Finite Group Databasebeta

← #13054428

Subgroups — #13054428

Order 1344.

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

normal subgroups only

This group has 320 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×#1004C211yesmaximal, characteristic
2#10621418C211yesmaximal, characteristic, complemented
3#10621105C212yesmaximal, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5#10620531C212yesmaximal, complemented
6#90015—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C22×Dic21C2211yescharacteristic, complemented
9C2×C7×#67—21no—
10C2×C7×#67C2212yes—
11C7×#237—22no—
12D3×C7×D4—22no—
13C7×#235—21no—
14#66636—21no—
15#66636C2211yescharacteristic, complemented
16#66629—21no—
17C3×C22⋊Dic7C2212yescomplemented
18Dic7⋊Dic3—22no—
19(C2×D3)⋊Dic7—22no—
20C7×H(5,𝔽2)—31no—
21#6498—31no—
22#6498—32no—
23#6408—32no—
24#6408D311yescharacteristic, complemented
25#4742—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×D3×C7—21no—
29C22×D3×C7D412yes—
30C3×C7×D4—22no—
31C2×C3×C4×C7D411yescharacteristic
32C2×Dic21—22no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C7×#67—22no—
36C7×#67—22no—
37C7×#67—22no—
38C2×C7×Dic3—21no—
39C2×C7×Dic3D412yes—
40C7×D12—21no—
41C4×D3×C7—22no—
42C7×Dic6—21no—
43C2×C3×Dic7—22no—
44C7×#41—62no—
45C7×#41—61no—
46C2×C7×D4—32no—
47C2×C7×D4—62no—
48C2×C7×D4—61no—
49C2×C7×D4C2×D311yescharacteristic
50C22×Dic7—31no—
51C22⋊Dic7—31no—
52C22⋊Dic7—62no—
53C22⋊Dic7—61no—
54C22⋊Dic7C2×D312yescomplemented
55Dic7⋊C4—62no—
56Dic7⋊C4—61no—
57#1004D711yescharacteristic
58#935—71no—
59#886—72no—
60C3×#99—71no—
61#806—72no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D411yescharacteristic
66C22×C3×C7C2×D412yes—
67C2×D3×C7—22no—
68C2×D3×C7—42no—
69C2×D3×C7—81no—
70C3×C4×C7—21no—
71Dic21—41no—
72Dic21—41no—
73C3×Dic7—42no—
74C7×Dic3—22no—
75#458—211noSylow
76C23×C7—32no—
77C23×C7—61no—
78C23×C7D1211yescharacteristic
79C23×C7C22×D311yescharacteristic
80C7×Q8—61no—
81C7×D4—22no—
82C7×D4—61no—
83C7×D4—62no—
84C7×D4—62no—
85C7×D4—62no—
86C2×C4×C7—32no—
87C2×C4×C7—62no—
88C2×C4×C7—61no—
89C2×C4×C7D1211yescharacteristic
90C2×Dic7—22no—
91C2×Dic7—61no—
92C2×Dic7—61no—
93C2×Dic7—62no—
94C2×C3×D4C2×D711yescharacteristic
95C2×#67—21no—
96C2×#67C2×D712yes—
97C22×Dic3—71no—
98#237—22no—
99D3×D4—22no—
100#235—21no—
101C3×C22⋊C4—72no—
102#217—71no—
103#217—141no—
104(C2×D3)⋊C4—142no—
105C4⋊Dic3—142no—
106Dic3⋊C4—141no—
107C2×C3×C7—22no—
108C2×C3×C7—21no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111D3×C7—42no—
112H(5,𝔽2)—31no—
113#123—212no—
114#123—211no—
115#99—71no—
116#99—212no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7—62no—
121C22×C7—122no—
122C22×C7—241no—
123C22×C7C2×D1211yescharacteristic
124C22×C7D3×D412yes—
125C4×C7—21no—
126C4×C7—62no—
127Dic7—42no—
128Dic7—121no—
129Dic7—121no—
130C23×C3C22⋊D711yescharacteristic
131C23×C3C22×D711yescharacteristic
132C22×D3—21no—
133C22×D3C22⋊D712yes—
134C3×D4—22no—
135C2×C3×C4—142no—
136C2×C3×C4C22⋊D711yescharacteristic
137#67—22no—
138#67—22no—
139#67—22no—
140C2×Dic3—21no—
141C2×Dic3—141no—
142C2×Dic3—141no—
143C2×Dic3—142no—
144C2×Dic3C22⋊D712yes—
145D12—21no—
146C4×D3—22no—
147Dic6—21no—
148C3×C7#45811yescharacteristic, complemented, Hall
149#41—62no—
150#41—61no—
151C2×D4—32no—
152C2×D4—62no—
153C2×D4—61no—
154C2×D4D3×D711yescharacteristic
155C22×C4—211no—
156C4⋊C4—422no—
157C4⋊C4—421no—
158C22⋊C4—72no—
159C22⋊C4—211no—
160C22⋊C4—422no—
161C22⋊C4—421no—
162C2×C7—22no—
163C2×C7—21no—
164C2×C7—41no—
165C2×C7—122no—
166C2×C7#88211yescharacteristic
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3C2×C22⋊D711yescharacteristic
171C22×C3C2×C22⋊D712yes—
172C2×D3—22no—
173C2×D3—42no—
174C2×D3—81no—
175C3×C4—21no—
176C3×C4—282no—
177Dic3—22no—
178Dic3—281no—
179Dic3—281no—
180C23—32no—
181C23—61no—
182C23#424711yescharacteristic
183C23C2×D3×D711yescharacteristic
184Q8—61no—
185D4—22no—
186D4—61no—
187D4—62no—
188D4—62no—
189D4—62no—
190C2×C4—32no—
191C2×C4—62no—
192C2×C4—61no—
193C2×C4—142no—
194C2×C4—421no—
195C2×C4—421no—
196C2×C4—422no—
197C2×C4#424711yescharacteristic
198C7#474211yescharacteristic, complemented, Sylow
199C2×C3—21no—
200C2×C3—22no—
201C2×C3—41no—
202C2×C3#651611yescharacteristic
203D3—42no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—62no—
208C22—122no—
209C22—241no—
210C22C2×#424711yescharacteristic
211C22D3×C22⋊D712yes—
212C4—21no—
213C4—62no—
214C4—282no—
215C4—841no—
216C4—841no—
217C3#9001511yescharacteristic, complemented, Sylow
218C2—22no—
219C2—21no—
220C2—41no—
221C2—122no—
222C2#1062113211yescharacteristic
subgroupG/H
centerC2#10621132
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radicalC7×#1004C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#458
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10621418C2semidirect
3. #10621105C2semidirect
4. C3×#6408C2semidirect
5. #10620531C2semidirect
8. C22×Dic21C22semidirect
15. #66636C22semidirect
17. C3×C22⋊Dic7C22semidirect
24. #6408D3semidirect
54. C22⋊Dic7C2×D3semidirect
148. C3×C7#458semidirect
198. C7#4742semidirect
217. C3#90015semidirect

Maximal subgroups

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

subgroupindex
C7×#10042
#106214182
#106211052
C3×#64082
#106205312
#900153
#47427