Finite Group Databasebeta

← #13056311

Subgroups — #13056311

Order 1344.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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×D84C211yesmaximal, characteristic
2C2×#66631C212yesmaximal, complemented
3C2×#66631C212yesmaximal, complemented
4C2×#66629C211yesmaximal, characteristic, complemented
5C3×C7×#95C211yesmaximal, characteristic, complemented
6#89444—31nomaximal, Hall
7C23×D21C2212yes—
8C22×C3×C4×C7C2212yes—
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C2×D84—24no—
13C2×D84—22no—
14#66631—24no—
15#66631—24no—
16#66629—22no—
17C22×D28—31no—
18C2×(C2×D7)⋊C4—32no—
19C2×(C2×D7)⋊C4—32no—
20C2×Dic7⋊C4—31no—
21C7×#95D311yescharacteristic, complemented
22#4672—71nomaximal, Hall
23C23×C3×C7C2311yescharacteristic
24C22×D21—22no—
25C22×D21—22no—
26C22×D21—22no—
27C22×D21—22no—
28C22×D21—22no—
29C22×D21—22no—
30C22×D21D414yes—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7—22no—
33C2×C3×C4×C7—22no—
34C2×C3×C4×C7—22no—
35C2×C3×C4×C7D412yes—
36C2×Dic21—21no—
37C2×Dic21—21no—
38C2×Dic21—21no—
39C2×Dic21—22no—
40C2×Dic21D412yescomplemented
41D84—28no—
42C23×D7—32no—
43C22×C4×C7C2×D312yes—
44C22×C4×C7C2×D311yescharacteristic
45C22×Dic7—31no—
46C22×Dic7—31no—
47C2×D28—64no—
48C2×D28—62no—
49(C2×D7)⋊C4—64no—
50(C2×D7)⋊C4—64no—
51Dic7⋊C4—62no—
52C22×D12—71no—
53C2×(C2×D3)⋊C4—72no—
54C2×(C2×D3)⋊C4—72no—
55C2×Dic3⋊C4—71no—
56C3×#95D711yescharacteristic, complemented
57C22×C3×C7C2×D411yescharacteristic
58C22×C3×C7C2×D412yes—
59C22×C3×C7C2×D411yescharacteristic
60C22×C3×C7#4112yes—
61C22×C3×C7#4111yescharacteristic
62C2×D21—24no—
63C2×D21—24no—
64C2×D21—24no—
65C2×D21—42no—
66C2×D21—42no—
67C2×D21—42no—
68C2×D21—42no—
69C3×C4×C7—22no—
70C3×C4×C7—42no—
71Dic21—22no—
72Dic21—41no—
73#394—211noSylow
74C23×C7C22×D311yescharacteristic
75C22×D7—34no—
76C22×D7—62no—
77C22×D7—62no—
78C22×D7—62no—
79C22×D7—62no—
80C22×D7—62no—
81C22×D7—62no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7D1212yes—
87C2×Dic7—32no—
88C2×Dic7—61no—
89C2×Dic7—61no—
90C2×Dic7—61no—
91C2×Dic7—62no—
92D28—68no—
93C23×D3—72no—
94C22×C3×C4C2×D712yes—
95C22×C3×C4C2×D711yescharacteristic
96C22×Dic3—71no—
97C22×Dic3—71no—
98C2×D12—144no—
99C2×D12—142no—
100(C2×D3)⋊C4—144no—
101(C2×D3)⋊C4—144no—
102Dic3⋊C4—142no—
103C2×C3×C7#12011yescharacteristic
104C2×C3×C7#12112yes—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#12312yes—
107C2×C3×C7#12411yescharacteristic
108D21—44no—
109C22×D4—211no—
110C2×C4⋊C4—211no—
111C2×C22⋊C4—212no—
112C2×C22⋊C4—212no—
113#95D2111yescharacteristic, complemented
114C22×C7C2×D1211yescharacteristic
115C22×C7#23512yes—
116C22×C7D3×D412yes—
117C22×C7D3×D411yescharacteristic
118C22×C7#23911yescharacteristic
119C2×D7—64no—
120C2×D7—64no—
