Finite Group Databasebeta

← C2×#10621362 #13061573

Subgroups — C2×#10621362 #13061573

Order 1344.

492 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 171 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×#66636C211yesmaximal, characteristic
3C2×#66636C211yesmaximal, characteristic
4C2×C4⋊Dic21C211yesmaximal, characteristic, complemented
5C2×#66629C211yesmaximal, characteristic, complemented
6C2×#66629C211yesmaximal, characteristic, complemented
7C2×C4×Dic21C211yesmaximal, characteristic, complemented
8#10621362C218yesmaximal, direct product factor
9C2×#6442—31nomaximal, Hall
10C24×C3×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C22×Dic21C2211yescharacteristic
17C3×C7×C22⋊C4C2214yes—
18#66636C2214yes—
19#66636C2214yes—
20C4⋊Dic21C2214yescomplemented
21#66629C2214yescomplemented
22#66629C2214yescomplemented
23C4×Dic21C2214yescomplemented
24C2×C7×C22⋊C4D311yescharacteristic
25C2×C22⋊Dic7—31no—
26C2×C22⋊Dic7—31no—
27C2×C4⋊Dic7—31no—
28C2×Dic7⋊C4—31no—
29C2×Dic7⋊C4—31no—
30C2×C4×Dic7—31no—
31#6442—38no—
32C2×#879—71nomaximal, Hall
33C23×C3×C7—22no—
34C23×C3×C7—22no—
35C23×C3×C7—22no—
36C23×C3×C7C2312yes—
37C23×C3×C7C2311yescharacteristic
38C2×C3×C4×C7—22no—
39C2×C3×C4×C7—22no—
40C2×C3×C4×C7C2312yes—
41C2×C3×C4×C7C2312yes—
42C2×Dic21—22no—
43C2×Dic21—22no—
44C2×Dic21—22no—
45C2×Dic21—22no—
46C2×Dic21C2312yes—
47C2×Dic21C2312yes—
48C2×Dic21C2312yes—
49C2×Dic21C2312yes—
50C24×C7C2×D311yescharacteristic
51C22×C4×C7C2×D311yescharacteristic
52C22×C4×C7C2×D311yescharacteristic
53C22×Dic7—31no—
54C22×Dic7—31no—
55C22×Dic7—31no—
56C22×Dic7—31no—
57C7×C22⋊C4C2×D314yes—
58C22⋊Dic7—34no—
59C22⋊Dic7—34no—
60C4⋊Dic7—34no—
61Dic7⋊C4—34no—
62Dic7⋊C4—34no—
63C4×Dic7—34no—
64C2×C3×C22⋊C4D711yescharacteristic
65C2×#217—71no—
66C2×#217—71no—
67C2×C4⋊Dic3—71no—
68C2×Dic3⋊C4—71no—
69C2×Dic3⋊C4—71no—
70C2×C4×Dic3—71no—
71#879—78no—
72C22×C3×C7—22no—
73C22×C3×C7—22no—
74C22×C3×C7—22no—
75C22×C3×C7—44no—
76C22×C3×C7#4112yes—
77C22×C3×C7#4112yes—
78C22×C3×C7#4112yes—
79C22×C3×C7C2411yescharacteristic
80C3×C4×C7—22no—
81C3×C4×C7—22no—
82Dic21—22no—
83Dic21—22no—
84Dic21—22no—
85Dic21—22no—
86C2×#126—211noSylow
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7C22×D312yes—
91C23×C7C22×D311yescharacteristic
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C22×D312yes—
95C2×C4×C7C22×D312yes—
96C2×Dic7—32no—
97C2×Dic7—32no—
98C2×Dic7—32no—
99C2×Dic7—32no—
100C2×Dic7—62no—
101C2×Dic7—62no—
102C2×Dic7—62no—
103C2×Dic7—62no—
104C24×C3C2×D711yescharacteristic
105C22×C3×C4C2×D711yescharacteristic
106C22×C3×C4C2×D711yescharacteristic
107C22×Dic3—71no—
108C22×Dic3—71no—
109C22×Dic3—71no—
110C22×Dic3—71no—
111C3×C22⋊C4C2×D714yes—
112#217—74no—
113#217—74no—
114C4⋊Dic3—74no—
115Dic3⋊C4—74no—
116Dic3⋊C4—74no—
117C4×Dic3—74no—
118C2×C3×C7—42no—
119C2×C3×C7#12614yescomplemented
120C2×C3×C7C2×#4111yescharacteristic
121C2×C3×C7C2×#4111yescharacteristic
122C2×C3×C7C2×#4111yescharacteristic
123#126—218no—
124C2×C4⋊C4—211no—
125C2×C4⋊C4—211no—
126C2×C4⋊C4—211no—
127C2×C22⋊C4—211no—
128C2×C22⋊C4—211no—
129C2×C22⋊C4D2111yescharacteristic
