Finite Group Databasebeta

← #13054427

Subgroups — #13054427

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
2C3×#6498C211yesmaximal, characteristic, complemented
3#10621101C212yesmaximal, complemented
4#10620673C211yesmaximal, characteristic, complemented
5#10620531C212yesmaximal, complemented
6#90015—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×C7×#67—21no—
9C2×C7×#67C2212yes—
10C7×#237—22no—
11D3×C7×D4—22no—
12C7×#235—21no—
13C22×C3×Dic7C2211yescharacteristic, complemented
14#66636C2212yescomplemented
15C3×C22⋊Dic7—21no—
16C3×C22⋊Dic7C2211yescharacteristic, complemented
17C3×Dic7⋊C4—21no—
18#66577—22no—
19(C2×D3)⋊Dic7—22no—
20C7×H(5,𝔽2)—31no—
21#6498—32no—
22#6498D311yescharacteristic, complemented
23#6408—31no—
24#6408—32no—
25#5053—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—
33C7×#67—22no—
34C7×#67—22no—
35C7×#67—22no—
36C2×C7×Dic3—21no—
37C2×C7×Dic3D412yes—
38C7×D12—21no—
39C4×D3×C7—22no—
40C7×Dic6—21no—
41C2×C3×Dic7—22no—
42C2×C3×Dic7—21no—
43C2×C3×Dic7—21no—
44C7×#41—62no—
45C7×#41—61no—
46C2×C7×D4—32no—
47C2×C7×D4—62no—
48C2×C7×D4—61no—
49C2×C7×D4C2×D311yescharacteristic
50C22×Dic7C2×D311yescharacteristic, complemented
51C22⋊Dic7—21no—
52C22⋊Dic7—32no—
53C22⋊Dic7—62no—
54C22⋊Dic7C2×D311yescharacteristic, complemented
55Dic7⋊C4—21no—
56Dic7⋊C4—62no—
57#1004D711yescharacteristic
58C3×#123—71no—
59#929—72no—
60#834—71no—
61#806—72no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D412yes—
66C22×C3×C7C2×D411yescharacteristic
67C2×D3×C7—22no—
68C2×D3×C7—42no—
69C2×D3×C7—81no—
70C3×C4×C7—21no—
71Dic21—42no—
72C3×Dic7—41no—
73C3×Dic7—41no—
74C7×Dic3—22no—
75#458—211noSylow
76C23×C7—32no—
77C23×C7—61no—
78C23×C7#6711yescharacteristic
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×C7#6711yescharacteristic
90C2×Dic7—22no—
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7—62no—
94C2×C3×D4C2×D711yescharacteristic
95C22×C3×C4—71no—
96C2×#67—21no—
97C2×#67C2×D712yes—
98#237—22no—
99D3×D4—22no—
100#235—21no—
101C3×C4⋊C4—141no—
102C3×C22⋊C4—71no—
103C3×C22⋊C4—141no—
104#217—72no—
105(C2×D3)⋊C4—142no—
106Dic3⋊C4—142no—
107C2×C3×C7—21no—
108C2×C3×C7—22no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111D3×C7—42no—
112H(5,𝔽2)—31no—
113#123—71no—
114#123—212no—
115#99—212no—
116#99—211no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7—62no—
121C22×C7—122no—
122C22×C7—241no—
123C22×C7D3×D412yes—
124C22×C7C2×#6711yescharacteristic
125C4×C7—21no—
126C4×C7—62no—
127Dic7—41no—
128Dic7—41no—
129Dic7—122no—
130C23×C3C22⋊D711yescharacteristic
131C23×C3C22×D711yescharacteristic
132C22×D3—21no—
133C22×D3C22⋊D712yes—
134C3×D4—22no—
135C2×C3×C4—142no—
136C2×C3×C4—141no—
137C2×C3×C4—141no—
138C2×C3×C4C22⋊D711yescharacteristic
139#67—22no—
140#67—22no—
141#67—22no—
142C2×Dic3—21no—
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—71no—
156C4⋊C4—141no—
157C4⋊C4—422no—
158C22⋊C4—71no—
159C22⋊C4—141no—
160C22⋊C4—212no—
161C22⋊C4—422no—
162C2×C7—21no—
163C2×C7—22no—
164C2×C7—41no—
165C2×C7—122no—
166C2×C7#93711yescharacteristic
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—281no—
177C3×C4—281no—
178Dic3—22no—
179Dic3—282no—
180C23—32no—
181C23—61no—
182C23#424511yescharacteristic
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—141no—
194C2×C4—141no—
195C2×C4—142no—
196C2×C4—422no—
197C2×C4#424511yescharacteristic
198C7#505311yescharacteristic, complemented, Sylow
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—41no—
202C2×C3#651611yescharacteristic
203D3—42no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—62no—
208C22—122no—
209C22—241no—
210C22C2×#424511yescharacteristic
211C22D3×C22⋊D712yes—
212C4—21no—
213C4—62no—
214C4—281no—
215C4—281no—
216C4—842no—
217C3#9001511yescharacteristic, complemented, Sylow
218C2—21no—
219C2—22no—
220C2—41no—
221C2—122no—
222C2#1062113011yescharacteristic
subgroupG/H
centerC2#10621130
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. C3×#6498C2semidirect
3. #10621101C2semidirect
4. #10620673C2semidirect
5. #10620531C2semidirect
13. C22×C3×Dic7C22semidirect
14. #66636C22semidirect
16. C3×C22⋊Dic7C22semidirect
22. #6498D3semidirect
50. C22×Dic7C2×D3semidirect
54. C22⋊Dic7C2×D3semidirect
148. C3×C7#458semidirect
198. C7#5053semidirect
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
C3×#64982
#106211012
#106206732
#106205312
#900153
#50537