121C2×D7—64no—
122C2×D7—122no—
123C2×D7—122no—
124C2×D7—122no—
125C2×D7—122no—
126C4×C7—22no—
127C4×C7—42no—
128Dic7—62no—
129Dic7—121no—
130C23×C3C22×D711yescharacteristic
131C22×D3—74no—
132C22×D3—142no—
133C22×D3—142no—
134C22×D3—142no—
135C22×D3—142no—
136C22×D3—142no—
137C22×D3—142no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4D2812yes—
143C2×Dic3—72no—
144C2×Dic3—141no—
145C2×Dic3—141no—
146C2×Dic3—141no—
147C2×Dic3—142no—
148D12—148no—
149C3×C7#39411yescharacteristic, complemented, Hall
150C24—212no—
151C2×D4—424no—
152C2×D4—422no—
153C22×C4—211no—
154C22×C4—211no—
155C22×C4C2×D2112yes—
156C22×C4C2×D2111yescharacteristic
157C4⋊C4—422no—
158C22⋊C4—424no—
159C22⋊C4—424no—
160C2×C7#87511yescharacteristic
161C2×C7#88211yescharacteristic
162C2×C7#88412yes—
163C2×C7#89412yes—
164C2×C7#89511yescharacteristic
165D7—124no—
166C22×C3C2×D2811yescharacteristic
167C22×C3#115712yes—
168C22×C3D4×D712yes—
169C22×C3D4×D711yescharacteristic
170C22×C3#116111yescharacteristic
171C2×D3—144no—
172C2×D3—144no—
173C2×D3—144no—
174C2×D3—282no—
175C2×D3—282no—
176C2×D3—282no—
177C2×D3—282no—
178C3×C4—22no—
179C3×C4—42no—
180Dic3—142no—
181Dic3—281no—
182C23—214no—
183C23—422no—
184C23—422no—
185C23—422no—
186C23—422no—
187C23—422no—
188C23—422no—
189C23C22×D2111yescharacteristic
190D4—428no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—212no—
196C2×C4—421no—
197C2×C4—421no—
198C2×C4—421no—
199C2×C4—422no—
200C2×C4D8412yes—
201C7#467211yescharacteristic, complemented, Sylow
202C2×C3#643811yescharacteristic
203C2×C3#644511yescharacteristic
204C2×C3#644712yes—
205C2×C3#645712yes—
206C2×C3#645811yescharacteristic
207D3—284no—
208C22—424no—
209C22—424no—
210C22—424no—
211C22—842no—
212C22—842no—
213C22—842no—
214C22—842no—
215C22C2×D8411yescharacteristic
216C22#6672812yes—
217C22D4×D2112yes—
218C22D4×D2111yescharacteristic
219C22#6673211yescharacteristic
220C4—22no—
221C4—42no—
222C4—422no—
223C4—841no—
224C3#8944411yescharacteristic, complemented, Sylow
225C2—844no—
226C2#1062135811yescharacteristic
227C2#1062136511yescharacteristic
228C2#1062136712yes—
229C2#1062137712yes—
230C2#1062137811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, cosocleC23×C3×C7C23
FittingC3×C7×#95C2
prime radicalC22×D84C2
hypercenter#95D21
hypocenterC3×C7#394
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×#66631C2semidirect
3. C2×#66631C2semidirect
4. C2×#66629C2semidirect
5. C3×C7×#95C2semidirect
21. C7×#95D3semidirect
40. C2×Dic21D4semidirect
56. C3×#95D7semidirect
113. #95D21semidirect
149. C3×C7#394semidirect
201. C7#4672semidirect
224. C3#89444semidirect

Maximal subgroups

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

subgroupindex
C22×D842
C2×#666312
C2×#666292
C3×C7×#952
#894443
#46727