Finite Group Databasebeta

← #13054347

Subgroups — #13054347

Order 1344.

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×#66730C211yesmaximal, characteristic
2#10621094C212yesmaximal, complemented
3C7×#834C211yesmaximal, characteristic, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5#10620532C212yesmaximal, complemented
6#89512—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268C411yescharacteristic, complemented
9C2×#4268C2211yescharacteristic
10C22×Dic21C411yescharacteristic, complemented
11C22×Dic21C2211yescharacteristic
12#66730—22no—
13#66730—22no—
14C2×C4×D21C411yescharacteristic, complemented
15C2×Dic42C411yescharacteristic, complemented
16C7×#217C2212yes—
17C3×C22⋊Dic7C2212yes—
18#66577—22no—
19Dic3×Dic7—22no—
20C2×#1159—31no—
21#6440—32no—
22C7×#99—31no—
23#6408D311yescharacteristic, complemented
24#6380—32no—
25#4740—71nomaximal, Hall
26C23×C3×C7C2×C411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×D21C2×C411yescharacteristic
29C3×C7×D4—22no—
30C2×C3×C4×C7C2×C411yescharacteristic
31#4268—22no—
32#4268—22no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21C2×C411yescharacteristic
36C2×Dic21C2×C411yescharacteristic
37C2×Dic21C2×C411yescharacteristic
38C2×Dic21D412yes—
39C2×Dic21D412yes—
40C4×D21—21no—
41C4×D21—21no—
42Dic42—22no—
43C2×C7×Dic3—22no—
44C2×C3×Dic7—22no—
45C2×C7×D4C2×D311yescharacteristic
46C2×C22⋊D7—31no—
47C2×C22⋊D7—31no—
48C22×Dic7—31no—
49C22×Dic7—31no—
50#1159—62no—
51#1159—62no—
52C2×C4×D7—31no—
53C2×Dic14—31no—
54C7×C22⋊C4—32no—
55C22⋊Dic7C2×D312yes—
56Dic7⋊C4—62no—
57C4×Dic7—62no—
58C2×#237—71no—
59#877—72no—
60C3×#99—71no—
61#834D711yescharacteristic, complemented
62#806—72no—
63C22×C3×C7—21no—
64C22×C3×C7—41no—
65C22×C3×C7—41no—
66C22×C3×C7C22×C411yescharacteristic
67C22×C3×C7C2×D412yes—
68C2×D21—21no—
69C2×D21—41no—
70C3×C4×C7—21no—
71Dic21—21no—
72Dic21C22⋊C412yes—
73Dic21C22⋊C411yescharacteristic
74Dic21C22⋊C411yescharacteristic
75C3×Dic7—42no—
76C7×Dic3—42no—
77#410—211noSylow
78C23×C7C4×D311yescharacteristic
79C23×C7C22×D311yescharacteristic
80C22×D7—31no—
81C7×D4—22no—
82C2×C4×C7—62no—
83C2×C4×C7C4×D311yescharacteristic
84C22⋊D7—62no—
85C22⋊D7—62no—
86C2×Dic7—22no—
87C2×Dic7—31no—
88C2×Dic7—31no—
89C2×Dic7—32no—
90C2×Dic7—32no—
91C2×Dic7—31no—
92C2×Dic7—61no—
93C2×Dic7—61no—
94C4×D7—61no—
95C4×D7—61no—
96Dic14—62no—
97C2×C3×D4C2×D711yescharacteristic
98C2×#67—71no—
99C2×#67—71no—
100C22×Dic3—71no—
101C22×Dic3—71no—
102#237—142no—
103#237—142no—
104C2×C4×D3—71no—
105C2×Dic6—71no—
106C3×C22⋊C4—72no—
107#217C2×D712yes—
108Dic3⋊C4—142no—
109C4×Dic3—142no—
110C2×C3×C7—22no—
111C2×C3×C7—21no—
112C2×C3×C7—41no—
113C2×C3×C7C2×C22⋊C411yescharacteristic
114D21—41no—
115C2×#41—211no—
116#117—212no—
117#99—31no—
118#99—71no—
