Finite Group Databasebeta

← C3×#89764 #13055411

Subgroups — C3×#89764 #13055411

Order 1344.

372 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 122 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×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
3C2×C3×(C2×D7)⋊C4C212yesmaximal
4C2×C3×Dic7⋊C4C212yesmaximal, complemented
5C3×C7⋊#95C211yesmaximal, characteristic, complemented
6#89764C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×D7C2211yescharacteristic
8C22×C3×C4×C7C2212yes—
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×Dic7C2211yescharacteristic
11C22×C3×Dic7C2212yes—
12C3×C7×C4⋊C4—22no—
13C3×(C2×D7)⋊C4—22no—
14C3×(C2×D7)⋊C4—24no—
15C3×Dic7⋊C4—24no—
16C2×C7×C4⋊C4C2×C311yescharacteristic, complemented
17C2×(C2×D7)⋊C4C2×C311yescharacteristic
18C2×(C2×D7)⋊C4C2×C312yes—
19C2×Dic7⋊C4C2×C312yescomplemented
20C7⋊#95C2×C311yescharacteristic, complemented
21C3×#399—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C22×C3×D7—22no—
24C22×C3×D7—21no—
25C22×C3×D7—22no—
26C22×C3×D7—21no—
27C22×C3×D7Q812yes—
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7D412yes—
33C2×C3×Dic7—22no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7—22no—
36C2×C3×Dic7—21no—
37C2×C3×Dic7D414yes—
38C23×D7C22×C311yescharacteristic
39C22×C4×C7C22×C312yes—
40C22×C4×C7C22×C311yescharacteristic
41C22×Dic7C22×C311yescharacteristic
42C22×Dic7C22×C312yes—
43C7×C4⋊C4—22no—
44(C2×D7)⋊C4—22no—
45(C2×D7)⋊C4—24no—
46Dic7⋊C4—24no—
47C2×C3×C4⋊C4—72no—
48C2×C3×C4⋊C4D711yescharacteristic, complemented
49C2×C3×C22⋊C4—72no—
50C2×C3×C22⋊C4—71no—
51C3×#95—71no—
52C22×C3×C7C2×D412yes—
53C22×C3×C7C2×D411yescharacteristic
54C22×C3×C7C2×Q811yescharacteristic
55C22×C3×C7#4112yes—
56C22×C3×C7#4111yescharacteristic
57C2×C3×D7—24no—
58C2×C3×D7—22no—
59C2×C3×D7—42no—
60C2×C3×D7—41no—
61C2×C3×D7—41no—
62C3×C4×C7—22no—
63C3×C4×C7—42no—
64C3×Dic7—24no—
65C3×Dic7—41no—
66#399—71noSylow
67C23×C7C23×C311yescharacteristic
68C22×D7—22no—
69C22×D7—21no—
70C22×D7—22no—
71C22×D7—21no—
72C22×D7C3×Q812yes—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7C3×D412yes—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—21no—
82C2×Dic7C3×D414yes—
83C24×C3—71no—
84C22×C3×C4—72no—
85C22×C3×C4—71no—
86C22×C3×C4C2×D712yes—
87C22×C3×C4C2×D711yescharacteristic
88C3×C4⋊C4—22no—
89C3×C4⋊C4—144no—
90C3×C22⋊C4—144no—
91C3×C22⋊C4—142no—
92C2×C3×C7C22⋊Q812yes—
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#12312yes—
95C2×C3×C7#12311yescharacteristic
96C2×C3×C7#12711yescharacteristic
97C3×D7—42no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4C3×D711yescharacteristic, complemented
100C2×C22⋊C4—72no—
101C2×C22⋊C4—71no—
102#95—71no—
103C22×C7C2×C3×D412yes—
104C22×C7C2×C3×D411yescharacteristic
105C22×C7C2×C3×Q811yescharacteristic
