Finite Group Databasebeta

← #13264983

Subgroups — #13264983

Order 1792.

296 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 75 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#11736014C211yesmaximal, characteristic
2C7×#2166C211yesmaximal, characteristic, complemented
3#11727638C211yesmaximal, characteristic, complemented
4#11725326C214yesmaximal, complemented
5#90226C412yes—
6C7×#405C2211yescharacteristic
7#89907—21no—
8#89887—21no—
9C2×#6386—21no—
10#89485C2212yes—
11#89483—21no—
12#89467C411yescharacteristic
13C4×D56—21no—
14#89455C411yescharacteristic
15C7×#364—21no—
16C7×#349—21no—
17C7×#347C2212yes—
18#89261—21no—
19#89261C2212yes—
20#11913—71nomaximal, Sylow
21C2×C7×M16D411yescharacteristic
22C22×C7×C8D411yescharacteristic
23C7×#117C2×C411yescharacteristic
24#6493—22no—
25C2×D56—21no—
26C2×D56—21no—
27#6457—42no—
28#6447—42no—
29#6437C2×C412yes—
30C4×D28—22no—
31#6433C2×C412yes—
32C7×M32—22no—
33C7×#97D411yescharacteristic
34C7×#97C2311yescharacteristic
35C4×C7×C8C2×C411yescharacteristic
36C4×C7×C8D411yescharacteristic
37#6395—22no—
38#6395—22no—
39#6392—21no—
40#6391—21no—
41#6386—22no—
42#6386—22no—
43C2×C7⋊C16—22no—
44#3176—71no—
45#2166D711yescharacteristic, complemented
46#2110—71no—
47#1355—74no—
48C22×C4×C7C22⋊C411yescharacteristic
49C2×C22⋊D7—42no—
50C2×D28—22no—
51C2×D28—42no—
52C2×C4×D7—42no—
53C7×M16—41no—
54C2×C7×C8—21no—
55C2×C7×C8—21no—
56C2×C7×C8C22⋊C411yescharacteristic
57C2×C7×C8C22⋊C411yescharacteristic
58C2×C7×C8C2×D411yescharacteristic
59C2×C7×C8C2×D411yescharacteristic
60C7×C4⋊C4—21no—
61C7×C22⋊C4—21no—
62C42×C7C22⋊C411yescharacteristic
63C42×C7C22×C411yescharacteristic
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66C4⋊Dic7—22no—
67Dic7⋊C4—42no—
68D56—42no—
69D56—41no—
70D56—41no—
71C7×C16—42no—
72C7⋊C16—24no—
73#540—72no—
74C2×M32—141no—
75#499—71no—
76#493—71no—
77#490—72no—
78#469—141no—
79#466—141no—
80#442—141no—
81C4×D8—141no—
82#405C2×D711yescharacteristic
83#364—21no—
84#349—21no—
85#347—72no—
86#347—141no—
87#347C2×D712yes—
88C23×C7#10011yescharacteristic
89C22×D7—42no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7#10011yescharacteristic
95C2×C4×C7C2×C22⋊C411yescharacteristic
96C22⋊D7—162no—
97C2×Dic7—42no—
98D28—42no—
99D28—42no—
100D28—162no—
101C4×D7—162no—
102C7×C8—21no—
103C7×C8—21no—
104C7×C8—21no—
105C7×C8#10411yescharacteristic
106C7×C8#10411yescharacteristic
107C2×D8—141no—
108C2×D8—141no—
109C2×M16C22⋊D711yescharacteristic
110C22×C8D2811yescharacteristic
111#127—72no—
112#125—72no—
113#123—282no—
114#121—142no—
115#121—282no—
116C4×D4—142no—
117#117C4×D711yescharacteristic
118M32—22no—
119M32—142no—
120M32—142no—
121C2×C16—142no—
122#107—141no—
123#106—141no—
124#102—142no—
125#102—142no—
126#97D2811yescharacteristic
127#97C22×D711yescharacteristic
128C4×C8C4×D711yescharacteristic
129C4×C8C22⋊D711yescharacteristic
130C22×C7—21no—
131C22×C7—41no—
132C22×C7C2×#10011yescharacteristic
133C2×D7—82no—
134C2×D7—82no—
135C2×D7—82no—
136C4×C7—21no—
137C4×C7—41no—
138C4×C7—41no—
139C4×C7C2×#10411yescharacteristic
140C4×C7#42111yescharacteristic
141Dic7—82no—
142C2×D4—142no—
143C2×D4—282no—
144C2×D4—282no—
145C22×C4—282no—
146C22×C4(C2×D7)⋊C411yescharacteristic
147D8—282no—
148D8—281no—
149D8—281no—
150M16—41no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8(C2×D7)⋊C411yescharacteristic
154C2×C8(C2×D7)⋊C411yescharacteristic
155C2×C8C2×D2811yescharacteristic
156C2×C8C2×C22⋊D711yescharacteristic
157C4⋊C4—21no—
158C4⋊C4—142no—
159C4⋊C4—282no—
160C22⋊C4—21no—
161C22⋊C4—282no—
162C22⋊C4—282no—
163C42(C2×D7)⋊C411yescharacteristic
164C42C2×C4×D711yescharacteristic
165C16—42no—
166C16—144no—
167C2×C7—21no—
168C2×C7—41no—
169C2×C7#153011yescharacteristic
170D7—162no—
171C23—282no—
172C23#639711yescharacteristic
173D4—282no—
174D4—282no—
175D4—1122no—
176D4—1122no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—282no—
182C2×C4—1122no—
183C2×C4#639711yescharacteristic
184C2×C4C2×(C2×D7)⋊C411yescharacteristic
185C8—21no—
186C8—21no—
187C8—21no—
188C8#639911yescharacteristic
189C8#639911yescharacteristic
190C7#1191311yescharacteristic, complemented, Sylow
191C22—21no—
192C22—41no—
193C22—562no—
194C22—562no—
195C22—562no—
196C22C2×#639711yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—562no—
201C4#8977711yescharacteristic
202C4C2×#639911yescharacteristic
203C2—21no—
204C2—41no—
205C2—1122no—
206C2#1172856011yescharacteristic
subgroupG/H
centerC2#11728560
commutatorC42×C7C22×C4
Frattini#97C22×D7
FittingC7×#2166C2
socleC2×C7#1530
prime radical#11736014C2
cosocleC7×#97C23
hypercenter#2166D7
hypocenterC7#11913
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. C7×#2166C2semidirect
3. #11727638C2semidirect
4. #11725326C2semidirect
45. #2166D7semidirect
190. C7#11913semidirect

Maximal subgroups

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

subgroupindex
#117360142
C7×#21662
#117276382
#117253262
#119137