Finite Group Databasebeta

← #13264343

Subgroups — #13264343

Order 1792.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3177C211yesmaximal, characteristic
2#11727649C211yesmaximal, characteristic
3#11727640C211yesmaximal, characteristic
4#11725587C212yesmaximal
5#11725586C212yesmaximal
6C7×#541—21no—
7C7×#499C411yescharacteristic, complemented
8C7×#494C411yescharacteristic, complemented
9C7×#491—21no—
10C7×#470C2212yes—
11C7×#467—21no—
12C7×#441C412yescomplemented
13C4×C7×Q16—21no—
14C7×#405C2211yescharacteristic
15#89873—21no—
16C7⋊#338C2211yescharacteristic
17#89352C2211yescharacteristic
18#89348C2212yes—
19#89326—21no—
20#89261—21no—
21#89258—21no—
22#11771—71nomaximal, Sylow
23C2×C7×Q16—21no—
24C2×C7×Q16C2×C412yes—
25C2×C7×M16C2311yescharacteristic
26C22×C7×C8D411yescharacteristic
27C7×C4⋊Q8—21no—
28C7×#126—42no—
29C7×#125—21no—
30C7×#125—42no—
31C7×C22⋊Q8—22no—
32C4×C7×Q8—22no—
33C7×#117D411yescharacteristic
34#6489—21no—
35C2×#1136—21no—
36C7×#107—21no—
37C7×#106C2×C412yes—
38C7×#103—22no—
39C7×#103—22no—
40C7×#97C2×C411yescharacteristic
41C7×#97D411yescharacteristic
42C4×C7×C8C2×C411yescharacteristic
43C4×C7×C8D411yescharacteristic
44#6389—22no—
45C8×Dic7—22no—
46#6386—22no—
47#3177D711yescharacteristic
48#2166—71no—
49#1747—71no—
50#1342—72no—
51#1340—72no—
52C2×C7×Q8—22no—
53C22×C4×C7C2×D411yescharacteristic
54C7×Q16—42no—
55C7×Q16—41no—
56C7×Q16—41no—
57C7×M16—41no—
58C2×C7×C8—21no—
59C2×C7×C8—21no—
60C2×C7×C8C22⋊C411yescharacteristic
61C2×C7×C8C22⋊C411yescharacteristic
62C2×C7×C8C22×C411yescharacteristic
63C2×C7×C8C2×D411yescharacteristic
64C7×C4⋊C4—21no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C22⋊C4—21no—
71C7×C22⋊C4—42no—
72C42×C7—42no—
73C42×C7C22⋊C411yescharacteristic
74C42×C7C22⋊C411yescharacteristic
75C22⋊Dic7—41no—
76C4⋊Dic7—41no—
77C4×Dic7—22no—
78#1136—41no—
79#1136—41no—
80C2×C7⋊C8—22no—
81C7⋊C16—42no—
82#541—21no—
83#499Dic711yescharacteristic, complemented
84#494Dic711yescharacteristic, complemented
85#491—21no—
86#470C2×D712yes—
87#467—21no—
88#441Dic712yescomplemented
89C4×Q16—21no—
90#405—141no—
91#405C2×D711yescharacteristic
92#364—141no—
93#349—72no—
94#347—141no—
95#338—71no—
96H(3,ℤ/4)—71no—
97H(3,ℤ/4)—141no—
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C7×Q8—42no—
100C7×Q8—42no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7C2×C22⋊C411yescharacteristic
108C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
109C2×Dic7—42no—
110C7×C8—21no—
111C7×C8—21no—
112C7×C8—21no—
113C7×C8#10411yescharacteristic
114C7×C8#10411yescharacteristic
115C7⋊C8—42no—
116C2×Q16—21no—
117C2×Q16C2×Dic712yes—
118C2×M16—141no—
119C2×M16C22×D711yescharacteristic
120C22×C8C22⋊D711yescharacteristic
121C4⋊Q8—21no—
122#126—42no—
123#125—21no—
124#125—42no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117—141no—
128#117C22⋊D711yescharacteristic
129M32—142no—
130#107—21no—
131#106C2×Dic712yes—
132#103—22no—
133#103—22no—
134#97—142no—
135#97C2×Dic711yescharacteristic
136#97C22⋊D711yescharacteristic
137C4×C8—142no—
138C4×C8C2×Dic711yescharacteristic
139C4×C8C22⋊D711yescharacteristic
140C22×C7—21no—
141C22×C7—41no—
142C22×C7#41811yescharacteristic
143C4×C7—21no—
144C4×C7—41no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7—82no—
148C4×C7#41011yescharacteristic
149C4×C7C2×#10411yescharacteristic
150Dic7—82no—
151C2×Q8—22no—
152C22×C4C2×C22⋊D711yescharacteristic
153Q16—42no—
154Q16—41no—
155Q16—41no—
156M16—41no—
157M16—281no—
158M16—281no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—142no—
162C2×C8C22⋊Dic711yescharacteristic
163C2×C8C22⋊Dic711yescharacteristic
164C2×C8C22×Dic711yescharacteristic
165C2×C8C2×C22⋊D711yescharacteristic
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—281no—
173C22⋊C4—21no—
174C22⋊C4—42no—
175C22⋊C4—281no—
176C42—42no—
177C42—142no—
178C42C22⋊Dic711yescharacteristic
179C42C22⋊Dic711yescharacteristic
180C16—282no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7#150511yescharacteristic
184C23#649511yescharacteristic
185Q8—42no—
186Q8—42no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—282no—
194C2×C4#649511yescharacteristic
195C2×C4C2×C22⋊Dic711yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—282no—
200C8#641111yescharacteristic
201C8#641111yescharacteristic
202C7#1177111yescharacteristic, complemented, Sylow
203C22—21no—
204C22—41no—
205C22#8998711yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—562no—
212C4C2×#641111yescharacteristic
213C4#9001211yescharacteristic
214C2—21no—
215C2—41no—
216C2#1172956611yescharacteristic
subgroupG/H
centerC2#11729566
commutatorC2×C7×C8C22×C4
FrattiniC2×M16C22×D7
FittingC7×#3177C2
socleC2×C7#1505
prime radicalC23×C7AGL(1,ℤ/8)
cosocleC2×C7×M16C23
hypercenter#3177D7
hypocenterC7#11771
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
7. C7×#499C4semidirect
8. C7×#494C4semidirect
12. C7×#441C4semidirect
83. #499Dic7semidirect
84. #494Dic7semidirect
88. #441Dic7semidirect
202. C7#11771semidirect

Maximal subgroups

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

subgroupindex
C7×#31772
#117276492
#117276402
#117255872
#117255862
#117717