Finite Group Databasebeta

← C3×#89725 #13055386

Subgroups — C3×#89725 #13055386

Order 1344.

340 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 118 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
4C3×C7⋊#95C212yesmaximal, complemented
5C3×C7⋊#95C211yesmaximal, characteristic, complemented
6#89725C311yesmaximal, 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—
15C2×C7×C22⋊C4C2×C311yescharacteristic
16C2×C22⋊Dic7C2×C312yes—
17C2×C4⋊Dic7C2×C311yescharacteristic, complemented
18C7⋊#95C2×C311yescharacteristic, complemented
19C7⋊#95C2×C312yescomplemented
20C3×#397—71nomaximal, Hall
21C23×C3×C7—21no—
22C23×C3×C7—21no—
23C23×C3×C7—22no—
24C23×C3×C7—22no—
25C23×C3×C7D412yes—
26C23×C3×C7C2311yescharacteristic
27C2×C3×C4×C7—22no—
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7D412yes—
31C2×C3×Dic7—22no—
32C2×C3×Dic7—22no—
33C2×C3×Dic7—22no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7—22no—
36C2×C3×Dic7—22no—
37C24×C7C22×C311yescharacteristic
38C22×C4×C7C22×C311yescharacteristic
39C22×C4×C7C22×C311yescharacteristic
40C22×Dic7C22×C312yes—
41C22×Dic7C22×C312yes—
42C7×C22⋊C4—22no—
43C22⋊Dic7—24no—
44C4⋊Dic7—22no—
45C2×C3×C4⋊C4—71no—
46C2×C3×C22⋊C4—72no—
47C2×C3×C22⋊C4D711yescharacteristic
48C3×#95—71no—
49C3×#95—72no—
50C22×C3×C7—22no—
51C22×C3×C7—24no—
52C22×C3×C7—41no—
53C22×C3×C7—41no—
54C22×C3×C7—42no—
55C22×C3×C7C2×D411yescharacteristic
56C22×C3×C7C2×D411yescharacteristic
57C22×C3×C7#4111yescharacteristic
58C22×C3×C7#4112yes—
59C22×C3×C7#4112yes—
60C3×C4×C7—22no—
61C3×C4×C7—41no—
62C3×Dic7—42no—
63C3×Dic7—42no—
64#397—71noSylow
65C23×C7—21no—
66C23×C7—21no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7C3×D412yes—
70C23×C7C23×C311yescharacteristic
71C2×C4×C7—22no—
72C2×C4×C7—21no—
73C2×C4×C7—22no—
74C2×C4×C7C3×D412yes—
75C2×Dic7—22no—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C24×C3C2×D711yescharacteristic
82C22×C3×C4—72no—
83C22×C3×C4—72no—
84C22×C3×C4C2×D711yescharacteristic
85C22×C3×C4C2×D711yescharacteristic
86C3×C4⋊C4—142no—
87C3×C22⋊C4—22no—
88C3×C22⋊C4—144no—
89C2×C3×C7—42no—
90C2×C3×C7#12111yescharacteristic
91C2×C3×C7#12312yes—
92C2×C3×C7#12311yescharacteristic
93C2×C3×C7#12411yescharacteristic
94C2×C3×C7#12612yes—
95C2×C4⋊C4—71no—
96C2×C22⋊C4—72no—
97C2×C22⋊C4C3×D711yescharacteristic
98#95—71no—
99#95—72no—
100C22×C7—22no—
101C22×C7—24no—
102C22×C7—41no—
103C22×C7—41no—
104C22×C7—42no—
105C22×C7C2×C3×D411yescharacteristic
106C22×C7C2×C3×D411yescharacteristic
107C22×C7C3×#4111yescharacteristic
108C22×C7C3×#4112yes—
109C22×C7C3×#4112yes—
110C4×C7—22no—
111C4×C7—41no—
112Dic7—42no—
113Dic7—42no—
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3D2812yes—
119C23×C3C22×D711yescharacteristic
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—142no—
124C2×C3×C4—142no—
125C2×C3×C4—142no—
126C2×C3×C4—142no—
127C2×C3×C4—142no—
128C2×C3×C4—142no—
129C2×C3×C4C22⋊D712yes—
130C3×C7#39711yescharacteristic, complemented, Hall
131C24C2×C3×D711yescharacteristic
132C22×C4—72no—
133C22×C4—72no—
134C22×C4C2×C3×D711yescharacteristic
135C22×C4C2×C3×D711yescharacteristic
136C4⋊C4—142no—
137C22⋊C4—22no—
138C22⋊C4—144no—
139C2×C7—42no—
140C2×C7C3×#12111yescharacteristic
141C2×C7C3×#12312yes—
142C2×C7C3×#12311yescharacteristic
143C2×C7C3×#12411yescharacteristic
144C2×C7C3×#12612yes—
145C22×C3—24no—
146C22×C3—22no—
147C22×C3—42no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3C2×D2811yescharacteristic
151C22×C3#115711yescharacteristic
152C22×C3#115912yes—
153C22×C3#115912yes—
154C22×C3C2×C22⋊D711yescharacteristic
155C3×C4—22no—
156C3×C4—41no—
157C3×C4—282no—
158C3×C4—282no—
159C23—21no—
160C23—22no—
161C23—22no—
162C23—21no—
163C23C3×D2812yes—
164C23C22×C3×D711yescharacteristic
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4C3×C22⋊D712yes—
175C7C3×#39711yescharacteristic, complemented, Sylow
176C2×C3—42no—
177C2×C3#644212yes—
178C2×C3#644912yes—
179C2×C3#649311yescharacteristic
180C2×C3#649811yescharacteristic
181C2×C3#649911yescharacteristic
182C22—24no—
183C22—22no—
184C22—42no—
185C22—41no—
186C22—41no—
187C22C2×C3×D2811yescharacteristic
188C22C3×#115711yescharacteristic
189C22C3×#115912yes—
190C22C3×#115912yes—
191C22C2×C3×C22⋊D711yescharacteristic
192C4—22no—
193C4—41no—
194C4—282no—
195C4—282no—
196C3#8972511yescharacteristic, direct product factor, Sylow
197C2—42no—
198C2C3×#644212yes—
199C2C3×#644912yes—
200C2C3×#649311yescharacteristic
201C2C3×#649811yescharacteristic
202C2C3×#649911yescharacteristic
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×#397
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. C3×C7⋊#95C2semidirect
5. C3×C7⋊#95C2semidirect
6. #89725C3direct (N × H)
17. C2×C4⋊Dic7C2×C3semidirect
18. C7⋊#95C2×C3semidirect
19. C7⋊#95C2×C3semidirect
130. C3×C7#397semidirect
175. C7C3×#397semidirect
196. C3#89725direct (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
C3×C7⋊#952
#897253
C3×#3977