Finite Group Databasebeta

← #13265314

Subgroups — #13265314

Order 1792.

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

normal subgroups only

This group has 292 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
1#11736014C211yesmaximal, characteristic
2C7×#2166C211yesmaximal, characteristic, complemented
3#11727639C211yesmaximal, characteristic, complemented
4#11725379C212yesmaximal, complemented
5#11725377C212yesmaximal, complemented
6#90226C412yes—
7C7×#405C2211yescharacteristic
8#89907—21no—
9#89887—21no—
10C2×#6386—21no—
11#89485C412yes—
12#89483—21no—
13#89467C2211yescharacteristic
14C4×D56—21no—
15#89455C2211yescharacteristic
16C7×#364C2212yes—
17C7×#349—21no—
18C7×#347—21no—
19#89258—21no—
20#89258C2212yes—
21#11989—71nomaximal, Sylow
22C2×C7×M16D411yescharacteristic
23C22×C7×C8D411yescharacteristic
24C7×#117C2×C411yescharacteristic
25#6493—22no—
26C2×D56—21no—
27C2×D56—21no—
28#6457—42no—
29#6447—42no—
30#6437C2×C412yes—
31C4×D28—22no—
32#6433C2×C412yes—
33C7×M32—22no—
34C7×#97C2×C411yescharacteristic
35C7×#97D411yescharacteristic
36C4×C7×C8D411yescharacteristic
37C4×C7×C8C2311yescharacteristic
38#6395—22no—
39#6395—22no—
40#6392—21no—
41#6391—21no—
42#6386—22no—
43#6386—41no—
44C2×C7⋊C16—41no—
45#3176—71no—
46#2166D711yescharacteristic, complemented
47#2153—71no—
48#1366—72no—
49#1357—72no—
50C22×C4×C7C22⋊C411yescharacteristic
51C2×C22⋊D7—42no—
52C2×D28—22no—
53C2×D28—42no—
54C2×C4×D7—42no—
55C7×M16—22no—
56C2×C7×C8—21no—
57C2×C7×C8—21no—
58C2×C7×C8C22⋊C411yescharacteristic
59C2×C7×C8C22⋊C411yescharacteristic
60C2×C7×C8C2×D411yescharacteristic
61C2×C7×C8C2×D411yescharacteristic
62C7×C4⋊C4—21no—
63C7×C22⋊C4—21no—
64C42×C7C22⋊C411yescharacteristic
65C42×C7C22×C411yescharacteristic
66(C2×D7)⋊C4—42no—
67(C2×D7)⋊C4—42no—
68C4⋊Dic7—22no—
69Dic7⋊C4—42no—
70D56—42no—
71D56—41no—
72D56—41no—
73C7×C16—42no—
74C7⋊C16—42no—
75#540—72no—
76C2×M32—141no—
77#499—71no—
78#493—71no—
79#490—72no—
80#469—141no—
81#466—141no—
82#442—141no—
83C4×D8—141no—
84#405C2×D711yescharacteristic
85#364—72no—
86#364—141no—
87#364C2×D712yes—
88#349—21no—
89#347—21no—
90C23×C7#10011yescharacteristic
91C22×D7—42no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7#10011yescharacteristic
97C2×C4×C7C2×C22⋊C411yescharacteristic
98C22⋊D7—162no—
99C2×Dic7—42no—
100D28—42no—
101D28—42no—
102D28—162no—
103C4×D7—162no—
104C7×C8—21no—
105C7×C8—21no—
106C7×C8#13512yes—
107C7×C8AGL(1,ℤ/8)11yescharacteristic
108C7×C8#13711yescharacteristic
109C2×D8—141no—
110C2×D8—141no—
111C2×M16C22⋊D711yescharacteristic
112C22×C8D2811yescharacteristic
113#127—72no—
114#125—72no—
115#123—282no—
116#121—142no—
117#121—282no—
118C4×D4—142no—
119#117C4×D711yescharacteristic
120M32—22no—
121M32—142no—
122M32—281no—
123C2×C16—281no—
124#107—141no—
125#106—141no—
126#102—142no—
127#102—142no—
128#97C4×D711yescharacteristic
129#97C22⋊D711yescharacteristic
130C4×C8D2811yescharacteristic
131C4×C8C22×D711yescharacteristic
132C22×C7—21no—
133C22×C7—41no—
134C22×C7C2×#10011yescharacteristic
135C2×D7—82no—
136C2×D7—82no—
137C2×D7—82no—
138C4×C7—21no—
139C4×C7—41no—
140C4×C7—41no—
141C4×C7#41611yescharacteristic
142C4×C7#41711yescharacteristic
143Dic7—82no—
144C2×D4—142no—
145C2×D4—282no—
146C2×D4—282no—
147C22×C4—282no—
148C22×C4(C2×D7)⋊C411yescharacteristic
149D8—281no—
150D8—281no—
151D8—282no—
152M16—22no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8(C2×D7)⋊C411yescharacteristic
156C2×C8(C2×D7)⋊C411yescharacteristic
157C2×C8C2×D2811yescharacteristic
158C2×C8C2×C22⋊D711yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—142no—
161C4⋊C4—282no—
162C22⋊C4—21no—
163C22⋊C4—282no—
164C22⋊C4—282no—
165C42(C2×D7)⋊C411yescharacteristic
166C42C2×C4×D711yescharacteristic
167C16—42no—
168C16—282no—
169C2×C7—21no—
170C2×C7—41no—
171C2×C7#155111yescharacteristic
172D7—162no—
173C23—282no—
174C23#639711yescharacteristic
175D4—282no—
176D4—282no—
177D4—1122no—
178D4—1122no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—282no—
184C2×C4—1122no—
185C2×C4#639711yescharacteristic
186C2×C4C2×(C2×D7)⋊C411yescharacteristic
187C8—21no—
188C8—21no—
189C8#647111yescharacteristic
190C8#647211yescharacteristic
191C8#651312yes—
192C7#1198911yescharacteristic, complemented, Sylow
193C22—21no—
194C22—41no—
195C22—562no—
196C22—562no—
197C22—562no—
198C22C2×#639711yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—562no—
203C4#8977411yescharacteristic
204C4#8990511yescharacteristic
205C2—21no—
206C2—41no—
207C2—1122no—
208C2#1172857011yescharacteristic
subgroupG/H
centerC2#11728570
commutatorC42×C7C22×C4
FrattiniC4×C8C22×D7
FittingC7×#2166C2
socleC2×C7#1551
prime radical#11736014C2
cosocleC4×C7×C8C23
hypercenter#2166D7
hypocenterC7#11989
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. #11727639C2semidirect
4. #11725379C2semidirect
5. #11725377C2semidirect
46. #2166D7semidirect
192. C7#11989semidirect

Maximal subgroups

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

subgroupindex
#117360142
C7×#21662
#117276392
#117253792
#117253772
#119897