Finite Group Databasebeta

← #13264342

Subgroups — #13264342

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×#1747C211yesmaximal, characteristic
3#11727640C211yesmaximal, characteristic
4#11725290C212yesmaximal
5#11725288C212yesmaximal
6#90209—21no—
7C7×#405—21no—
8C7×#405C2211yescharacteristic
9#89909C2212yes—
10#89888—21no—
11#89488C412yescomplemented
12#89475—21no—
13#89471C411yescharacteristic, complemented
14C4×Dic28—21no—
15#89455C411yescharacteristic, complemented
16C7×#338C2211yescharacteristic
17C7×H(3,ℤ/4)—21no—
18C7×H(3,ℤ/4)C2211yescharacteristic
19#89348C2212yes—
20#89261—21no—
21#89258—21no—
22#11771—71nomaximal, Sylow
23C2×C7×M16—21no—
24C2×C7×M16C2311yescharacteristic
25C22×C7×C8D411yescharacteristic
26C7×#117—21no—
27C7×#117D411yescharacteristic
28#6487—22no—
29C2×Dic28—21no—
30C2×Dic28C2×C412yes—
31#6452—42no—
32#6442—42no—
33#6433—21no—
34#6432—21no—
35C4×Dic14—22no—
36C7×#97—22no—
37C7×#97C2×C411yescharacteristic
38C7×#97D411yescharacteristic
39C4×C7×C8—22no—
40C4×C7×C8C2×C411yescharacteristic
41C4×C7×C8D411yescharacteristic
42#6392—21no—
43#6391C2×C412yes—
44#6390—22no—
45#6390—22no—
46#6386—22no—
47#3177—71no—
48#2166—71no—
49#1747D711yescharacteristic
50#1342—72no—
51#1340—72no—
52C22×C4×C7C2×D411yescharacteristic
53C2×Dic14—22no—
54C7×M16—41no—
55C7×M16—41no—
56C7×M16—41no—
57C2×C7×C8—21no—
58C2×C7×C8—21no—
59C2×C7×C8—22no—
60C2×C7×C8C22⋊C411yescharacteristic
61C2×C7×C8C22⋊C411yescharacteristic
62C2×C7×C8C22×C411yescharacteristic
63C2×C7×C8C2×D411yescharacteristic
64C7×C4⋊C4—21no—
65C7×C4⋊C4—41no—
66C7×C22⋊C4—21no—
67C7×C22⋊C4—41no—
68C42×C7—22no—
69C42×C7C22⋊C411yescharacteristic
70C42×C7C22⋊C411yescharacteristic
71C22⋊Dic7—42no—
72C4⋊Dic7—22no—
73C4⋊Dic7—42no—
74Dic7⋊C4—42no—
75Dic7⋊C4—42no—
76Dic7⋊C4—42no—
77C4×Dic7—42no—
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—21no—
91#405C2×D711yescharacteristic
92#364—141no—
93#349—72no—
94#347—141no—
95#338C2×D711yescharacteristic
96H(3,ℤ/4)—21no—
97H(3,ℤ/4)C2×D711yescharacteristic
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—42no—
104C2×C4×C7C2×C22⋊C411yescharacteristic
105C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
106C2×Dic7—42no—
107C2×Dic7—42no—
108Dic14—42no—
109Dic14—42no—
110C7×C8—21no—
111C7×C8—21no—
112C7×C8—21no—
113C7×C8—42no—
114C7×C8#10411yescharacteristic
115C7×C8#10411yescharacteristic
116C2×Q16—72no—
117C2×Q16—141no—
118C2×M16—21no—
119C2×M16C22×D711yescharacteristic
120C22×C8C22⋊D711yescharacteristic
121C4⋊Q8—141no—
122#126—282no—
123#125—141no—
124#125—282no—
125C22⋊Q8—142no—
126C4×Q8—142no—
127#117—21no—
128#117C22⋊D711yescharacteristic
129M32—142no—
130#107—141no—
131#106—72no—
132#103—142no—
133#103—142no—
134#97—22no—
135#97C4×D711yescharacteristic
136#97D2811yescharacteristic
137C4×C8—22no—
138C4×C8C4×D711yescharacteristic
139C4×C8D2811yescharacteristic
140C22×C7—21no—
141C22×C7—41no—
142C22×C7#41811yescharacteristic
143C4×C7—21no—
144C4×C7—41no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7#41011yescharacteristic
148C4×C7C2×#10411yescharacteristic
149Dic7—82no—
150Dic7—82no—
151C2×Q8—142no—
152C22×C4C2×C22⋊D711yescharacteristic
153Q16—282no—
154Q16—281no—
155Q16—281no—
156M16—41no—
157M16—41no—
158M16—41no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8(C2×D7)⋊C411yescharacteristic
163C2×C8(C2×D7)⋊C411yescharacteristic
164C2×C8C2×C4×D711yescharacteristic
165C2×C8C2×D2811yescharacteristic
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—142no—
169C4⋊C4—282no—
170C4⋊C4—282no—
171C4⋊C4—282no—
172C4⋊C4—282no—
173C22⋊C4—21no—
174C22⋊C4—41no—
175C22⋊C4—282no—
176C42—22no—
177C42—282no—
178C42(C2×D7)⋊C411yescharacteristic
179C42(C2×D7)⋊C411yescharacteristic
180C16—282no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7#150511yescharacteristic
184C23#651211yescharacteristic
185Q8—282no—
186Q8—282no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—282no—
193C2×C4—282no—
194C2×C4C2×(C2×D7)⋊C411yescharacteristic
195C2×C4#651211yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—42no—
200C8#637911yescharacteristic
201C8#637911yescharacteristic
202C7#1177111yescharacteristic, complemented, Sylow
203C22—21no—
204C22—41no—
205C22#8977311yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—562no—
211C4—562no—
212C4C2×#637911yescharacteristic
213C4#8977811yescharacteristic
214C2—21no—
215C2—41no—
216C2#1172718611yescharacteristic
subgroupG/H
centerC2#11727186
commutatorC2×C7×C8C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1747C2
socleC2×C7#1505
prime radicalC23×C7AGL(1,ℤ/8)
cosocleC2×C7×M16C23
hypercenter#1747D7
hypocenterC7#11771
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

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×#17472
#117276402
#117252902
#117252882
#117717