Finite Group Databasebeta

← #13268209

Subgroups — #13268209

Order 1792.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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#11729112C211yesmaximal, characteristic
2#11729107C211yesmaximal, characteristic, complemented
3#11727060C211yesmaximal, characteristic
4#11727055C211yesmaximal, characteristic
5C7×#1366C211yesmaximal, characteristic, complemented
6#11725378C211yesmaximal, characteristic, complemented
7#11725373C211yesmaximal, characteristic, complemented
8C7×#499C2211yescharacteristic
9#89681—21no—
10#89679—21no—
11#89472C2211yescharacteristic
12#89471C2211yescharacteristic
13C7×#364C2211yescharacteristic
14C7×#364C2211yescharacteristic
15#89312—21no—
16#89310—21no—
17#89287—21no—
18#89287—21no—
19#89283—21no—
20#89282—21no—
21#89258C2211yescharacteristic
22#89256C2211yescharacteristic
23#12466—71nomaximal, Sylow
24C7×#125—21no—
25C2×Dic28—21no—
26C2×Dic28—21no—
27#6467D411yescharacteristic
28#6464—21no—
29#6432—21no—
30C7×M32—21no—
31C7×M32—21no—
32C7×#107—21no—
33C7×#106—21no—
34C4×C7×C8C2311yescharacteristic
35#6393D411yescharacteristic
36#6386—21no—
37C2×C7⋊C16D411yescharacteristic
38C2×C7⋊C16D411yescharacteristic
39#6379—21no—
40Dic56—41no—
41#6374—41no—
42#6372—41no—
43D7×C16—41no—
44#2245—71no—
45#2231—71no—
46#2214—71no—
47#2171—71no—
48#1369—71no—
49#1366D711yescharacteristic, complemented
50#1358—71no—
51#1157—21no—
52C2×Dic14—41no—
53C2×C7×C8C2×D411yescharacteristic
54C2×C7×C8C2×D411yescharacteristic
55C7×C4⋊C4—41no—
56C7×C4⋊C4—81no—
57C42×C7#4111yescharacteristic
58C4⋊Dic7—81no—
59#1136—21no—
60Dic28—41no—
61Dic28—41no—
62Dic28—41no—
63Dic28—41no—
64Dic28SD1611yescharacteristic
65D56SD1611yescharacteristic
66#1132—21no—
67#1131—41no—
68C8×D7—41no—
69C7×C16—41no—
70C7×C16—41no—
71C7⋊C16—21no—
72C7⋊C16—21no—
73C7⋊C16—41no—
74#510—141no—
75#504—141no—
76#499C2×D711yescharacteristic
77#494—71no—
78#443—71no—
79#365—141no—
80#364—71no—
81#364C2×D711yescharacteristic
82#364C2×D711yescharacteristic
83#362—141no—
84#360—141no—
85#358—141no—
86#358—141no—
87#350—141no—
88#346—71no—
89C2×C4×C7—21no—
90C2×C4×C7—81no—
91C2×C4×C7#12111yescharacteristic
92C22⋊D7—41no—
93C2×Dic7—81no—
94D28—21no—
95C4×D7—41no—
96Dic14—21no—
97Dic14—81no—
98Dic14—81no—
99C7×C8C2×SD1611yescharacteristic
100C7×C8#13511yescharacteristic
101C7×C8#13511yescharacteristic
102C7×C8AGL(1,ℤ/8)11yescharacteristic
103C7⋊C8—41no—
104#135—71no—
105C2×Q16—141no—
106C2×Q16—141no—
107#131—141no—
108C4⋊Q8—141no—
109#125—21no—
110Q32—281no—
111SD32—281no—
112M32—21no—
113M32—21no—
114M32—141no—
115M32—281no—
116C2×C16—71no—
117C2×C16—71no—
118C2×C16—281no—
119#108—71no—
120#107—21no—
121#106—21no—
122#104—141no—
123C4×C8C22×D711yescharacteristic
124C22×C7#47211yescharacteristic
125C2×D7—41no—
126C4×C7—41no—
127C4×C7—161no—
128C4×C7#46311yescharacteristic
129C4×C7#46511yescharacteristic
130Dic7—41no—
131Dic7—161no—
132#41—141no—
133C2×Q8—281no—
134Q16—71no—
135Q16—281no—
136Q16—281no—
137Q16—281no—
138Q16—281no—
139SD16—141no—
140D8—71no—
141M16—141no—
142M16—281no—
143C2×C8—281no—
144C2×C8D4×D711yescharacteristic
145C2×C8D4×D711yescharacteristic
146C4⋊C4—41no—
147C4⋊C4—81no—
148C4⋊C4—561no—
149C42#115711yescharacteristic
150C16—41no—
151C16—41no—
152C16—141no—
153C16—141no—
154C16—281no—
155C2×C7—21no—
156C2×C7#167711yescharacteristic
157D7—81no—
158Q8—141no—
159Q8—561no—
160Q8—561no—
161D4—141no—
162D4—281no—
163C2×C4—21no—
164C2×C4—81no—
165C2×C4—281no—
166C2×C4—561no—
167C2×C4#644711yescharacteristic
168C8—281no—
169C8D7×SD1611yescharacteristic
170C8#647711yescharacteristic
171C8#648211yescharacteristic
172C8#648211yescharacteristic
173C7#1246611yescharacteristic, complemented, Sylow
174C22—281no—
175C22#8952111yescharacteristic
176C4—41no—
177C4—161no—
178C4—281no—
179C4—1121no—
180C4#8957911yescharacteristic
181C4#8959111yescharacteristic
182C2—21no—
183C2—561no—
184C2#1172625211yescharacteristic
subgroupG/H
centerC2#11726252
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1366C2
socleC2×C7#1677
prime radical#6467D4
hypercenter#1366D7
hypocenterC7#12466
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
2. #11729107C2semidirect
5. C7×#1366C2semidirect
6. #11725378C2semidirect
7. #11725373C2semidirect
49. #1366D7semidirect
173. C7#12466semidirect

Maximal subgroups

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

subgroupindex
#117291122
#117291072
#117270602
#117270552
C7×#13662
#117253782
#117253732
#124667