Finite Group Databasebeta

← C2×#10620896 #13059371

Subgroups — C2×#10620896 #13059371

Order 1344.

452 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 176 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×C4⋊Dic21C211yesmaximal, characteristic
2C2×C7×Dic3⋊C4C211yesmaximal, characteristic
3C2×C3×Dic7⋊C4C211yesmaximal, characteristic
4C2×#66577C211yesmaximal, characteristic
5C2×Dic7⋊Dic3C211yesmaximal, characteristic
6C2×Dic7⋊Dic3C211yesmaximal, characteristic
7C2×Dic3×Dic7C211yesmaximal, characteristic
8#10620896C218yesmaximal, direct product factor
9C2×#6452—31nomaximal, Hall
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C22×C3×Dic7C2211yescharacteristic
17C4⋊Dic21C2214yes—
18C7×Dic3⋊C4C2214yes—
19C3×Dic7⋊C4C2214yes—
20#66577C2214yes—
21Dic7⋊Dic3C2214yes—
22Dic7⋊Dic3C2214yes—
23Dic3×Dic7C2214yes—
24C2×C7×C4⋊C4—31no—
25C2×C4⋊Dic7—31no—
26C2×Dic7⋊C4—31no—
27C2×Dic7⋊C4—31no—
28C2×Dic7⋊C4—31no—
29C2×Dic7⋊C4D311yescharacteristic
30C2×C4×Dic7—31no—
31#6452—38no—
32C2×#890—71nomaximal, Hall
33C23×C3×C7C2311yescharacteristic
34C2×C3×C4×C7—22no—
35C2×C3×C4×C7C2312yes—
36C2×Dic21—22no—
37C2×Dic21—22no—
38C2×Dic21C2312yes—
39C2×Dic21C2312yes—
40C2×C7×Dic3—22no—
41C2×C7×Dic3—22no—
42C2×C7×Dic3C2312yes—
43C2×C7×Dic3C2312yes—
44C2×C3×Dic7—22no—
45C2×C3×Dic7Q814yes—
46C2×C3×Dic7C2312yes—
47C2×C3×Dic7C2312yes—
48C22×C4×C7—31no—
49C22×C4×C7—31no—
50C22×C4×C7C2×D311yescharacteristic
51C22×Dic7—31no—
52C22×Dic7—31no—
53C22×Dic7C2×D311yescharacteristic
54C22×Dic7C2×D311yescharacteristic
55C7×C4⋊C4—34no—
56C4⋊Dic7—34no—
57Dic7⋊C4—34no—
58Dic7⋊C4—34no—
59Dic7⋊C4—34no—
60Dic7⋊C4C2×D314yes—
61C4×Dic7—34no—
62C2×C3×C4⋊C4—71no—
63C2×C4⋊Dic3—71no—
64C2×C4⋊Dic3—71no—
65C2×C4⋊Dic3—71no—
66C2×Dic3⋊C4—71no—
67C2×Dic3⋊C4D711yescharacteristic
68C2×C4×Dic3—71no—
69#890—78no—
70C22×C3×C7C2×Q812yes—
71C22×C3×C7#4112yes—
72C22×C3×C7#4112yes—
73C22×C3×C7C2411yescharacteristic
74C3×C4×C7—22no—
75Dic21—22no—
76Dic21—22no—
77C3×Dic7—22no—
78C3×Dic7C2×Q814yes—
79C7×Dic3—22no—
80C7×Dic3—22no—
81C2×#125—211noSylow
82C23×C7C22×D311yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—32no—
85C2×C4×C7—32no—
86C2×C4×C7—62no—
87C2×C4×C7—62no—
88C2×C4×C7C22×D312yes—
89C2×Dic7—22no—
90C2×Dic7—32no—
91C2×Dic7—32no—
92C2×Dic7—62no—
93C2×Dic7—62no—
94C2×Dic7Dic614yes—
95C2×Dic7C22×D312yes—
96C2×Dic7C22×D312yes—
97C22×C3×C4—71no—
98C22×C3×C4—71no—
99C22×C3×C4C2×D711yescharacteristic
100C22×Dic3—71no—
101C22×Dic3—71no—
102C22×Dic3C2×D711yescharacteristic
103C22×Dic3C2×D711yescharacteristic
104C3×C4⋊C4—74no—
105C4⋊Dic3—74no—
106C4⋊Dic3—74no—
107C4⋊Dic3—74no—
108Dic3⋊C4—74no—
109Dic3⋊C4C2×D714yes—
110C4×Dic3—74no—
111C2×C3×C7#12514yescomplemented
112C2×C3×C7C22×Q811yescharacteristic
113C2×C3×C7C2×#4111yescharacteristic
114C2×C3×C7C2×#4111yescharacteristic
