Finite Group Databasebeta

← C3×#89721 #13055384

Subgroups — C3×#89721 #13055384

Order 1344.

372 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 126 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×C3×C7×C22⋊C4C211yesmaximal, characteristic
2C2×C3×C22⋊Dic7C212yesmaximal
3C2×C3×C4⋊Dic7C211yesmaximal, characteristic, complemented
4C2×C3×Dic7⋊C4C212yesmaximal, complemented
5C3×C7⋊#95C211yesmaximal, characteristic, complemented
6#89721C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×C7C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×Dic7C2212yes—
11C22×C3×Dic7C2212yes—
12C3×C7×C22⋊C4—22no—
13C3×C22⋊Dic7—24no—
14C3×C4⋊Dic7—22no—
15C3×Dic7⋊C4—24no—
16C2×C7×C22⋊C4C2×C311yescharacteristic
17C2×C22⋊Dic7C2×C312yes—
18C2×C4⋊Dic7C2×C311yescharacteristic, complemented
19C2×Dic7⋊C4C2×C312yescomplemented
20C7⋊#95C2×C311yescharacteristic, complemented
21C3×#399—71nomaximal, Hall
22C23×C3×C7—22no—
23C23×C3×C7—21no—
24C23×C3×C7—21no—
25C23×C3×C7—22no—
26C23×C3×C7Q812yes—
27C23×C3×C7C2311yescharacteristic
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7D412yes—
32C2×C3×Dic7—22no—
33C2×C3×Dic7—22no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7—22no—
36C2×C3×Dic7—22no—
37C2×C3×Dic7D414yes—
38C24×C7C22×C311yescharacteristic
39C22×C4×C7C22×C311yescharacteristic
40C22×C4×C7C22×C311yescharacteristic
41C22×Dic7C22×C312yes—
42C22×Dic7C22×C312yes—
43C7×C22⋊C4—22no—
44C22⋊Dic7—24no—
45C4⋊Dic7—22no—
46Dic7⋊C4—24no—
47C2×C3×C4⋊C4—72no—
48C2×C3×C4⋊C4—71no—
49C2×C3×C22⋊C4—72no—
50C2×C3×C22⋊C4D711yescharacteristic
51C3×#95—71no—
52C22×C3×C7—24no—
53C22×C3×C7—22no—
54C22×C3×C7—42no—
55C22×C3×C7—41no—
56C22×C3×C7—41no—
57C22×C3×C7C2×D412yes—
58C22×C3×C7C2×D411yescharacteristic
59C22×C3×C7C2×Q811yescharacteristic
60C22×C3×C7#4111yescharacteristic
61C22×C3×C7#4112yes—
62C3×C4×C7—22no—
63C3×C4×C7—41no—
64C3×Dic7—24no—
65C3×Dic7—42no—
66#399—71noSylow
67C23×C7—22no—
68C23×C7—21no—
69C23×C7—21no—
70C23×C7—22no—
71C23×C7C3×Q812yes—
72C23×C7C23×C311yescharacteristic
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—21no—
76C2×C4×C7C3×D412yes—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7C3×D414yes—
83C24×C3C2×D711yescharacteristic
84C22×C3×C4—72no—
85C22×C3×C4—72no—
86C22×C3×C4C2×D711yescharacteristic
87C22×C3×C4C2×D711yescharacteristic
88C3×C4⋊C4—144no—
89C3×C4⋊C4—142no—
90C3×C22⋊C4—22no—
91C3×C22⋊C4—144no—
92C2×C3×C7—42no—
93C2×C3×C7C22⋊Q812yes—
94C2×C3×C7C22⋊Q811yescharacteristic
95C2×C3×C7#12312yes—
96C2×C3×C7#12311yescharacteristic
97C2×C3×C7#12711yescharacteristic
98C2×C4⋊C4—72no—
99C2×C4⋊C4—71no—
100C2×C22⋊C4—72no—
101C2×C22⋊C4C3×D711yescharacteristic
102#95—71no—
103C22×C7—24no—
104C22×C7—22no—
105C22×C7—42no—
106C22×C7—41no—
107C22×C7—41no—
108C22×C7C2×C3×D412yes—
109C22×C7C2×C3×D411yescharacteristic
110C22×C7C2×C3×Q811yescharacteristic
111C22×C7C3×#4111yescharacteristic
112C22×C7C3×#4112yes—
113C4×C7—22no—
114C4×C7—41no—
115Dic7—24no—
116Dic7—42no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3Dic1412yes—
122C23×C3C22×D711yescharacteristic
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—74no—
127C2×C3×C4—142no—
128C2×C3×C4—142no—
129C2×C3×C4—142no—
130C2×C3×C4—142no—
131C2×C3×C4—142no—
132C2×C3×C4C22⋊D712yes—
133C3×C7#39911yescharacteristic, complemented, Hall
134C24C2×C3×D711yescharacteristic
135C22×C4—72no—
136C22×C4—72no—
137C22×C4C2×C3×D711yescharacteristic
138C22×C4C2×C3×D711yescharacteristic
139C4⋊C4—144no—
140C4⋊C4—142no—
141C22⋊C4—22no—
142C22⋊C4—144no—
143C2×C7—42no—
144C2×C7C3×C22⋊Q812yes—
145C2×C7C3×C22⋊Q811yescharacteristic
146C2×C7C3×#12312yes—
147C2×C7C3×#12311yescharacteristic
148C2×C7C3×#12711yescharacteristic
149C22×C3—24no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3C2×Dic1411yescharacteristic
155C22×C3#115711yescharacteristic
156C22×C3D4×D712yes—
157C22×C3#115912yes—
158C22×C3C2×C22⋊D711yescharacteristic
159C3×C4—22no—
160C3×C4—41no—
161C3×C4—144no—
162C3×C4—282no—
163C23—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23C3×Dic1412yes—
168C23C22×C3×D711yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—74no—
173C2×C4—142no—
174C2×C4—142no—
175C2×C4—142no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4C3×C22⋊D712yes—
179C7C3×#39911yescharacteristic, complemented, Sylow
180C2×C3—42no—
181C2×C3#644112yes—
182C2×C3#644612yes—
183C2×C3#648711yescharacteristic
184C2×C3#649811yescharacteristic
185C2×C3#650311yescharacteristic
186C22—24no—
187C22—22no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22C2×C3×Dic1411yescharacteristic
192C22C3×#115711yescharacteristic
193C22C3×D4×D712yes—
194C22C3×#115912yes—
195C22C2×C3×C22⋊D711yescharacteristic
196C4—22no—
197C4—41no—
198C4—144no—
199C4—282no—
200C3#8972111yescharacteristic, direct product factor, Sylow
201C2—42no—
202C2C3×#644112yes—
203C2C3×#644612yes—
204C2C3×#648711yescharacteristic
205C2C3×#649811yescharacteristic
206C2C3×#650311yescharacteristic
subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC24×C3×C7C22
hypercenterC2×C3×C22⋊C4D7
hypocenterC7C3×#399
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊Dic7C2semidirect
4. C2×C3×Dic7⋊C4C2semidirect
5. C3×C7⋊#95C2semidirect
6. #89721C3direct (N × H)
18. C2×C4⋊Dic7C2×C3semidirect
19. C2×Dic7⋊C4C2×C3semidirect
20. C7⋊#95C2×C3semidirect
133. C3×C7#399semidirect
179. C7C3×#399semidirect
200. C3#89721direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×C22⋊C42
C2×C3×C22⋊Dic72
C2×C3×C4⋊Dic72
C2×C3×Dic7⋊C42
C3×C7⋊#952
#897213
C3×#3997