130C2×C42—211no—
131C22×C7—22no—
132C22×C7—22no—
133C22×C7—22no—
134C22×C7—44no—
135C22×C7#23512yes—
136C22×C7#23712yes—
137C22×C7#23712yes—
138C22×C7C23×D311yescharacteristic
139C4×C7—22no—
140C4×C7—22no—
141Dic7—62no—
142Dic7—62no—
143Dic7—62no—
144Dic7—62no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3C22×D712yes—
149C23×C3C22×D711yescharacteristic
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4C22×D712yes—
153C2×C3×C4C22×D712yes—
154C2×Dic3—72no—
155C2×Dic3—72no—
156C2×Dic3—72no—
157C2×Dic3—72no—
158C2×Dic3—142no—
159C2×Dic3—142no—
160C2×Dic3—142no—
161C2×Dic3—142no—
162C3×C7C2×#12611yescharacteristic, complemented, Hall
163C24C2×D2111yescharacteristic
164C22×C4—211no—
165C22×C4—211no—
166C22×C4—211no—
167C22×C4—211no—
168C22×C4C2×D2111yescharacteristic
169C22×C4C2×D2111yescharacteristic
170C4⋊C4—214no—
171C4⋊C4—214no—
172C4⋊C4—214no—
173C22⋊C4—214no—
174C22⋊C4—214no—
175C22⋊C4C2×D2114yes—
176C42—214no—
177C2×C7—42no—
178C2×C7#87914yescomplemented
179C2×C7C2×#23511yescharacteristic
180C2×C7C2×#23711yescharacteristic
181C2×C7C2×#23711yescharacteristic
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—44no—
186C22×C3#115712yes—
187C22×C3#115912yes—
188C22×C3#115912yes—
189C22×C3C23×D711yescharacteristic
190C3×C4—22no—
191C3×C4—22no—
192Dic3—142no—
193Dic3—142no—
194Dic3—142no—
195Dic3—142no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23C22×D2112yes—
200C23C22×D2111yescharacteristic
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—212no—
204C2×C4—212no—
205C2×C4—212no—
206C2×C4—212no—
207C2×C4—422no—
208C2×C4—422no—
209C2×C4—422no—
210C2×C4—422no—
211C2×C4C22×D2112yes—
212C2×C4C22×D2112yes—
213C7C2×#87911yescharacteristic, complemented, Sylow
214C2×C3—42no—
215C2×C3#644214yescomplemented
216C2×C3C2×#115711yescharacteristic
217C2×C3C2×#115911yescharacteristic
218C2×C3C2×#115911yescharacteristic
219C22—22no—
220C22—22no—
221C22—22no—
222C22—44no—
223C22#6672812yes—
224C22#6673012yes—
225C22#6673012yes—
226C22C23×D2111yescharacteristic
227C4—22no—
228C4—22no—
229C4—422no—
230C4—422no—
231C4—422no—
232C4—422no—
233C3C2×#644211yescharacteristic, complemented, Sylow
234C2—42no—
235C2#1062136214yesdirect product factor
236C2C2×#6672811yescharacteristic
237C2C2×#6673011yescharacteristic
238C2C2×#6673011yescharacteristic
subgroupG/H
centerC23C22×D21
commutator, cosocleC22×C3×C7C24
FrattiniC22C23×D21
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC24×C3×C7C22
hypercenterC2×C22⋊C4D21
hypocenterC3×C7C2×#126
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic21C2semidirect
5. C2×#66629C2semidirect
6. C2×#66629C2semidirect
7. C2×C4×Dic21C2semidirect
8. #10621362C2direct (N × H)
20. C4⋊Dic21C22semidirect
21. #66629C22semidirect
22. #66629C22semidirect
23. C4×Dic21C22semidirect
119. C2×C3×C7#126semidirect
162. C3×C7C2×#126semidirect
178. C2×C7#879semidirect
213. C7C2×#879semidirect
215. C2×C3#6442semidirect
233. C3C2×#6442semidirect
235. C2#10621362direct (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×#666362
C2×C4⋊Dic212
C2×#666292
C2×C4×Dic212
#106213622
C2×#64423
C2×#8797