115#125—218no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—211no—
119C2×C4⋊C4—211no—
120C2×C4⋊C4—211no—
121C2×C4⋊C4—211no—
122C2×C42—211no—
123C22×C7C2×Dic612yes—
124C22×C7#23712yes—
125C22×C7#23912yes—
126C22×C7C23×D311yescharacteristic
127C4×C7—22no—
128C4×C7—62no—
129C4×C7—62no—
130Dic7—22no—
131Dic7—62no—
132Dic7—62no—
133Dic7C2×Dic614yes—
134C23×C3C22×D711yescharacteristic
135C2×C3×C4—22no—
136C2×C3×C4—74no—
137C2×C3×C4—72no—
138C2×C3×C4—72no—
139C2×C3×C4—142no—
140C2×C3×C4C22×D712yes—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—72no—
144C2×Dic3—72no—
145C2×Dic3—142no—
146C2×Dic3—142no—
147C2×Dic3C22×D712yes—
148C2×Dic3C22×D712yes—
149C3×C7C2×#12511yescharacteristic, complemented, Hall
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—71no—
153C22×C4—71no—
154C22×C4—211no—
155C22×C4—211no—
156C22×C4D3×D711yescharacteristic
157C4⋊C4—34no—
158C4⋊C4—74no—
159C4⋊C4—214no—
160C4⋊C4—214no—
161C4⋊C4—214no—
162C4⋊C4—214no—
163C42—214no—
164C2×C7#89014yescomplemented
165C2×C7C22×Dic611yescharacteristic
166C2×C7C2×#23711yescharacteristic
167C2×C7C2×#23911yescharacteristic
168C22×C3#115712yes—
169C22×C3#115912yes—
170C22×C3Q8×D712yes—
171C22×C3C23×D711yescharacteristic
172C3×C4—22no—
173C3×C4—74no—
174C3×C4—142no—
175Dic3—22no—
176Dic3—22no—
177Dic3—142no—
178Dic3—142no—
179C23C2×D3×D711yescharacteristic
180C2×C4—22no—
181C2×C4—32no—
182C2×C4—32no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—74no—
186C2×C4—72no—
187C2×C4—72no—
188C2×C4—142no—
189C2×C4—212no—
190C2×C4—212no—
191C2×C4—422no—
192C2×C4—422no—
193C2×C4C2×D3×D712yes—
194C7C2×#89011yescharacteristic, complemented, Sylow
195C2×C3#645214yescomplemented
196C2×C3C2×#115711yescharacteristic
197C2×C3C2×#115911yescharacteristic
198C2×C3C2×Q8×D711yescharacteristic
199C22D7×Dic612yes—
200C22#6667312yes—
201C22#6668412yes—
202C22C22×D3×D711yescharacteristic
203C4—22no—
204C4—62no—
205C4—62no—
206C4—74no—
207C4—142no—
208C4—422no—
209C4—422no—
210C3C2×#645211yescharacteristic, complemented, Sylow
211C2#1062089614yesdirect product factor
212C2C2×D7×Dic611yescharacteristic
213C2C2×#6667311yescharacteristic
214C2C2×#6668411yescharacteristic
subgroupG/H
centerC23C2×D3×D7
commutator, cosocleC22×C3×C7C24
FrattiniC22C22×D3×D7
FittingC22×C3×C4×C7C22
socle, prime radicalC23×C3×C7C23
hypercenterC22×C4D3×D7
hypocenterC3×C7C2×#125
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. #10620896C2direct (N × H)
111. C2×C3×C7#125semidirect
149. C3×C7C2×#125semidirect
164. C2×C7#890semidirect
194. C7C2×#890semidirect
195. C2×C3#6452semidirect
210. C3C2×#6452semidirect
211. C2#10620896direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Dic212
C2×C7×Dic3⋊C42
C2×C3×Dic7⋊C42
C2×#665772
C2×Dic7⋊Dic32
C2×Dic3×Dic72
#106208962
C2×#64523
C2×#8907