Finite Group Databasebeta

← C3×#89498 #13055160

Subgroups — C3×#89498 #13055160

Order 1344.

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

normal subgroups only

This group has 380 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
1C22×C3×C4×D7C211yesmaximal, characteristic
2C2×C3×#1131C212yesmaximal, complemented
3C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5C3×(C2×D7)⋊C8C212yesmaximal, complemented
6#89498C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×D7C411yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×Dic7C411yescharacteristic
10C2×C3×C4×D7—24no—
11C2×C3×C4×D7C411yescharacteristic
12C2×C3×C4×D7C411yescharacteristic
13C2×C3×C4×D7C2211yescharacteristic
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×C7×C8C2212yescomplemented
16C2×C3×C7⋊C8C2212yescomplemented
17C3×#1131—24no—
18C22×C4×D7C2×C311yescharacteristic
19C2×#1131C2×C312yescomplemented
20C7×C22⋊C8C2×C311yescharacteristic, complemented
21C22⋊(C7⋊C8)C2×C311yescharacteristic, complemented
22(C2×D7)⋊C8C2×C312yescomplemented
23C3×#407—71nomaximal, Hall
24C23×C3×C7C2×C411yescharacteristic
25C22×C3×D7—22no—
26C22×C3×D7—22no—
27C22×C3×D7—22no—
28C22×C3×D7C2×C411yescharacteristic
29C22×C3×D7C2×C411yescharacteristic
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7C2×C411yescharacteristic
32C2×C3×C4×C7C2311yescharacteristic
33C2×C3×Dic7—22no—
34C2×C3×Dic7C2×C411yescharacteristic
35C2×C3×Dic7C2×C411yescharacteristic
36C3×C4×D7—22no—
37C3×C4×D7—22no—
38C3×C4×D7—22no—
39C3×C4×D7D414yes—
40C3×C7×C8—22no—
41C3×C7⋊C8—22no—
42C23×D7C3×C411yescharacteristic
43C22×C4×C7C22×C311yescharacteristic
44C22×Dic7C3×C411yescharacteristic
45C2×C4×D7—24no—
46C2×C4×D7C3×C411yescharacteristic
47C2×C4×D7C3×C411yescharacteristic
48C2×C4×D7C22×C311yescharacteristic
49C2×C4×D7C22×C311yescharacteristic
50C2×C7×C8C22×C312yescomplemented
51C2×C7⋊C8C22×C312yescomplemented
52#1131—24no—
53C23×C3×C4—71no—
54C2×C3×M16—72no—
55C3×C22⋊C8—71no—
56C3×C22⋊C8—72no—
57C3×C22⋊C8D711yescharacteristic, complemented
58C22×C3×C7—21no—
59C22×C3×C7—21no—
60C22×C3×C7M1612yescomplemented
61C22×C3×C7C22×C411yescharacteristic
62C2×C3×D7—28no—
63C2×C3×D7—21no—
64C2×C3×D7—21no—
65C2×C3×D7—21no—
66C2×C3×D7—21no—
67C2×C3×D7C22⋊C412yes—
68C2×C3×D7M1612yescomplemented
69C3×C4×C7—21no—
70C3×C4×C7C2×D412yes—
71C3×Dic7—21no—
72C3×Dic7C22⋊C412yes—
73#407—71noSylow
74C23×C7C2×C3×C411yescharacteristic
75C22×D7—22no—
76C22×D7—22no—
77C22×D7—22no—
78C22×D7C2×C3×C411yescharacteristic
79C22×D7C2×C3×C411yescharacteristic
80C2×C4×C7—22no—
81C2×C4×C7C2×C3×C411yescharacteristic
82C2×C4×C7C23×C311yescharacteristic
83C2×Dic7—22no—
84C2×Dic7C2×C3×C411yescharacteristic
85C2×Dic7C2×C3×C411yescharacteristic
86C4×D7—22no—
87C4×D7—22no—
88C4×D7—22no—
89C4×D7C3×D414yes—
90C7×C8—22no—
91C7⋊C8—22no—
92C24×C3—71no—
93C22×C3×C4—71no—
94C22×C3×C4—71no—
95C22×C3×C4—71no—
96C22×C3×C4—71no—
97C22×C3×C4—71no—
98C22×C3×C4—144no—
99C22×C3×C4C2×D711yescharacteristic
100C3×M16—144no—
101C2×C3×C8—72no—
102C2×C3×C8C2×D712yescomplemented
103C2×C3×C7—22no—
104C2×C3×C7C2×C22⋊C411yescharacteristic
105C2×C3×C7C2×M1611yescharacteristic
106C2×C3×C7C2×M1611yescharacteristic
107C3×D7—22no—
108C3×D7—22no—
109C23×C4—71no—
110C2×M16—72no—
111C22⋊C8—71no—
112C22⋊C8—72no—
113C22⋊C8C3×D711yescharacteristic, complemented
114C22×C7—21no—
115C22×C7—21no—
116C22×C7C3×M1612yescomplemented
117C22×C7C22×C3×C411yescharacteristic
