Finite Group Databasebeta

← #13054705

Subgroups — #13054705

Order 1344.

228 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 64 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#10621429C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620952C211yesmaximal, characteristic
4C2×D21⋊C8C211yesmaximal, characteristic
5C7×#835C211yesmaximal, characteristic, complemented
6#10620523C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89585—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66631—21no—
13C4⋊Dic21C2211yescharacteristic
14#66629—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1144—22no—
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic3⋊Dic7—21no—
20#66575—21no—
21D21⋊C8—22no—
22#6509—31no—
23C2×#1144D311yescharacteristic, complemented
24C2×C8×D7—31no—
25#6460—31no—
26C7×#103—31no—
27#6390—31no—
28#6381—31no—
29#5188—71nomaximal, Hall
30C22×D21D411yescharacteristic
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C4×D21—21no—
38C4×D21—21no—
39C2×C7×Dic3—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3×C7⋊C8D412yes—
44C7×C3⋊C8—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×C4×D7—31no—
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4—31no—
50#1144—22no—
51(C2×D7)⋊C4—61no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—31no—
54C4⋊Dic7—61no—
55Dic7⋊C4—61no—
56C2×C7⋊C8C2×D311yescharacteristic
57C8×D7—62no—
58C2×C3×Q16—71no—
59#946—71no—
60#946—71no—
61C2×D3×C8—71no—
62#835—71no—
63#835D711yescharacteristic, complemented
64#818—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D21—21no—
67C2×D21—21no—
68C2×D21Q1612yescomplemented
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—41no—
74C3×Dic7—41no—
75C7×Dic3—41no—
76#467—211noSylow
77C22×D7—31no—
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7—61no—
82C2×C4×C7C22×D311yescharacteristic
83C2×Dic7—21no—
84C2×Dic7—31no—
85C2×Dic7—61no—
86C4×D7—61no—
87C4×D7—61no—
88Dic14—21no—
89Dic14—21no—
90C7×C8—61no—
91C7⋊C8#6712yes—
92C2×C3×Q8—71no—
93C2×C3×Q8C2×D711yescharacteristic
94C2×C4×D3—71no—
95C3×Q16—142no—
96C2×C3×C8—71no—
97(C2×D3)⋊C4—141no—
98(C2×D3)⋊C4—141no—
99C4⋊Dic3—71no—
100C4⋊Dic3C2×D711yescharacteristic
101Dic3⋊C4—141no—
102Dic3⋊C4—141no—
103C2×C3⋊C8C2×D711yescharacteristic
104D3×C8—142no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7C2×Q1611yescharacteristic
107C2×C3×C7#13511yescharacteristic
108D21—22no—
109C2×Q16—71no—
110C22×C8—211no—
111C22⋊Q8—211no—
112C22⋊Q8—211no—
113#107—211no—
114#103—31no—
115#103—211no—
116C22×C7D3×D411yescharacteristic
117C2×D7—32no—
118C2×D7—61no—
119C2×D7—61no—
120C4×C7—41no—
121C4×C7—121no—
122C4×C7#23711yescharacteristic
123C4×C7C2×#6711yescharacteristic
124Dic7—41no—
125Dic7—61no—
126Dic7—121no—
127C22×D3—71no—
128C3×Q8—21no—
129C3×Q8—21no—
130C3×Q8—141no—
131C3×Q8—141no—
132C2×C3×C4—21no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—21no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138C4×D3—141no—
139C4×D3—141no—
140C3×C8—72no—
141C3⋊C8—21no—
142C3×C7#46711yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8D3×D711yescharacteristic
145C22×C4—211no—
146Q16—142no—
147C2×C8—31no—
148C2×C8—71no—
149C2×C8—422no—
150C4⋊C4—31no—
151C4⋊C4—211no—
152C4⋊C4—421no—
153C4⋊C4—421no—
154C22⋊C4—421no—
155C22⋊C4—421no—
156C2×C7D3×Q1611yescharacteristic
157C2×C7#91911yescharacteristic
158C2×C7#93811yescharacteristic
159D7—62no—
160C22×C3D4×D711yescharacteristic
161C2×D3—72no—
162C2×D3—141no—
163C2×D3—141no—
164C3×C4—41no—
165C3×C4—281no—
166C3×C4#115711yescharacteristic
167C3×C4D4×D711yescharacteristic
168Dic3—41no—
169Dic3—141no—
170Dic3—281no—
171C23—211no—
172Q8—21no—
173Q8—21no—
174Q8—141no—
175Q8—141no—
176C2×C4—21no—
177C2×C4—61no—
178C2×C4—141no—
179C2×C4—211no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—72no—
186C7#518811yescharacteristic, complemented, Sylow
187C2×C3#644711yescharacteristic
188C2×C3#647911yescharacteristic
189C2×C3D7×Q1611yescharacteristic
190D3—142no—
191C22—212no—
192C22—421no—
193C22—421no—
194C22#6669411yescharacteristic
195C4—41no—
196C4—121no—
197C4—281no—
198C4—421no—
199C4—841no—
200C4#6668311yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8958511yescharacteristic, complemented, Sylow
203C2—422no—
204C2#1062107311yescharacteristic
205C2#1062108511yescharacteristic
206C2#1062109511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#467
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1144C2semidirect
5. C7×#835C2semidirect
6. #10620523C2semidirect
7. #10620506C2semidirect
23. C2×#1144D3semidirect
63. #835D7semidirect
68. C2×D21Q16semidirect
142. C3×C7#467semidirect
186. C7#5188semidirect
202. C3#89585semidirect

Maximal subgroups

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

subgroupindex
#106214292
C2×C3×#11442
#106209522
C2×D21⋊C82
C7×#8352
#106205232
#106205062
#895853
#51887