8C2×D311yescharacteristic
51C7×C4⋊C4C2×D311yescharacteristic
52C7×C4⋊C4C2×D311yescharacteristic
53(C2×D7)⋊C4—61no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—61no—
56C2×C7⋊C8—31no—
57#1132—62no—
58#1131—62no—
59C2×#204—71no—
60C2×ΓL(1,𝔽25)—71no—
61#897—71no—
62#895—71no—
63C3×#107D711yescharacteristic, complemented
64#810—71no—
65#809—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—21no—
68C2×D21—21no—
69C2×D21—21no—
70C2×D21—41no—
71C2×D21—41no—
72C2×D21—41no—
73C3×C4×C7—41no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7#4111yescharacteristic
77Dic21—21no—
78Dic21—41no—
79#468—211noSylow
80C22×D7—31no—
81C22×D7—61no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—31no—
86C2×Dic7—61no—
87D28—61no—
88D28—61no—
89D28—122no—
90C4×D7—61no—
91C4×D7—61no—
92Dic14—61no—
93Dic14—61no—
94C7×C8D1212yescomplemented
95C7⋊C8—61no—
96C2×D12—71no—
97C2×D12—141no—
98C2×C4×D3—71no—
99C2×Dic6—71no—
100C2×C3×C8C2×D711yescharacteristic
101C3×C4⋊C4C2×D711yescharacteristic
102C3×C4⋊C4C2×D711yescharacteristic
103(C2×D3)⋊C4—141no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—141no—
106C2×C3⋊C8—71no—
107#204—142no—
108ΓL(1,𝔽25)—142no—
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C3×C7#13711yescharacteristic
112D21—41no—
113D21—81no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—211no—
118#107D2111yescharacteristic, complemented
119#103—211no—
120#102—211no—
121C22×C7D3×D411yescharacteristic
122C2×D7—61no—
123C2×D7—61no—
124C2×D7—61no—
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7—41no—
130C4×C7C2×D1211yescharacteristic
131C4×C7#23911yescharacteristic
132Dic7—61no—
133Dic7—121no—
134C22×D3—71no—
135C22×D3—141no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4C22×D711yescharacteristic
139C2×Dic3—71no—
140C2×Dic3—141no—
141D12—141no—
142D12—141no—
143D12—282no—
144C4×D3—141no—
145C4×D3—141no—
146Dic6—141no—
147Dic6—141no—
148C3×C8D2812yescomplemented
149C3⋊C8—141no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—211no—
153C2×D4—421no—
154C22×C4—211no—
155SD16—422no—
156M16—422no—
157C2×C8—211no—
158C2×C8C2×D2111yescharacteristic
159C4⋊C4—421no—
160C4⋊C4C2×D2111yescharacteristic
161C4⋊C4C2×D2111yescharacteristic
162C22⋊C4—421no—
163C22⋊C4—421no—
164C2×C7#89511yescharacteristic
165C2×C7#91111yescharacteristic
166C2×C7#91811yescharacteristic
167D7—121no—
168D7—241no—
169C22×C3D4×D711yescharacteristic
170C2×D3—141no—
171C2×D3—141no—
172C2×D3—141no—
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4C2×D2811yescharacteristic
179C3×C4#116111yescharacteristic
180Dic3—141no—
181Dic3—281no—
182C23—211no—
183C23—421no—
184Q8—421no—
185Q8—421no—
186D4—421no—
187D4—421no—
188D4—842no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—211no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4—421no—
195C2×C4C22×D2111yescharacteristic
196C8—421no—
197C8D8412yescomplemented
198C7#488611yescharacteristic, complemented, Sylow
199C2×C3#645811yescharacteristic
200C2×C3#647411yescharacteristic
201C2×C3#648111yescharacteristic
202D3—281no—
203D3—561no—
204C22—421no—
205C22—421no—
206C22—421no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22D4×D2111yescharacteristic
211C4—41no—
212C4—41no—
213C4—421no—
214C4—841no—
215C4C2×D8411yescharacteristic
216C4#6673211yescharacteristic
217C3#8965811yescharacteristic, complemented, Sylow
218C2—841no—
219C2—1681no—
220C2#1062137811yescharacteristic
221C2#1062139411yescharacteristic
222C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radical#10621378C2
hypercenter#107D21
hypocenterC3×C7#468
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×#66623C2semidirect
2. C2×#66622C2semidirect
3. #10621380C2semidirect
5. C3×C7×#107C2semidirect
6. #10620649C2semidirect
7. #10620648C2semidirect
26. C7×#107D3semidirect
44. C3×C7×C8D4semidirect
63. C3×#107D7semidirect
94. C7×C8D12semidirect
118. #107D21semidirect
148. C3×C8D28semidirect
150. C3×C7#468semidirect
197. C8D84semidirect
198. C7#4886semidirect
217. C3#89658semidirect

Maximal subgroups

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

subgroupindex
C2×#666232
C2×#666222
#106213802
#106213782
C3×C7×#1072
#106206492
#106206482
#896583
#48867