Finite Group Databasebeta

← #13056679

Subgroups — #13056679

Order 1344.

316 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 79 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×Dic42C211yesmaximal, characteristic, complemented
2C3×C7×#121C211yesmaximal, characteristic
3C2×#66633C212yesmaximal, complemented
4#10620669C211yesmaximal, characteristic, complemented
5#10620649C212yesmaximal, complemented
6#89812—31nomaximal, Hall
7C2×C3×C7×D4—21no—
8C2×C3×C7×D4C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21—21no—
11C2×Dic42—21no—
12C2×Dic42—24no—
13C2×Dic42C2212yes—
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16#66633—24no—
17C2×C3⋊(C7⋊C8)C2212yes—
18C22×Dic14—31no—
19C7×#121D311yescharacteristic
20C2×#1142—32no—
21C22⋊(C7⋊C8)—31no—
22#6384—32no—
23#5040—71nomaximal, Hall
24C23×C3×C7—21no—
25C23×C3×C7D411yescharacteristic
26C3×C7×D4—22no—
27C3×C7×D4—42no—
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7D411yescharacteristic
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—24no—
33C2×Dic21—22no—
34Dic42—24no—
35Dic42—22no—
36Dic42D414yescomplemented
37C3⋊(C7⋊C8)—22no—
38C2×C7×D4—21no—
39C2×C7×D4C2×D311yescharacteristic
40C22×C4×C7C2×D311yescharacteristic
41C22×Dic7—61no—
42C2×Dic14—32no—
43C2×Dic14—64no—
44C2×Dic14—61no—
45C7×C4⋊C4C2×D311yescharacteristic
46C7×C22⋊C4—21no—
47#1142—64no—
48C2×C7⋊C8—32no—
49C22×Dic6—71no—
50C3×#121D711yescharacteristic
51C2×#214—72no—
52#830—71no—
53#810—72no—
54C22×C3×C7—21no—
55C22×C3×C7—21no—
56C22×C3×C7—41no—
57C22×C3×C7—41no—
58C22×C3×C7—41no—
59C22×C3×C7SD1612yescomplemented
60C22×C3×C7C2×D411yescharacteristic
61C3×C4×C7—21no—
62C3×C4×C7—41no—
63C3×C4×C7C2×D412yes—
64Dic21—24no—
65#448—211noSylow
66C23×C7—21no—
67C23×C7#6711yescharacteristic
68C7×D4—22no—
69C7×D4—42no—
70C2×C4×C7—22no—
71C2×C4×C7—21no—
72C2×C4×C7#6711yescharacteristic
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—64no—
75C2×Dic7—62no—
76Dic14—34no—
77Dic14—64no—
78Dic14—62no—
79C7⋊C8—62no—
80C2×C3×D4—21no—
81C2×C3×D4C2×D711yescharacteristic
82C22×C3×C4C2×D711yescharacteristic
83C22×Dic3—141no—
84C2×Dic6—72no—
85C2×Dic6—144no—
86C2×Dic6—141no—
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C22⋊C4—21no—
89#214—144no—
90C2×C3⋊C8—72no—
91C2×C3×C7—22no—
92C2×C3×C7—81no—
93C2×C3×C7#12011yescharacteristic
94C2×C3×C7C2×SD1611yescharacteristic
95C2×C3×C7#13711yescharacteristic
96C22×Q8—211no—
97C2×SD16—212no—
98#121D2111yescharacteristic
99#103—212no—
100C22⋊C8—211no—
101C22×C7—21no—
102C22×C7—21no—
103C22×C7—41no—
104C22×C7—41no—
105C22×C7—41no—
106C22×C7#21412yescomplemented
107C22×C7C2×#6711yescharacteristic
108C4×C7—21no—
109C4×C7—41no—
110C4×C7D3×D412yes—
111Dic7—64no—
112C23×C3—21no—
113C23×C3C22⋊D711yescharacteristic
114C3×D4—22no—
115C3×D4—42no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4C22⋊D711yescharacteristic
119C2×C3×C4C22×D711yescharacteristic
120C2×Dic3—144no—
121C2×Dic3—142no—
122Dic6—74no—
123Dic6—144no—
124Dic6—142no—
125C3⋊C8—142no—
126C3×C7#44811yescharacteristic, complemented, Hall
127C2×Q8—212no—
128C2×Q8—424no—
129C2×Q8—421no—
130C2×D4—21no—
131C2×D4C2×D2111yescharacteristic
132C22×C4—421no—
133C22×C4C2×D2111yescharacteristic
134SD16—424no—
135C2×C8—212no—
136C4⋊C4C2×D2111yescharacteristic
137C22⋊C4—21no—
138C2×C7—22no—
139C2×C7—81no—
140C2×C7C2×#21411yescharacteristic
141C2×C7#93711yescharacteristic
142C2×C7#95111yescharacteristic
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3#114212yescomplemented
149C22×C3C2×C22⋊D711yescharacteristic
150C3×C4—21no—
151C3×C4—41no—
152C3×C4D4×D712yes—
153Dic3—144no—
154C23—21no—
155C23#426811yescharacteristic
156Q8—214no—
157Q8—424no—
158Q8—422no—
159D4—22no—
160D4—42no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—424no—
164C2×C4—422no—
165C2×C4#426811yescharacteristic
166C2×C4C22×D2111yescharacteristic
167C8—422no—
168C7#504011yescharacteristic, complemented, Sylow
169C2×C3—22no—
170C2×C3—81no—
171C2×C3C2×#114211yescharacteristic
172C2×C3#650011yescharacteristic
173C2×C3#651411yescharacteristic
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22#6663312yescomplemented
180C22C2×#426811yescharacteristic
181C4—21no—
182C4—41no—
183C4—424no—
184C4D4×D2112yes—
185C3#8981211yescharacteristic, complemented, Sylow
186C2—22no—
187C2—81no—
188C2C2×#6663311yescharacteristic
189C2#1062142011yescharacteristic
190C2#1062143411yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
Fitting, prime radicalC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#448
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×Dic42C2semidirect
3. C2×#66633C2semidirect
4. #10620669C2semidirect
5. #10620649C2semidirect
36. Dic42D4semidirect
59. C22×C3×C7SD16semidirect
106. C22×C7#214semidirect
126. C3×C7#448semidirect
148. C22×C3#1142semidirect
168. C7#5040semidirect
179. C22#66633semidirect
185. C3#89812semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C3×C7×#1212
C2×#666332
#106206692
#106206492
#898123
#50407