118C2×D7—28no—
119C2×D7—21no—
120C2×D7—21no—
121C2×D7—21no—
122C2×D7—21no—
123C2×D7C3×C22⋊C412yes—
124C2×D7C3×M1612yescomplemented
125C4×C7—21no—
126C4×C7C2×C3×D412yes—
127Dic7—21no—
128Dic7C3×C22⋊C412yes—
129C23×C3—71no—
130C23×C3—71no—
131C23×C3—142no—
132C23×C3—142no—
133C23×C3—142no—
134C23×C3C4×D711yescharacteristic
135C2×C3×C4—22no—
136C2×C3×C4—71no—
137C2×C3×C4—74no—
138C2×C3×C4—71no—
139C2×C3×C4—142no—
140C2×C3×C4—142no—
141C2×C3×C4—142no—
142C2×C3×C4—142no—
143C2×C3×C4C4×D711yescharacteristic
144C2×C3×C4C22×D711yescharacteristic
145C3×C8—22no—
146C3×C8—142no—
147C3×C7#40711yescharacteristic, complemented, Hall
148C24—71no—
149C22×C4—71no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—71no—
153C22×C4—71no—
154C22×C4—144no—
155C22×C4C2×C3×D711yescharacteristic
156M16—144no—
157C2×C8—72no—
158C2×C8C2×C3×D712yescomplemented
159C2×C7—22no—
160C2×C7C2×C3×C22⋊C411yescharacteristic
161C2×C7C2×C3×M1611yescharacteristic
162C2×C7C2×C3×M1611yescharacteristic
163D7—22no—
164D7—22no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—72no—
168C22×C3—72no—
169C22×C3—148no—
170C22×C3—141no—
171C22×C3—141no—
172C22×C3—141no—
173C22×C3—141no—
174C22×C3#113112yescomplemented
175C22×C3C2×C4×D711yescharacteristic
176C3×C4—21no—
177C3×C4—72no—
178C3×C4—141no—
179C3×C4D4×D712yes—
180C23—71no—
181C23—71no—
182C23—142no—
183C23—142no—
184C23—142no—
185C23C3×C4×D711yescharacteristic
186C2×C4—22no—
187C2×C4—71no—
188C2×C4—74no—
189C2×C4—71no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4C3×C4×D711yescharacteristic
195C2×C4C22×C3×D711yescharacteristic
196C8—22no—
197C8—142no—
198C7C3×#40711yescharacteristic, complemented, Sylow
199C2×C3—22no—
200C2×C3—142no—
201C2×C3—142no—
202C2×C3D7×C22⋊C411yescharacteristic
203C2×C3C2×#113111yescharacteristic
204C2×C3D7×M1611yescharacteristic
205C22—21no—
206C22—21no—
207C22—72no—
208C22—72no—
209C22—148no—
210C22—141no—
211C22—141no—
212C22—141no—
213C22—141no—
214C22C3×#113112yescomplemented
215C22C2×C3×C4×D711yescharacteristic
216C4—21no—
217C4—72no—
218C4—141no—
219C4C3×D4×D712yes—
220C3#8949811yescharacteristic, direct product factor, Sylow
221C2—22no—
222C2—142no—
223C2—142no—
224C2C3×D7×C22⋊C411yescharacteristic
225C2C2×C3×#113111yescharacteristic
226C2C3×D7×M1611yescharacteristic
subgroupG/H
centerC2×C3×C4C22×D7
commutatorC22×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C22×C4
prime radicalC23×C3×D7C4
cosocleC2×C4×C7C23×C3
hypercenterC3×C22⋊C8D7
hypocenterC7C3×#407
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1131C2semidirect
3. C3×C7×C22⋊C8C2semidirect
4. C3×C22⋊(C7⋊C8)C2semidirect
5. C3×(C2×D7)⋊C8C2semidirect
6. #89498C3direct (N × H)
15. C2×C3×C7×C8C22semidirect
16. C2×C3×C7⋊C8C22semidirect
19. C2×#1131C2×C3semidirect
20. C7×C22⋊C8C2×C3semidirect
21. C22⋊(C7⋊C8)C2×C3semidirect
22. (C2×D7)⋊C8C2×C3semidirect
50. C2×C7×C8C22×C3semidirect
51. C2×C7⋊C8C22×C3semidirect
57. C3×C22⋊C8D7semidirect
60. C22×C3×C7M16semidirect
68. C2×C3×D7M16semidirect
102. C2×C3×C8C2×D7semidirect
113. C22⋊C8C3×D7semidirect
116. C22×C7C3×M16semidirect
124. C2×D7C3×M16semidirect
147. C3×C7#407semidirect
158. C2×C8C2×C3×D7semidirect
174. C22×C3#1131semidirect
198. C7C3×#407semidirect
214. C22C3×#1131semidirect
220. C3#89498direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D72
C2×C3×#11312
C3×C7×C22⋊C82
C3×C22⋊(C7⋊C8)2
C3×(C2×D7)⋊C82
#894983
C3×#4077