106C22×C7C3×#4112yes—
107C22×C7C3×#4111yescharacteristic
108C2×D7—24no—
109C2×D7—22no—
110C2×D7—42no—
111C2×D7—41no—
112C2×D7—41no—
113C4×C7—22no—
114C4×C7—42no—
115Dic7—24no—
116Dic7—41no—
117C23×C3—72no—
118C23×C3—142no—
119C23×C3—142no—
120C23×C3—141no—
121C23×C3—141no—
122C23×C3C22×D711yescharacteristic
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—74no—
128C2×C3×C4—142no—
129C2×C3×C4—142no—
130C2×C3×C4—142no—
131C2×C3×C4—141no—
132C2×C3×C4C22⋊D712yes—
133C3×C7#39911yescharacteristic, complemented, Hall
134C24—71no—
135C22×C4—72no—
136C22×C4—71no—
137C22×C4C2×C3×D712yes—
138C22×C4C2×C3×D711yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—144no—
141C22⋊C4—144no—
142C22⋊C4—142no—
143C2×C7C3×C22⋊Q812yes—
144C2×C7C3×C22⋊Q811yescharacteristic
145C2×C7C3×#12312yes—
146C2×C7C3×#12311yescharacteristic
147C2×C7C3×#12711yescharacteristic
148D7—42no—
149C22×C3—144no—
150C22×C3—142no—
151C22×C3—282no—
152C22×C3—281no—
153C22×C3—281no—
154C22×C3#115712yes—
155C22×C3D4×D712yes—
156C22×C3Q8×D711yescharacteristic
157C22×C3#116111yescharacteristic
158C22×C3C2×C22⋊D711yescharacteristic
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—144no—
162C3×C4—281no—
163C23—72no—
164C23—142no—
165C23—142no—
166C23—141no—
167C23—141no—
168C23C22×C3×D711yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—74no—
174C2×C4—142no—
175C2×C4—142no—
176C2×C4—142no—
177C2×C4—141no—
178C2×C4C3×C22⋊D712yes—
179C7C3×#39911yescharacteristic, complemented, Sylow
180C2×C3—282no—
181C2×C3#645712yes—
182C2×C3#645912yes—
183C2×C3#649211yescharacteristic
184C2×C3#650311yescharacteristic
185C2×C3#650911yescharacteristic
186C22—144no—
187C22—142no—
188C22—282no—
189C22—281no—
190C22—281no—
191C22C3×#115712yes—
192C22C3×D4×D712yes—
193C22C3×Q8×D711yescharacteristic
194C22C3×#116111yescharacteristic
195C22C2×C3×C22⋊D711yescharacteristic
196C4—22no—
197C4—42no—
198C4—144no—
199C4—281no—
200C3#8976411yescharacteristic, direct product factor, Sylow
201C2—282no—
202C2C3×#645712yes—
203C2C3×#645912yes—
204C2C3×#649211yescharacteristic
205C2C3×#650311yescharacteristic
206C2C3×#650911yescharacteristic
subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C4⋊C4C2
socleC23×C3×C7C23
prime radicalC23×C3×D7C22
hypercenterC2×C3×C4⋊C4D7
hypocenterC7C3×#399
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×C4⋊C4C2semidirect
4. C2×C3×Dic7⋊C4C2semidirect
5. C3×C7⋊#95C2semidirect
6. #89764C3direct (N × H)
16. C2×C7×C4⋊C4C2×C3semidirect
19. C2×Dic7⋊C4C2×C3semidirect
20. C7⋊#95C2×C3semidirect
48. C2×C3×C4⋊C4D7semidirect
99. C2×C4⋊C4C3×D7semidirect
133. C3×C7#399semidirect
179. C7C3×#399semidirect
200. C3#89764direct (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×C4⋊C42
C2×C3×(C2×D7)⋊C42
C2×C3×Dic7⋊C42
C3×C7⋊#952
#897643
C3×#3997