Finite Group Databasebeta

← #13274674

Subgroups — #13274674

Order 1792.

184 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#11729261C211yesmaximal, characteristic, complemented
2C2×#89316C211yesmaximal, characteristic, complemented
3#11727126C211yesmaximal, characteristic, complemented
4#11727115C211yesmaximal, characteristic
5C7×#1420C211yesmaximal, characteristic, complemented
6#11725410C211yesmaximal, characteristic, complemented
7#11725387C211yesmaximal, characteristic, complemented
8C2×C7×Q32C2211yescharacteristic
9#89964—21no—
10C2×D112C2211yescharacteristic
11#89673C2211yescharacteristic
12#89669—21no—
13C7×M64C2211yescharacteristic
14C7×#368C2211yescharacteristic
15#89360—21no—
16#89316—21no—
17#89316—21no—
18#89301C2211yescharacteristic
19C2×C7⋊C32C2211yescharacteristic
20#89290—21no—
21#89242—21no—
22D7×C32—21no—
23#13702—71nomaximal, Sylow
24C2×C7×Q16—21no—
25#6505—41no—
26#6471—41no—
27C2×D56—21no—
28#6464D411yescharacteristic
29C7×Q32—21no—
30C7×Q32—21no—
31C2×C7×C16C2311yescharacteristic
32C7×#108—21no—
33#6410—41no—
34#6397—41no—
35#6393—21no—
36#6386D411yescharacteristic
37D112—21no—
38D112—21no—
39#6372—21no—
40D7×C16—21no—
41C7×C32—21no—
42C7⋊C32D411yescharacteristic
43C7⋊C32D411yescharacteristic
44C2×SD64—71no—
45#2258—71no—
46#2222—71no—
47#2221—71no—
48#1425—71no—
49#1420D711yescharacteristic, complemented
50#1419—71no—
51C2×C7×Q8—41no—
52#1157D811yescharacteristic
53C2×D28—41no—
54C7×Q16—41no—
55C7×Q16—41no—
56C7×M16—41no—
57C2×C7×C8C2×D411yescharacteristic
58#1144—81no—
59#1143—81no—
60#1136—41no—
61#1136D811yescharacteristic
62D56—41no—
63D56—41no—
64D56—81no—
65#1132—81no—
66#1131—21no—
67C8×D7—21no—
68C7×C16C2×D411yescharacteristic
69C7×C16#4111yescharacteristic
70C7⋊C16—21no—
71C2×Q32C2×D711yescharacteristic
72C2×D16—71no—
73#504—71no—
74#473—141no—
75#472—141no—
76SD64—141no—
77SD64—141no—
78M64—141no—
79M64C2×D711yescharacteristic
80C2×C32—71no—
81C2×C32—141no—
82#368—71no—
83#368C2×D711yescharacteristic
84#362—141no—
85#361—141no—
86C22×D7—81no—
87C7×Q8—81no—
88C7×Q8—81no—
89C2×C4×C7—81no—
90C2×C4×C7C2×D811yescharacteristic
91C22⋊D7—21no—
92D28—81no—
93D28—81no—
94D28D1611yescharacteristic
95C4×D7—21no—
96Dic14D1611yescharacteristic
97C7×C8—81no—
98C7×C8#12111yescharacteristic
99C7×C8#13511yescharacteristic
100C7⋊C8—21no—
101C7⋊C8—81no—
102#137—281no—
103AGL(1,ℤ/8)—281no—
104C2×Q16—21no—
105C2×D8—141no—
106#131—71no—
107Q32—21no—
108Q32—21no—
109D16—141no—
110D16—141no—
111M32—71no—
112M32—141no—
113C2×C16—141no—
114C2×C16C22×D711yescharacteristic
115#108—21no—
116#108—141no—
117#101—281no—
118#100—281no—
119C32—21no—
120C32—71no—
121C32—71no—
122C22×C7#50811yescharacteristic
123C2×D7—21no—
124C2×D7—161no—
125C2×D7—321no—
126C4×C7—161no—
127C4×C7#46611yescharacteristic
128C4×C7C2×D1611yescharacteristic
129Dic7—21no—
130#41—71no—
131C2×Q8—41no—
132C2×D4—281no—
133Q16—41no—
134Q16—41no—
135Q16—561no—
136SD16—561no—
137SD16—561no—
138D8—281no—
139D8—281no—
140D8—561no—
141M16—41no—
142M16—71no—
143M16—141no—
144M16—281no—
145C2×C8—141no—
146C2×C8D4×D711yescharacteristic
147C16—141no—
148C16#115711yescharacteristic
149C16D4×D711yescharacteristic
150C2×C7—21no—
151C2×C7#221511yescharacteristic
152D7—41no—
153D7—321no—
154C23—561no—
155Q8—71no—
156Q8—81no—
157Q8—81no—
158D4—71no—
159D4—141no—
160D4—561no—
161D4—561no—
162C2×C4—81no—
163C2×C4—141no—
164C2×C4D7×D811yescharacteristic
165C8—81no—
166C8—141no—
167C8—561no—
168C8#644711yescharacteristic
169C8#647911yescharacteristic
170C7#1370211yescharacteristic, complemented, Sylow
171C22—141no—
172C22—1121no—
173C22—2241no—
174C22#8968811yescharacteristic
175C4—141no—
176C4—161no—
177C4#8954711yescharacteristic
178C4D7×D1611yescharacteristic
179C2—21no—
180C2—281no—
181C2—2241no—
182C2#1172694911yescharacteristic
subgroupG/H
centerC2#11726949
commutator, cosocleC2×C7×C16C23
FrattiniC2×C16C22×D7
FittingC7×#1420C2
socleC2×C7#2215
prime radical#11727115C2
hypercenter#1420D7
hypocenterC7#13702
LCSG ▷ C2×C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1420

Complements

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

NH ≅ G/Ntype
1. #11729261C2semidirect
2. C2×#89316C2semidirect
3. #11727126C2semidirect
5. C7×#1420C2semidirect
6. #11725410C2semidirect
7. #11725387C2semidirect
49. #1420D7semidirect
170. C7#13702semidirect

Maximal subgroups

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

subgroupindex
#117292612
C2×#893162
#117271262
#117271152
C7×#14202
#117254102
#117253872
#137027