Finite Group Databasebeta

← C3×#89765 #13055427

Subgroups — C3×#89765 #13055427

Order 1344.

340 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 114 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)⋊C4C212yesmaximal
3C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
4C3×C7⋊#95C212yesmaximal, complemented
5C3×C7⋊#95C211yesmaximal, characteristic, complemented
6#89765C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×D7C2211yescharacteristic
8C22×C3×C4×C7C2212yes—
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×Dic7C2212yes—
11C22×C3×Dic7C2211yescharacteristic
12C3×C7×C4⋊C4—22no—
13C3×(C2×D7)⋊C4—24no—
14C3×(C2×D7)⋊C4—22no—
15C2×C7×C4⋊C4C2×C311yescharacteristic, complemented
16C2×(C2×D7)⋊C4C2×C312yes—
17C2×(C2×D7)⋊C4C2×C311yescharacteristic
18C7⋊#95C2×C312yescomplemented
19C7⋊#95C2×C311yescharacteristic, complemented
20C3×#397—71nomaximal, Hall
21C23×C3×C7C2311yescharacteristic
22C22×C3×D7—21no—
23C22×C3×D7—22no—
24C22×C3×D7—21no—
25C22×C3×D7—22no—
26C22×C3×D7D412yes—
27C2×C3×C4×C7—22no—
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7D412yes—
32C2×C3×Dic7—22no—
33C2×C3×Dic7—21no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7—22no—
36C2×C3×Dic7—22no—
37C23×D7C22×C311yescharacteristic
38C22×C4×C7C22×C312yes—
39C22×C4×C7C22×C311yescharacteristic
40C22×Dic7C22×C312yes—
41C22×Dic7C22×C311yescharacteristic
42C7×C4⋊C4—22no—
43(C2×D7)⋊C4—24no—
44(C2×D7)⋊C4—22no—
45C2×C3×C4⋊C4D711yescharacteristic, complemented
46C2×C3×C22⋊C4—72no—
47C2×C3×C22⋊C4—71no—
48C3×#95—72no—
49C3×#95—71no—
50C22×C3×C7C2×D411yescharacteristic
51C22×C3×C7C2×D411yescharacteristic
52C22×C3×C7#4112yes—
53C22×C3×C7#4111yescharacteristic
54C22×C3×C7#4112yes—
55C2×C3×D7—22no—
56C2×C3×D7—24no—
57C2×C3×D7—41no—
58C2×C3×D7—41no—
59C2×C3×D7—42no—
60C3×C4×C7—22no—
61C3×C4×C7—42no—
62C3×Dic7—42no—
63C3×Dic7—41no—
64#397—71noSylow
65C23×C7C23×C311yescharacteristic
66C22×D7—21no—
67C22×D7—22no—
68C22×D7—21no—
69C22×D7—22no—
70C22×D7C3×D412yes—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7C3×D412yes—
76C2×Dic7—22no—
77C2×Dic7—21no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C24×C3—71no—
82C22×C3×C4—72no—
83C22×C3×C4—71no—
84C22×C3×C4C2×D712yes—
85C22×C3×C4C2×D711yescharacteristic
86C3×C4⋊C4—22no—
87C3×C22⋊C4—144no—
88C3×C22⋊C4—142no—
89C2×C3×C7#12111yescharacteristic
90C2×C3×C7#12312yes—
91C2×C3×C7#12311yescharacteristic
92C2×C3×C7#12411yescharacteristic
93C2×C3×C7#12612yes—
94C3×D7—42no—
95C2×C4⋊C4C3×D711yescharacteristic, complemented
96C2×C22⋊C4—72no—
97C2×C22⋊C4—71no—
98#95—72no—
99#95—71no—
100C22×C7C2×C3×D411yescharacteristic
101C22×C7C2×C3×D411yescharacteristic
102C22×C7C3×#4112yes—
103C22×C7C3×#4111yescharacteristic
104C22×C7C3×#4112yes—
105C2×D7—22no—
106C2×D7—24no—
107C2×D7—41no—
108C2×D7—41no—
109C2×D7—42no—
110C4×C7—22no—
111C4×C7—42no—
112Dic7—42no—
113Dic7—41no—
114C23×C3—72no—
115C23×C3—142no—
116C23×C3—141no—
117C23×C3—142no—
118C23×C3—141no—
119C23×C3C22×D711yescharacteristic
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—142no—
125C2×C3×C4—142no—
126C2×C3×C4—142no—
127C2×C3×C4—142no—
128C2×C3×C4—141no—
129C2×C3×C4C22⋊D712yes—
130C3×C7#39711yescharacteristic, complemented, Hall
131C24—71no—
132C22×C4—72no—
133C22×C4—71no—
134C22×C4C2×C3×D712yes—
135C22×C4C2×C3×D711yescharacteristic
136C4⋊C4—22no—
137C22⋊C4—144no—
138C22⋊C4—142no—
139C2×C7C3×#12111yescharacteristic
140C2×C7C3×#12312yes—
141C2×C7C3×#12311yescharacteristic
142C2×C7C3×#12411yescharacteristic
143C2×C7C3×#12612yes—
144D7—42no—
145C22×C3—144no—
146C22×C3—142no—
147C22×C3—282no—
148C22×C3—281no—
149C22×C3—281no—
150C22×C3#115712yes—
151C22×C3D4×D711yescharacteristic
152C22×C3#115911yescharacteristic
153C22×C3#116112yes—
154C22×C3C2×C22⋊D711yescharacteristic
155C3×C4—22no—
156C3×C4—42no—
157C3×C4—282no—
158C3×C4—281no—
159C23—72no—
160C23—142no—
161C23—141no—
162C23—142no—
163C23—141no—
164C23C22×C3×D711yescharacteristic
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—141no—
174C2×C4C3×C22⋊D712yes—
175C7C3×#39711yescharacteristic, complemented, Sylow
176C2×C3—282no—
177C2×C3#645712yes—
178C2×C3#646112yes—
179C2×C3#649211yescharacteristic
180C2×C3#650111yescharacteristic
181C2×C3#651011yescharacteristic
182C22—144no—
183C22—142no—
184C22—282no—
185C22—281no—
186C22—281no—
187C22C3×#115712yes—
188C22C3×D4×D711yescharacteristic
189C22C3×#115911yescharacteristic
190C22C3×#116112yes—
191C22C2×C3×C22⋊D711yescharacteristic
192C4—22no—
193C4—42no—
194C4—282no—
195C4—281no—
196C3#8976511yescharacteristic, direct product factor, Sylow
197C2—282no—
198C2C3×#645712yes—
199C2C3×#646112yes—
200C2C3×#649211yescharacteristic
201C2C3×#650111yescharacteristic
202C2C3×#651011yescharacteristic
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×#397
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. C3×C7⋊#95C2semidirect
5. C3×C7⋊#95C2semidirect
6. #89765C3direct (N × H)
15. C2×C7×C4⋊C4C2×C3semidirect
18. C7⋊#95C2×C3semidirect
19. C7⋊#95C2×C3semidirect
45. C2×C3×C4⋊C4D7semidirect
95. C2×C4⋊C4C3×D7semidirect
130. C3×C7#397semidirect
175. C7C3×#397semidirect
196. C3#89765direct (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
C3×C7⋊#952
#897653
C3×#3977