Finite Group Databasebeta

← #13056463

Subgroups — #13056463

Order 1344.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621663C211yesmaximal, characteristic
2#10621434C211yesmaximal, characteristic, complemented
3#10621392C211yesmaximal, characteristic, complemented
4#10621352C211yesmaximal, characteristic, complemented
5C3×C7×#104C211yesmaximal, characteristic, complemented
6#10620663C211yesmaximal, characteristic, complemented
7#10620644C211yesmaximal, characteristic, complemented
8#89598—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10Q8×D21—21no—
11#66730—21no—
12#66730—22no—
13#66728—21no—
14#66728C2211yescharacteristic
15C2×Dic42—21no—
16C2×Dic42C2211yescharacteristic
17C2×Dic42C2211yescharacteristic
18C3×C7×M16C2211yescharacteristic
19C3×C42×C7C2211yescharacteristic
20#66635—21no—
21#66633—21no—
22C4⋊Dic21—22no—
23#66627C2211yescharacteristic
24Dic84—21no—
25#66623—21no—
26#6554—31no—
27#6514—31no—
28#6472—31no—
29#6432—31no—
30C7×#104D311yescharacteristic, complemented
31#6398—31no—
32#6379—31no—
33#4826—71nomaximal, Hall
34C3×C7×Q8D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4D411yescharacteristic
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2311yescharacteristic
40#4268—22no—
41#4268—21no—
42C2×Dic21—22no—
43C2×Dic21—21no—
44C2×Dic21D412yes—
45D84D411yescharacteristic
46C4×D21—22no—
47C4×D21—21no—
48Dic42—22no—
49Dic42—21no—
50Dic42—21no—
51Dic42—21no—
52Dic42—21no—
53Dic42D411yescharacteristic
54C3×C7×C8—21no—
55C3⋊(C7⋊C8)—21no—
56C7×#41C2×D311yescharacteristic
57Q8×D7—61no—
58#1159—62no—
59#1159—61no—
60#1157—31no—
61#1157—61no—
62C2×Dic14—31no—
63C2×Dic14—31no—
64C2×Dic14—61no—
65C7×M16C2×D311yescharacteristic
66C42×C7C2×D311yescharacteristic
67#1144—61no—
68#1142—61no—
69C4⋊Dic7—62no—
70#1136—31no—
71Dic28—61no—
72#1132—61no—
73#1010—71no—
74#951—71no—
75#909—71no—
76#869—71no—
77C3×#104D711yescharacteristic, complemented
78#824—71no—
79#805—71no—
80C22×C3×C7—21no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—21no—
83C3×C4×C7—22no—
84C3×C4×C7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—22no—
88Dic21—21no—
89Dic21—41no—
90#456—211noSylow
91C7×Q8D1211yescharacteristic
92C7×D4—21no—
93C7×D4D1211yescharacteristic
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—62no—
98C22⋊D7—61no—
99C2×Dic7—32no—
100C2×Dic7—62no—
101C2×Dic7—61no—
102D28—31no—
103C4×D7—62no—
104C4×D7—61no—
105Dic14—31no—
106Dic14—62no—
107Dic14—61no—
108Dic14—61no—
109Dic14—61no—
110Dic14—61no—
111C7×C8—21no—
112C7⋊C8—61no—
113C3×#41C2×D711yescharacteristic
114D3×Q8—141no—
115#237—142no—
116#237—141no—
117#235—71no—
118#235—141no—
119C2×Dic6—71no—
120C2×Dic6—71no—
121C2×Dic6—141no—
122C3×M16C2×D711yescharacteristic
123C3×C42C2×D711yescharacteristic
124#216—141no—
125#214—141no—
126C4⋊Dic3—142no—
127#208—71no—
128Dic12—141no—
129#204—141no—
130C2×C3×C7—21no—
131C2×C3×C7—41no—
132C2×C3×C7#12011yescharacteristic
133D21—41no—
1342−1+4—211no—
135#137—211no—
136#137—211no—
137C4⋊Q8—211no—
138#104—211no—
139#104D2111yescharacteristic, complemented
140#101—211no—
141C22×C7—21no—
142C22×C7D3×D411yescharacteristic
143C2×D7—61no—
144C4×C7—22no—
145C4×C7—21no—
146C4×C7C2×D1211yescharacteristic
147C4×C7D3×D411yescharacteristic
148Dic7—61no—
149Dic7—62no—
150Dic7—121no—
151C3×Q8D2811yescharacteristic
152C3×D4—21no—
153C3×D4D2811yescharacteristic
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4C22×D711yescharacteristic
157#67—142no—
158#67—141no—
159C2×Dic3—72no—
160C2×Dic3—142no—
161C2×Dic3—141no—
162D12—71no—
163C4×D3—142no—
164C4×D3—141no—
165Dic6—71no—
166Dic6—142no—
167Dic6—141no—
168Dic6—141no—
169Dic6—141no—
170Dic6—141no—
171C3×C8—21no—
172C3⋊C8—141no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—211no—
175#41—421no—
176#41—422no—
177#41—421no—
178#41C2×D2111yescharacteristic
179C2×Q8—211no—
180C2×Q8—211no—
181C2×Q8—421no—
182C2×Q8—421no—
183Q16—421no—
184Q16—421no—
185SD16—421no—
186SD16—421no—
187M16—211no—
188M16C2×D2111yescharacteristic
189C4⋊C4—422no—
190C42C2×D2111yescharacteristic
191C2×C7—21no—
192C2×C7—41no—
193C2×C7#88211yescharacteristic
194D7—121no—
195C22×C3—21no—
196C22×C3D4×D711yescharacteristic
197C2×D3—141no—
198C3×C4—22no—
199C3×C4—21no—
200C3×C4C2×D2811yescharacteristic
201C3×C4D4×D711yescharacteristic
202Dic3—141no—
203Dic3—142no—
204Dic3—281no—
205Q8—211no—
206Q8—422no—
207Q8—421no—
208Q8—421no—
209Q8—421no—
210Q8—421no—
211Q8D8411yescharacteristic
212D4—21no—
213D4—211no—
214D4—422no—
215D4—421no—
216D4D8411yescharacteristic
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—212no—
220C2×C4—422no—
221C2×C4—422no—
222C2×C4—421no—
223C2×C4—421no—
224C2×C4C22×D2111yescharacteristic
225C8—21no—
226C8—421no—
227C7#482611yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—41no—
230C2×C3#644511yescharacteristic
231D3—281no—
232C22—21no—
233C22—421no—
234C22D4×D2111yescharacteristic
235C4—22no—
236C4—21no—
237C4—422no—
238C4—421no—
239C4—841no—
240C4C2×D8411yescharacteristic
241C4D4×D2111yescharacteristic
242C3#8959811yescharacteristic, complemented, Sylow
243C2—21no—
244C2—41no—
245C2—841no—
246C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#104C2
socleC2×C3×C7#120
prime radical#10621663C2
hypercenter#104D21
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #10621434C2semidirect
3. #10621392C2semidirect
4. #10621352C2semidirect
5. C3×C7×#104C2semidirect
6. #10620663C2semidirect
7. #10620644C2semidirect
30. C7×#104D3semidirect
77. C3×#104D7semidirect
139. #104D21semidirect
173. C3×C7#456semidirect
227. C7#4826semidirect
242. C3#89598semidirect

Maximal subgroups

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

subgroupindex
#106216632
#106214342
#106213922
#106213522
C3×C7×#1042
#106206632
#106206442
#895983
#48267