Finite Group Databasebeta

← C3×#89823 #13055479

Subgroups — C3×#89823 #13055479

Order 1344.

228 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C3×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3C3×#6487C211yesmaximal, characteristic
4C22×C3×C7⋊C8C211yesmaximal, characteristic
5C3×#6409C211yesmaximal, characteristic, complemented
6C3×#6384C211yesmaximal, characteristic, complemented
7C3×#6381C211yesmaximal, characteristic, complemented
8#89823C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C3×C7×C4⋊C4—21no—
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×C7×C22⋊C4—21no—
15C3×C22⋊Dic7—21no—
16C3×#1144—22no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×Dic7⋊C4—21no—
19C2×C3×C7⋊C8—22no—
20C2×C3×C7⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22C7×C22⋊Q8C2×C311yescharacteristic
23C2×#1144C2×C311yescharacteristic, complemented
24#6487C2×C311yescharacteristic
25C22×C7⋊C8C2×C311yescharacteristic
26#6409C2×C311yescharacteristic, complemented
27#6384C2×C311yescharacteristic, complemented
28#6381C2×C311yescharacteristic, complemented
29C3×#467—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7D411yescharacteristic
38C2×C3×C4×C7C2311yescharacteristic
39C2×C3×Dic7—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3×C7⋊C8—21no—
44C3×C7⋊C8D412yes—
45C2×C7×Q8C22×C311yescharacteristic
46C22×C4×C7C22×C311yescharacteristic
47C2×Dic14C22×C311yescharacteristic
48C7×C4⋊C4—21no—
49C7×C4⋊C4C22×C311yescharacteristic
50C7×C22⋊C4—21no—
51C22⋊Dic7—21no—
52#1144—22no—
53C4⋊Dic7C22×C311yescharacteristic
54Dic7⋊C4—21no—
55C2×C7⋊C8—22no—
56C2×C7⋊C8C22×C311yescharacteristic
57C2×C7⋊C8C22×C311yescharacteristic
58C2×C3×Q16—71no—
59C22×C3×C8—71no—
60C3×C22⋊Q8—71no—
61C3×C22⋊Q8D711yescharacteristic
62C3×#107—71no—
63C3×#103—71no—
64C3×#103—71no—
65C22×C3×C7—21no—
66C22×C3×C7—21no—
67C22×C3×C7Q1612yescomplemented
68C22×C3×C7C2×D411yescharacteristic
69C3×C4×C7—21no—
70C3×C4×C7—41no—
71C3×C4×C7—41no—
72C3×C4×C7C2×D411yescharacteristic
73C3×C4×C7#4111yescharacteristic
74C3×Dic7—41no—
75C3×Dic7—41no—
76#467—71noSylow
77C23×C7C3×D411yescharacteristic
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C3×D411yescharacteristic
85C2×C4×C7C23×C311yescharacteristic
86C2×Dic7—21no—
87C2×Dic7—21no—
88Dic14—21no—
89Dic14—21no—
90C7⋊C8—21no—
91C7⋊C8C3×D412yes—
92C2×C3×Q8—71no—
93C2×C3×Q8C2×D711yescharacteristic
94C22×C3×C4C2×D711yescharacteristic
95C3×Q16—142no—
96C2×C3×C8—71no—
97C2×C3×C8—71no—
98C2×C3×C8—142no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—71no—
101C3×C4⋊C4—141no—
102C3×C4⋊C4C2×D711yescharacteristic
103C3×C22⋊C4—21no—
104C3×C22⋊C4—141no—
105C2×C3×C7—22no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7C2×Q1611yescharacteristic
108C2×C3×C7#13511yescharacteristic
109C2×Q16—71no—
110C22×C8—71no—
111C22⋊Q8—71no—
112C22⋊Q8C3×D711yescharacteristic
113#107—71no—
114#103—71no—
115#103—71no—
116C22×C7—21no—
117C22×C7—21no—
118C22×C7C3×Q1612yescomplemented
119C22×C7C2×C3×D411yescharacteristic
120C4×C7—21no—
121C4×C7—41no—
122C4×C7—41no—
123C4×C7C2×C3×D411yescharacteristic
124C4×C7C3×#4111yescharacteristic
125Dic7—41no—
126Dic7—41no—
127C23×C3C22⋊D711yescharacteristic
128C3×Q8—21no—
129C3×Q8—21no—
130C3×Q8—141no—
131C3×Q8—141no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—141no—
137C2×C3×C4—141no—
138C2×C3×C4C22⋊D711yescharacteristic
139C2×C3×C4C22×D711yescharacteristic
140C3×C8—72no—
141C3×C8—141no—
142C3×C7#46711yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8C2×C3×D711yescharacteristic
145C22×C4C2×C3×D711yescharacteristic
146Q16—142no—
147C2×C8—71no—
148C2×C8—71no—
149C2×C8—142no—
150C4⋊C4—21no—
151C4⋊C4—71no—
152C4⋊C4—141no—
153C4⋊C4C2×C3×D711yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—141no—
156C2×C7—22no—
157C2×C7C3×#12111yescharacteristic
158C2×C7C2×C3×Q1611yescharacteristic
159C2×C7C3×#13511yescharacteristic
160C22×C3—21no—
161C22×C3—21no—
162C22×C3#114412yescomplemented
163C22×C3C2×C22⋊D711yescharacteristic
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—281no—
168C3×C4—281no—
169C3×C4D4×D711yescharacteristic
170C3×C4#115911yescharacteristic
171C23C3×C22⋊D711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—141no—
175Q8—141no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4C3×C22⋊D711yescharacteristic
183C2×C4C22×C3×D711yescharacteristic
184C8—72no—
185C8—141no—
186C7C3×#46711yescharacteristic, complemented, Sylow
187C2×C3—22no—
188C2×C3#650211yescharacteristic
189C2×C3C2×#114411yescharacteristic
190C2×C3#651311yescharacteristic
191C22—21no—
192C22—21no—
193C22C3×#114412yescomplemented
194C22C2×C3×C22⋊D711yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—281no—
199C4—281no—
200C4C3×D4×D711yescharacteristic
201C4C3×#115911yescharacteristic
202C3#8982311yescharacteristic, direct product factor, Sylow
203C2—22no—
204C2C3×#650211yescharacteristic
205C2C2×C3×#114411yescharacteristic
206C2C3×#651311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#467
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1144C2semidirect
5. C3×#6409C2semidirect
6. C3×#6384C2semidirect
7. C3×#6381C2semidirect
8. #89823C3direct (N × H)
23. C2×#1144C2×C3semidirect
26. #6409C2×C3semidirect
27. #6384C2×C3semidirect
28. #6381C2×C3semidirect
67. C22×C3×C7Q16semidirect
118. C22×C7C3×Q16semidirect
142. C3×C7#467semidirect
162. C22×C3#1144semidirect
186. C7C3×#467semidirect
193. C22C3×#1144semidirect
202. C3#89823direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
C2×C3×#11442
C3×#64872
C22×C3×C7⋊C82
C3×#64092
C3×#63842
C3×#63812
#898233
C3×#4677