Finite Group Databasebeta

← #13054712

Subgroups — #13054712

Order 1344.

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

normal subgroups only

This group has 220 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
1C7×#946C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620931C211yesmaximal, characteristic
4C2×#66558C211yesmaximal, characteristic
5#10620674C211yesmaximal, characteristic, complemented
6#10620520C211yesmaximal, characteristic, complemented
7#10620505C211yesmaximal, characteristic, complemented
8#89824—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D3×C7C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×(C2×D3)⋊C4—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C7×Dic3⋊C4—21no—
17C3×#1144—22no—
18C2×C3×C7⋊C8C2211yescharacteristic
19#66577—21no—
20(C2×D3)⋊Dic7—21no—
21#66558—22no—
22C7×C22⋊Q8—31no—
23C2×#1144D311yescharacteristic, complemented
24#6487—31no—
25C2×#1136—31no—
26#6409—31no—
27#6384—31no—
28#6382—31no—
29#5189—71nomaximal, Hall
30C22×D3×C7D411yescharacteristic
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×C7×Dic3—21no—
37C2×C7×Dic3D411yescharacteristic
38C4×D3×C7—21no—
39C4×D3×C7—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8D412yes—
45C2×C7×Q8C2×D311yescharacteristic
46C22×C4×C7—31no—
47C2×Dic14C2×D311yescharacteristic
48C7×C4⋊C4—31no—
49C7×C4⋊C4—61no—
50C7×C22⋊C4—61no—
51C22⋊Dic7—61no—
52#1144—22no—
53C4⋊Dic7—31no—
54Dic7⋊C4—61no—
55#1136—62no—
56C2×C7⋊C8—31no—
57C2×C7⋊C8C2×D311yescharacteristic
58C2×C3×Q16—71no—
59#946—71no—
60#946D711yescharacteristic
61C2×ΓL(1,𝔽25)—71no—
62#835—71no—
63#835—71no—
64#817—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D3×C7—21no—
67C2×D3×C7—21no—
68C2×D3×C7—21no—
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73C3×Dic7—41no—
74C7×Dic3—21no—
75C7×Dic3—41no—
76#470—211noSylow
77C23×C7—31no—
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7—31no—
82C2×C4×C7—61no—
83C2×C4×C7—61no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—21no—
87C2×Dic7—61no—
88Dic14—21no—
89Dic14—21no—
90C7⋊C8—61no—
91C7⋊C8#6712yes—
92C2×C3×Q8—71no—
93C2×C3×Q8C2×D711yescharacteristic
94C2×C4×D3C2×D711yescharacteristic
95C3×Q16—142no—
96C2×C3×C8—71no—
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—141no—
99C4⋊Dic3—71no—
100C4⋊Dic3C2×D711yescharacteristic
101Dic3⋊C4—21no—
102Dic3⋊C4—141no—
103C2×C3⋊C8—71no—
104ΓL(1,𝔽25)—142no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13711yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D3×C7—41no—
109C2×Q16—71no—
110C2×M16—211no—
111C22⋊Q8—31no—
112C22⋊Q8—211no—
113#106—211no—
114#103—211no—
115#103—211no—
116C22×C7—61no—
117C22×C7—61no—
118C22×C7—61no—
119C22×C7D3×D411yescharacteristic
120C4×C7—41no—
121C4×C7—61no—
122C4×C7—121no—
123C4×C7#23711yescharacteristic
124C4×C7C2×#6711yescharacteristic
125Dic7—41no—
126Dic7—121no—
127C22×D3C22⋊D711yescharacteristic
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—141no—
137C2×Dic3C22⋊D711yescharacteristic
138C4×D3—21no—
139C4×D3—21no—
140C3×C8—72no—
141C3⋊C8—141no—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8D3×D711yescharacteristic
145C22×C4—31no—
146Q16—142no—
147M16—422no—
148C2×C8—71no—
149C2×C8—211no—
150C4⋊C4—31no—
151C4⋊C4—61no—
152C4⋊C4—211no—
153C4⋊C4—421no—
154C22⋊C4—61no—
155C22⋊C4—421no—
156C2×C7—121no—
157C2×C7#91811yescharacteristic
158C2×C7#91811yescharacteristic
159C2×C7#93811yescharacteristic
160C22×C3C2×C22⋊D711yescharacteristic
161C2×D3—21no—
162C2×D3—21no—
163C2×D3—21no—
164C3×C4—41no—
165C3×C4—281no—
166C3×C4D4×D711yescharacteristic
167C3×C4#115911yescharacteristic
168Dic3—21no—
169Dic3—41no—
170Dic3—281no—
171C23—31no—
172Q8—21no—
173Q8—21no—
174Q8—141no—
175Q8—141no—
176C2×C4—21no—
177C2×C4—31no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—72no—
185C8—421no—
186C7#518911yescharacteristic, complemented, Sylow
187C2×C3#650211yescharacteristic
188C2×C3#650511yescharacteristic
189C2×C3#651411yescharacteristic
190D3—41no—
191C22—61no—
192C22—61no—
193C22—61no—
194C22D3×C22⋊D711yescharacteristic
195C4—41no—
196C4—61no—
197C4—121no—
198C4—281no—
199C4—841no—
200C4#6668411yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8982411yescharacteristic, complemented, Sylow
203C2—121no—
204C2#1062107211yescharacteristic
205C2#1062108211yescharacteristic
206C2#1062112711yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D3×C7D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#470
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. #10620674C2semidirect
6. #10620520C2semidirect
7. #10620505C2semidirect
23. C2×#1144D3semidirect
142. C3×C7#470semidirect
186. C7#5189semidirect
202. C3#89824semidirect

Maximal subgroups

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

subgroupindex
C7×#9462
C2×C3×#11442
#106209312
C2×#665582
#106206742
#106205202
#106205052
#898243
#51897