Finite Group Databasebeta

← C3×#89821 #13055482

Subgroups — C3×#89821 #13055482

Order 1344.

240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 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, complemented
2C2×C3×#1143C211yesmaximal, characteristic, complemented
3C3×#6493C211yesmaximal, characteristic
4C2×C3×#1136C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6C3×#6383C211yesmaximal, characteristic, complemented
7C3×#6381C211yesmaximal, characteristic, complemented
8#89821C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C2×C3×D28C2211yescharacteristic
13C3×C7×C4⋊C4—21no—
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16C3×#1143—22no—
17C3×(C2×D7)⋊C4—21no—
18C3×C4⋊Dic7C2211yescharacteristic
19C3×#1136—22no—
20C2×C3×C7⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22C7×C22⋊Q8C2×C311yescharacteristic, complemented
23C2×#1143C2×C311yescharacteristic, complemented
24#6493C2×C311yescharacteristic
25C2×#1136C2×C311yescharacteristic, complemented
26#6409C2×C311yescharacteristic, complemented
27#6383C2×C311yescharacteristic, complemented
28#6381C2×C311yescharacteristic, complemented
29C3×#468—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C22×C3×D7—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7D411yescharacteristic
39C2×C3×C4×C7C2311yescharacteristic
40C3×C22⋊D7—42no—
41C2×C3×Dic7—21no—
42C3×D28—21no—
43C3×D28—21no—
44C3×C7⋊C8—21no—
45C3×C7⋊C8D412yescomplemented
46C2×C7×Q8C22×C311yescharacteristic
47C22×C4×C7C22×C311yescharacteristic
48C2×C22⋊D7—21no—
49C2×D28C22×C311yescharacteristic
50C7×C4⋊C4—21no—
51C7×C4⋊C4C22×C311yescharacteristic
52C7×C22⋊C4—21no—
53#1143—22no—
54(C2×D7)⋊C4—21no—
55C4⋊Dic7C22×C311yescharacteristic
56#1136—22no—
57C2×C7⋊C8C22×C311yescharacteristic
58C2×C7⋊C8C22×C311yescharacteristic
59C2×C3×SD16—71no—
60C2×C3×M16—71no—
61C3×C22⋊Q8D711yescharacteristic, complemented
62C3×#121—71no—
63C3×#107—71no—
64C3×#103—71no—
65C3×#102—71no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—41no—
71C2×C3×D7—41no—
72C2×C3×D7—41no—
73C3×C4×C7—21no—
74C3×C4×C7—41no—
75C3×C4×C7—41no—
76C3×C4×C7C2×D411yescharacteristic
77C3×C4×C7#4111yescharacteristic
78C3×Dic7—41no—
79#468—71noSylow
80C23×C7C3×D411yescharacteristic
81C22×D7—21no—
82C7×Q8—21no—
83C7×Q8—21no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7C3×D411yescharacteristic
89C2×C4×C7C23×C311yescharacteristic
90C22⋊D7—42no—
91C2×Dic7—21no—
92D28—21no—
93D28—21no—
94C7⋊C8—21no—
95C7⋊C8C3×D412yescomplemented
96C2×C3×Q8C2×D711yescharacteristic
97C2×C3×D4—71no—
98C2×C3×D4—141no—
99C22×C3×C4C2×D711yescharacteristic
100C3×SD16—142no—
101C3×M16—142no—
102C2×C3×C8—71no—
103C2×C3×C8—71no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—71no—
106C3×C4⋊C4C2×D711yescharacteristic
107C3×C22⋊C4—21no—
108C3×C22⋊C4—141no—
109C2×C3×C7—41no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113C3×D7—81no—
114C2×SD16—71no—
115C2×M16—71no—
116C22⋊Q8C3×D711yescharacteristic, complemented
117#121—71no—
118#107—71no—
119#103—71no—
120#102—71no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7—21no—
124C22×C7C2×C3×D411yescharacteristic
125C2×D7—41no—
126C2×D7—41no—
127C2×D7—41no—
128C4×C7—21no—
129C4×C7—41no—
130C4×C7—41no—
131C4×C7C2×C3×D411yescharacteristic
132C4×C7C3×#4111yescharacteristic
133Dic7—41no—
134C23×C3—141no—
135C23×C3C22⋊D711yescharacteristic
136C3×Q8—21no—
137C3×Q8—21no—
138C3×D4—141no—
139C3×D4—141no—
140C3×D4—282no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—141no—
146C2×C3×C4C22⋊D711yescharacteristic
147C2×C3×C4C22×D711yescharacteristic
148C3×C8—72no—
149C3×C8—141no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8C2×C3×D711yescharacteristic
152C2×D4—71no—
153C2×D4—141no—
154C22×C4C2×C3×D711yescharacteristic
155SD16—142no—
156M16—142no—
157C2×C8—71no—
158C2×C8—71no—
159C4⋊C4—21no—
160C4⋊C4—71no—
161C4⋊C4C2×C3×D711yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—141no—
164C2×C7—41no—
165C2×C7C3×#12111yescharacteristic
166C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
167C2×C7C3×#13711yescharacteristic
168D7—81no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—281no—
173C22×C3—281no—
174C22×C3—281no—
175C22×C3C2×C22⋊D711yescharacteristic
176C3×C4—21no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—281no—
180C3×C4D4×D711yescharacteristic
181C3×C4#115911yescharacteristic
182C23—141no—
183C23C3×C22⋊D711yescharacteristic
184Q8—21no—
185Q8—21no—
186D4—141no—
187D4—141no—
188D4—282no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—141no—
194C2×C4C3×C22⋊D711yescharacteristic
195C2×C4C22×C3×D711yescharacteristic
196C8—72no—
197C8—141no—
198C7C3×#46811yescharacteristic, complemented, Sylow
199C2×C3—41no—
200C2×C3—561no—
201C2×C3#650211yescharacteristic
202C2×C3#650511yescharacteristic
203C2×C3#651211yescharacteristic
204C22—21no—
205C22—21no—
206C22—21no—
207C22—281no—
208C22—281no—
209C22—281no—
210C22C2×C3×C22⋊D711yescharacteristic
211C4—21no—
212C4—41no—
213C4—41no—
214C4—281no—
215C4C3×D4×D711yescharacteristic
216C4C3×#115911yescharacteristic
217C3#8982111yescharacteristic, direct product factor, Sylow
218C2—41no—
219C2—561no—
220C2C3×#650211yescharacteristic
221C2C3×#650511yescharacteristic
222C2C3×#651211yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC3×#6493C2
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#468
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
1. C3×C7×C22⋊Q8C2semidirect
2. C2×C3×#1143C2semidirect
4. C2×C3×#1136C2semidirect
5. C3×#6409C2semidirect
6. C3×#6383C2semidirect
7. C3×#6381C2semidirect
8. #89821C3direct (N × H)
22. C7×C22⋊Q8C2×C3semidirect
23. C2×#1143C2×C3semidirect
25. C2×#1136C2×C3semidirect
26. #6409C2×C3semidirect
27. #6383C2×C3semidirect
28. #6381C2×C3semidirect
45. C3×C7⋊C8D4semidirect
61. C3×C22⋊Q8D7semidirect
95. C7⋊C8C3×D4semidirect
116. C22⋊Q8C3×D7semidirect
150. C3×C7#468semidirect
198. C7C3×#468semidirect
217. C3#89821direct (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×#11432
C3×#64932
C2×C3×#11362
C3×#64092
C3×#63832
C3×#63812
#898213
C3×#4687