Finite Group Databasebeta

← #13264345

Subgroups — #13264345

Order 1792.

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

normal subgroups only

This group has 292 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#11736022C211yesmaximal, characteristic
2C7×#2166C211yesmaximal, characteristic
3#11727649C211yesmaximal, characteristic
4#11725343C212yesmaximal
5#11725342C212yesmaximal
6#90209—21no—
7C7×#405C2211yescharacteristic
8#89909C2212yes—
9#89888—21no—
10#89873—21no—
11#89488C412yescomplemented
12#89475—21no—
13#89471C411yescharacteristic, complemented
14C4×Dic28—21no—
15#89455C411yescharacteristic, complemented
16C7×#364—21no—
17C7×#349C2212yes—
18C7×#347—21no—
19C7⋊#338C2211yescharacteristic
20#89352C2211yescharacteristic
21#89326—21no—
22#11771—71nomaximal, Sylow
23C2×C7×M16C2311yescharacteristic
24C22×C7×C8D411yescharacteristic
25C7×#117D411yescharacteristic
26#6489—21no—
27#6487—22no—
28C2×#1136—21no—
29C2×Dic28—21no—
30C2×Dic28C2×C412yes—
31#6452—42no—
32#6442—42no—
33#6433—21no—
34#6432—21no—
35C4×Dic14—22no—
36C7×M32—22no—
37C7×#97C2×C411yescharacteristic
38C7×#97D411yescharacteristic
39C4×C7×C8C2×C411yescharacteristic
40C4×C7×C8D411yescharacteristic
41#6392—21no—
42#6391C2×C412yes—
43#6390—22no—
44#6390—22no—
45#6389—22no—
46C8×Dic7—22no—
47#3177—71no—
48#2166D711yescharacteristic
49#1747—71no—
50#1342—72no—
51#1340—72no—
52C22×C4×C7C2×D411yescharacteristic
53C2×Dic14—22no—
54C7×M16—41no—
55C2×C7×C8—21no—
56C2×C7×C8—21no—
57C2×C7×C8C22⋊C411yescharacteristic
58C2×C7×C8C22⋊C411yescharacteristic
59C2×C7×C8C22×C411yescharacteristic
60C2×C7×C8C2×D411yescharacteristic
61C7×C4⋊C4—21no—
62C7×C22⋊C4—21no—
63C42×C7C22⋊C411yescharacteristic
64C42×C7C22⋊C411yescharacteristic
65C22⋊Dic7—42no—
66C22⋊Dic7—41no—
67C4⋊Dic7—22no—
68C4⋊Dic7—42no—
69C4⋊Dic7—41no—
70Dic7⋊C4—42no—
71Dic7⋊C4—42no—
72Dic7⋊C4—42no—
73C4×Dic7—22no—
74C4×Dic7—42no—
75#1136—41no—
76#1136—41no—
77C2×C7⋊C8—22no—
78Dic28—42no—
79Dic28—41no—
80Dic28—41no—
81C7×C16—42no—
82#541—141no—
83#499—71no—
84#494—71no—
85#491—141no—
86#470—72no—
87#467—141no—
88#441—72no—
89C4×Q16—141no—
90#405—141no—
91#405C2×D711yescharacteristic
92#364—21no—
93#349C2×D712yes—
94#347—21no—
95#338—71no—
96H(3,ℤ/4)—71no—
97H(3,ℤ/4)—141no—
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7C2×C22⋊C411yescharacteristic
104C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108Dic14—42no—
109Dic14—42no—
110C7×C8—21no—
111C7×C8—21no—
112C7×C8—21no—
113C7×C8#10411yescharacteristic
114C7×C8#10411yescharacteristic
115C7⋊C8—42no—
116C2×Q16—72no—
117C2×Q16—141no—
118C2×M16—141no—
119C2×M16C22×D711yescharacteristic
120C22×C8D2811yescharacteristic
121C4⋊Q8—141no—
122#126—282no—
123#125—141no—
124#125—282no—
125C22⋊Q8—142no—
126C4×Q8—142no—
127#117—141no—
128#117D2811yescharacteristic
129M32—22no—
130#107—141no—
131#106—72no—
132#103—142no—
133#103—142no—
134#97—142no—
135#97C4×D711yescharacteristic
136#97C22⋊D711yescharacteristic
137C4×C8—142no—
138C4×C8C4×D711yescharacteristic
139C4×C8C22⋊D711yescharacteristic
140C22×C7—21no—
141C22×C7—41no—
142C22×C7#41811yescharacteristic
143C4×C7—21no—
144C4×C7—41no—
145C4×C7—41no—
146C4×C7#41011yescharacteristic
147C4×C7C2×#10411yescharacteristic
148Dic7—82no—
149Dic7—82no—
150Dic7—82no—
151C2×Q8—142no—
152C22×C4C2×D2811yescharacteristic
153Q16—282no—
154Q16—281no—
155Q16—281no—
156M16—41no—
157M16—281no—
158M16—281no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—142no—
162C2×C8(C2×D7)⋊C411yescharacteristic
163C2×C8(C2×D7)⋊C411yescharacteristic
164C2×C8C2×C4×D711yescharacteristic
165C2×C8C2×C22⋊D711yescharacteristic
166C4⋊C4—21no—
167C4⋊C4—142no—
168C4⋊C4—282no—
169C4⋊C4—282no—
170C4⋊C4—281no—
171C4⋊C4—282no—
172C4⋊C4—282no—
173C22⋊C4—21no—
174C22⋊C4—281no—
175C22⋊C4—282no—
176C42—142no—
177C42—282no—
178C42(C2×D7)⋊C411yescharacteristic
179C42(C2×D7)⋊C411yescharacteristic
180C16—42no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7#150511yescharacteristic
184C23#647111yescharacteristic
185Q8—282no—
186Q8—282no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—282no—
192C2×C4—282no—
193C2×C4—282no—
194C2×C4#647111yescharacteristic
195C2×C4C2×(C2×D7)⋊C411yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—282no—
200C8#639911yescharacteristic
201C8#639911yescharacteristic
202C7#1177111yescharacteristic, complemented, Sylow
203C22—21no—
204C22—41no—
205C22#8990311yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—562no—
210C4—562no—
211C4—562no—
212C4#8977811yescharacteristic
213C4C2×#639911yescharacteristic
214C2—21no—
215C2—41no—
216C2#1172858611yescharacteristic
subgroupG/H
centerC2#11728586
commutatorC2×C7×C8C22×C4
FrattiniC2×M16C22×D7
FittingC7×#2166C2
socleC2×C7#1505
prime radicalC23×C7AGL(1,ℤ/8)
cosocleC2×C7×M16C23
hypercenter#2166D7
hypocenterC7#11771
LCSG ▷ C2×C7×C8 ▷ 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
11. #89488C4semidirect
13. #89471C4semidirect
15. #89455C4semidirect
202. C7#11771semidirect

Maximal subgroups

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

subgroupindex
#117360222
C7×#21662
#117276492
#117253432
#117253422
#117717