Finite Group Databasebeta

← #13264989

Subgroups — #13264989

Order 1792.

326 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 71 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#11735971C211yesmaximal, characteristic
2C7×#2165C211yesmaximal, characteristic, complemented
3#11727618C211yesmaximal, characteristic, complemented
4#11725326C214yesmaximal, complemented
5C2×#6437C411yescharacteristic
6#90161C411yescharacteristic
7C4×C7×M16C2211yescharacteristic
8#89903—21no—
9#89903—21no—
10#89485C412yes—
11#89485C2212yes—
12#89466—21no—
13#89466—21no—
14C7×#364—21no—
15C7×#349—21no—
16C7×#347C2212yescomplemented
17#89348—21no—
18#89261C2212yescomplemented
19#89258—21no—
20#11915—71nomaximal, Sylow
21C22×D28—21no—
22C22×D28—41no—
23C2×C7×M16D411yescharacteristic
24C2×C7×M16D411yescharacteristic
25C2×C42×C7C2×C411yescharacteristic
26#6489—21no—
27#6487—41no—
28#6437—21no—
29#6437—41no—
30#6437C2×C412yes—
31#6433C2×C412yes—
32#6432—21no—
33C4×Dic14—41no—
34C7×M32—22no—
35C7×#97C2×C411yescharacteristic
36C7×#97C2311yescharacteristic
37C4×C7×C8D411yescharacteristic
38C4×C7×C8D411yescharacteristic
39#6395—22no—
40#6395—22no—
41#6395—22no—
42#6395—22no—
43#6386—22no—
44#3138—71no—
45#2165—71no—
46#2165D711yescharacteristic, complemented
47#1355—74no—
48C23×D7—41no—
49C22×C4×C7—21no—
50C22×C4×C7C22⋊C411yescharacteristic
51C2×D28—22no—
52C2×D28—42no—
53C2×D28—42no—
54C2×D28—42no—
55C2×D28—42no—
56C2×D28—41no—
57C2×D28—84no—
58C2×Dic14—41no—
59C7×M16—41no—
60C7×M16—41no—
61C2×C7×C8C22⋊C411yescharacteristic
62C2×C7×C8C22⋊C411yescharacteristic
63C2×C7×C8C2×D411yescharacteristic
64C2×C7×C8C2×D411yescharacteristic
65C42×C7—21no—
66C42×C7C22⋊C411yescharacteristic
67C42×C7C22×C411yescharacteristic
68C22⋊Dic7—41no—
69C4⋊Dic7—22no—
70C4⋊Dic7—41no—
71Dic7⋊C4—42no—
72C4×Dic7—41no—
73C7×C16—42no—
74C7⋊C16—42no—
75#533—71no—
76C2×#127—71no—
77#490—72no—
78#490—72no—
79#486—141no—
80#486—141no—
81#418—141no—
82#418—141no—
83C4×M16C2×D711yescharacteristic
84#364—21no—
85#364—141no—
86#349—21no—
87#349—141no—
88#347—72no—
89#347C2×D712yescomplemented
90C23×C7#10011yescharacteristic
91C22×D7—42no—
92C22×D7—82no—
93C22×D7—84no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—42no—
99C2×C4×C7#10011yescharacteristic
100C2×C4×C7C2×C22⋊C411yescharacteristic
101C2×Dic7—42no—
102D28—42no—
103D28—42no—
104D28—82no—
105D28—84no—
106D28—84no—
107Dic14—82no—
108C7×C8—21no—
109C7×C8—21no—
110C7×C8—21no—
111C7×C8—21no—
112C22×D4—141no—
113C22×D4—281no—
114C2×M16D2811yescharacteristic
115C2×M16C22⋊D711yescharacteristic
116C4⋊Q8—141no—
117#127—72no—
118#127—141no—
119#127—281no—
120#125—72no—
121C22⋊Q8—281no—
122C4×Q8—281no—
123#117—141no—
124C2×C42C4×D711yescharacteristic
125M32—22no—
126M32—142no—
127#102—142no—
128#102—142no—
129#102—142no—
130#102—142no—
131#97C4×D711yescharacteristic
132#97C22×D711yescharacteristic
133C4×C8D2811yescharacteristic
134C4×C8C22⋊D711yescharacteristic
135C22×C7—21no—
136C22×C7—41no—
137C22×C7C2×#10011yescharacteristic
138C2×D7—84no—
139C2×D7—82no—
140C2×D7—164no—
141C4×C7—22no—
142C4×C7—21no—
143C4×C7—41no—
144C4×C7#42111yescharacteristic
145C4×C7#42111yescharacteristic
146Dic7—82no—
147C24—281no—
148C2×Q8—281no—
149C2×D4—142no—
150C2×D4—282no—
151C2×D4—282no—
152C2×D4—282no—
153C2×D4—282no—
154C2×D4—281no—
155C2×D4—564no—
156C22×C4—21no—
157C22×C4(C2×D7)⋊C411yescharacteristic
158M16—41no—
159M16—41no—
160C2×C8(C2×D7)⋊C411yescharacteristic
161C2×C8(C2×D7)⋊C411yescharacteristic
162C2×C8C2×D2811yescharacteristic
163C2×C8C2×C22⋊D711yescharacteristic
164C4⋊C4—142no—
165C4⋊C4—282no—
166C4⋊C4—281no—
167C22⋊C4—281no—
168C42—21no—
169C42—281no—
170C42(C2×D7)⋊C411yescharacteristic
171C42C2×C4×D711yescharacteristic
172C16—42no—
173C16—282no—
174C2×C7—21no—
175C2×C7—41no—
176C2×C7#153311yescharacteristic
177D7—162no—
178C23—282no—
179C23—562no—
180C23—564no—
181C23#639711yescharacteristic
182Q8—562no—
183D4—282no—
184D4—282no—
185D4—562no—
186D4—564no—
187D4—564no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—282no—
194C2×C4#639711yescharacteristic
195C2×C4C2×(C2×D7)⋊C411yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—21no—
200C7#1191511yescharacteristic, complemented, Sylow
201C22—21no—
202C22—41no—
203C22—564no—
204C22—562no—
205C22—1124no—
206C22C2×#639711yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—562no—
211C4#8977711yescharacteristic
212C4#8991111yescharacteristic
213C2—21no—
214C2—41no—
215C2—1122no—
216C2#1172856211yescharacteristic
subgroupG/H
centerC2#11728562
commutatorC42×C7C22×C4
Frattini#97C22×D7
FittingC7×#2165C2
socleC2×C7#1533
prime radicalC2×#6437C4
cosocleC7×#97C23
hypercenter#2165D7
hypocenterC7#11915
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. C7×#2165C2semidirect
3. #11727618C2semidirect
4. #11725326C2semidirect
16. C7×#347C22semidirect
18. #89261C22semidirect
46. #2165D7semidirect
89. #347C2×D7semidirect
200. C7#11915semidirect

Maximal subgroups

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

subgroupindex
#117359712
C7×#21652
#117276182
#117253262
#119157