119#99—212no—
120C22×C7—21no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7C2×C4×D311yescharacteristic
124C22×C7D3×D412yes—
125C2×D7—61no—
126C2×D7—121no—
127C4×C7—21no—
128C4×C7—122no—
129Dic7—32no—
130Dic7—31no—
131Dic7—31no—
132Dic7—42no—
133Dic7—61no—
134C23×C3C4×D711yescharacteristic
135C23×C3C22×D711yescharacteristic
136C22×D3—71no—
137C3×D4—22no—
138C2×C3×C4—142no—
139C2×C3×C4C4×D711yescharacteristic
140#67—142no—
141#67—142no—
142C2×Dic3—22no—
143C2×Dic3—71no—
144C2×Dic3—71no—
145C2×Dic3—72no—
146C2×Dic3—72no—
147C2×Dic3—71no—
148C2×Dic3—141no—
149C2×Dic3—141no—
150C4×D3—141no—
151C4×D3—141no—
152Dic6—142no—
153C3×C7#41011yescharacteristic, complemented, Hall
154#41—422no—
155#41—422no—
156C2×Q8—211no—
157C2×D4—211no—
158C2×D4—211no—
159C2×D4D3×D711yescharacteristic
160C22×C4—211no—
161C22×C4—211no—
162C22×C4—211no—
163C4⋊C4—422no—
164C22⋊C4—32no—
165C22⋊C4—72no—
166C42—422no—
167C2×C7—22no—
168C2×C7—21no—
169C2×C7—41no—
170C2×C7D3×C22⋊C411yescharacteristic
171D7—121no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3C2×C4×D711yescharacteristic
176C22×C3D4×D712yes—
177C2×D3—141no—
178C2×D3—281no—
179C3×C4—21no—
180C3×C4—282no—
181Dic3—42no—
182Dic3—72no—
183Dic3—71no—
184Dic3—71no—
185Dic3—141no—
186C23—211no—
187C23D21⋊C411yescharacteristic
188C23C2×D3×D711yescharacteristic
189Q8—422no—
190D4—22no—
191D4—422no—
192D4—422no—
193C2×C4—62no—
194C2×C4—142no—
195C2×C4—211no—
196C2×C4—211no—
197C2×C4—212no—
198C2×C4—212no—
199C2×C4—211no—
200C2×C4—421no—
201C2×C4—421no—
202C2×C4—421no—
203C2×C4—421no—
204C2×C4D21⋊C411yescharacteristic
205C7#474011yescharacteristic, complemented, Sylow
206C2×C3—22no—
207C2×C3—21no—
208C2×C3—41no—
209C2×C3D7×C22⋊C411yescharacteristic
210D3—281no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—421no—
215C22—841no—
216C22C2×D21⋊C411yescharacteristic
217C22#6669412yes—
218C4—21no—
219C4—122no—
220C4—211no—
221C4—211no—
222C4—212no—
223C4—282no—
224C4—421no—
225C3#8951211yescharacteristic, complemented, Sylow
226C2—22no—
227C2—21no—
228C2—41no—
229C2—841no—
230C2#1062112311yescharacteristic
subgroupG/H
centerC2#10621123
commutatorC22×C3×C7C22×C4
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×#66730C2
cosocleC23×C3×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#410
LCSG ▷ C22×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. #10621094C2semidirect
3. C7×#834C2semidirect
4. C3×#6408C2semidirect
5. #10620532C2semidirect
8. C2×#4268C4semidirect
10. C22×Dic21C4semidirect
14. C2×C4×D21C4semidirect
15. C2×Dic42C4semidirect
23. #6408D3semidirect
61. #834D7semidirect
153. C3×C7#410semidirect
205. C7#4740semidirect
225. C3#89512semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
#106210942
C7×#8342
C3×#64082
#106205322
#895123
#47407