Finite Group Databasebeta

← #13264648

Subgroups — #13264648

Order 1792.

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

normal subgroups only

This group has 280 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
1C7×#3185C211yesmaximal, characteristic
2#11727625C211yesmaximal, characteristic
3#11727618C211yesmaximal, characteristic
4#11725587C212yesmaximal
5#11725586C212yesmaximal
6C7×#533—21no—
7C7×#498C411yescharacteristic, complemented
8C7×#494C411yescharacteristic, complemented
9C7×#487—21no—
10C7×#470—21no—
11C7×#470C2212yes—
12C7×#441—21no—
13C7×#441C412yescomplemented
14C4×C7×M16C2211yescharacteristic
15C7⋊#338C2211yescharacteristic
16#89352C2211yescharacteristic
17#89351—21no—
18#89348C2212yes—
19#89318—21no—
20#89261—21no—
21#89258—21no—
22#11838—71nomaximal, Sylow
23C2×C7×Q16—21no—
24C2×C7×Q16C2×C412yes—
25C2×C7×M16D411yescharacteristic
26C2×C7×M16C2311yescharacteristic
27C7×C4⋊Q8—21no—
28C7×C4⋊Q8—21no—
29C7×#125—41no—
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—41no—
32C4×C7×Q8—22no—
33C4×C7×Q8—41no—
34C7×#117—41no—
35C2×C42×C7D411yescharacteristic
36#6489—21no—
37C2×#1136—21no—
38C7×#106—21no—
39C7×#106C2×C412yes—
40C7×#103—22no—
41C7×#103—22no—
42C7×#97C2×C411yescharacteristic
43C7×#97D411yescharacteristic
44C4×C7×C8C2×C411yescharacteristic
45C4×C7×C8D411yescharacteristic
46#6386—22no—
47#3185D711yescharacteristic
48#2165—71no—
49#1841—71no—
50#1342—72no—
51#1340—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—41no—
54C22×C4×C7—21no—
55C22×C4×C7C2×D411yescharacteristic
56C7×Q16—42no—
57C7×Q16—42no—
58C7×M16—41no—
59C7×M16—41no—
60C2×C7×C8C22⋊C411yescharacteristic
61C2×C7×C8C22⋊C411yescharacteristic
62C2×C7×C8C22×C411yescharacteristic
63C2×C7×C8C2×D411yescharacteristic
64C7×C4⋊C4—22no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—41no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—42no—
70C7×C22⋊C4—42no—
71C42×C7—21no—
72C42×C7—42no—
73C42×C7C22⋊C411yescharacteristic
74C42×C7C22⋊C411yescharacteristic
75C22⋊Dic7—41no—
76C4⋊Dic7—41no—
77C4×Dic7—41no—
78#1136—41no—
79#1136—41no—
80C2×C7⋊C8—41no—
81C7⋊C16—42no—
82#533—21no—
83#498Dic711yescharacteristic, complemented
84#494Dic711yescharacteristic, complemented
85#487—21no—
86#470—21no—
87#470C2×D712yes—
88#441—21no—
89#441Dic712yescomplemented
90C4×M16C2×D711yescharacteristic
91#364—141no—
92#349—72no—
93#347—141no—
94#344—141no—
95#339—141no—
96#338—71no—
97H(3,ℤ/4)—71no—
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C7×Q8—42no—
100C7×Q8—42no—
101C7×Q8—82no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7C2×C22⋊C411yescharacteristic
110C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
111C2×Dic7—81no—
112C7×C8—21no—
113C7×C8—21no—
114C7×C8—21no—
115C7×C8—21no—
116C7⋊C8—81no—
117C2×Q16—21no—
118C2×Q16C2×Dic712yes—
119C2×M16—141no—
120C2×M16C22⋊D711yescharacteristic
121C2×M16C22×D711yescharacteristic
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C4×Q8—22no—
128C4×Q8—41no—
129#117—41no—
130#117—141no—
131C2×C42C22⋊D711yescharacteristic
132M32—142no—
133#106—21no—
134#106C2×Dic712yes—
135#103—22no—
136#103—22no—
137#97C2×Dic711yescharacteristic
138#97C22⋊D711yescharacteristic
139C4×C8C2×Dic711yescharacteristic
140C4×C8C22⋊D711yescharacteristic
141C22×C7—21no—
142C22×C7—41no—
143C22×C7#41811yescharacteristic
144C4×C7—22no—
145C4×C7—21no—
146C4×C7—41no—
147C4×C7—82no—
148C4×C7—82no—
149C4×C7#41011yescharacteristic
150C4×C7#42111yescharacteristic
151Dic7—161no—
152C2×Q8—22no—
153C2×Q8—41no—
154C22×C4—21no—
155C22×C4C2×C22⋊D711yescharacteristic
156Q16—42no—
157Q16—42no—
158M16—41no—
159M16—41no—
160M16—281no—
161M16—281no—
162C2×C8—281no—
163C2×C8C22⋊Dic711yescharacteristic
164C2×C8C22⋊Dic711yescharacteristic
165C2×C8C22×Dic711yescharacteristic
166C2×C8C2×C22⋊D711yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—281no—
174C22⋊C4—42no—
175C22⋊C4—281no—
176C42—21no—
177C42—42no—
178C42—281no—
179C42C22⋊Dic711yescharacteristic
180C42C22⋊Dic711yescharacteristic
181C16—282no—
182C2×C7—21no—
183C2×C7—41no—
184C2×C7#151411yescharacteristic
185C23#649511yescharacteristic
186Q8—42no—
187Q8—42no—
188Q8—82no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—561no—
197C2×C4#649511yescharacteristic
198C2×C4C2×C22⋊Dic711yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—21no—
203C8—561no—
204C7#1183811yescharacteristic, complemented, Sylow
205C22—21no—
206C22—41no—
207C22#8998711yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—1121no—
214C4#9000811yescharacteristic
215C4#9001211yescharacteristic
216C2—21no—
217C2—41no—
218C2#1172956711yescharacteristic
subgroupG/H
centerC2#11729567
commutatorC2×C7×C8C22×C4
FrattiniC2×M16C22×D7
FittingC7×#3185C2
socleC2×C7#1514
prime radicalC23×C7AGL(1,ℤ/8)
cosocleC2×C7×M16C23
hypercenter#3185D7
hypocenterC7#11838
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
7. C7×#498C4semidirect
8. C7×#494C4semidirect
13. C7×#441C4semidirect
83. #498Dic7semidirect
84. #494Dic7semidirect
89. #441Dic7semidirect
204. C7#11838semidirect

Maximal subgroups

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

subgroupindex
C7×#31852
#117276252
#117276182
#117255872
#117255862
#118387