Finite Group Databasebeta

← #13264337

Subgroups — #13264337

Order 1792.

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

normal subgroups only

This group has 316 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#11736014C211yesmaximal, characteristic
2C7×#1747C211yesmaximal, characteristic, complemented
3#11727640C211yesmaximal, characteristic, complemented
4#11725289C212yesmaximal, complemented
5#11725287C212yesmaximal, complemented
6#90226—21no—
7C7×#405—21no—
8C7×#405C2211yescharacteristic
9#89907C2212yes—
10#89887—21no—
11#89485—21no—
12#89483C412yescomplemented
13#89467C411yescharacteristic, complemented
14C4×D56—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#11770—71nomaximal, Sylow
23C2×C7×M16—21no—
24C2×C7×M16C2311yescharacteristic
25C22×C7×C8D411yescharacteristic
26C7×#117—21no—
27C7×#117D411yescharacteristic
28#6493—22no—
29C2×D56—21no—
30C2×D56C2×C412yes—
31#6457—42no—
32#6447—42no—
33#6437—21no—
34C4×D28—22no—
35#6433—21no—
36C7×#97—22no—
37C7×#97C2×C411yescharacteristic
38C7×#97D411yescharacteristic
39C4×C7×C8—22no—
40C4×C7×C8C2×C411yescharacteristic
41C4×C7×C8D411yescharacteristic
42#6395—22no—
43#6395—22no—
44#6392—21no—
45#6391C2×C412yes—
46#6386—22no—
47#3176—71no—
48#2166—71no—
49#1747D711yescharacteristic, complemented
50#1341—72no—
51#1339—72no—
52C22×C4×C7C2×D411yescharacteristic
53C2×C22⋊D7—42no—
54C2×D28—22no—
55C2×D28—42no—
56C2×C4×D7—42no—
57C7×M16—41no—
58C7×M16—41no—
59C7×M16—41no—
60C2×C7×C8—21no—
61C2×C7×C8—21no—
62C2×C7×C8—22no—
63C2×C7×C8C22⋊C411yescharacteristic
64C2×C7×C8C22⋊C411yescharacteristic
65C2×C7×C8C22×C411yescharacteristic
66C2×C7×C8C2×D411yescharacteristic
67C7×C4⋊C4—21no—
68C7×C4⋊C4—41no—
69C7×C22⋊C4—21no—
70C7×C22⋊C4—41no—
71C42×C7—22no—
72C42×C7C22⋊C411yescharacteristic
73C42×C7C22⋊C411yescharacteristic
74(C2×D7)⋊C4—42no—
75(C2×D7)⋊C4—42no—
76C4⋊Dic7—22no—
77Dic7⋊C4—42no—
78D56—42no—
79D56—41no—
80D56—41no—
81C7⋊C16—42no—
82#540—141no—
83#499—71no—
84#493—71no—
85#490—141no—
86#469—72no—
87#466—141no—
88#442—72no—
89C4×D8—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
99C22×D7—42no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—42no—
105C2×C4×C7C2×C22⋊C411yescharacteristic
106C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
107C22⋊D7—162no—
108C2×Dic7—42no—
109D28—42no—
110D28—42no—
111D28—162no—
112C4×D7—162no—
113C7×C8—21no—
114C7×C8—21no—
115C7×C8—21no—
116C7×C8—42no—
117C7×C8#10411yescharacteristic
118C7×C8#10411yescharacteristic
119C2×D8—72no—
120C2×D8—141no—
121C2×M16—21no—
122C2×M16C22×D711yescharacteristic
123C22×C8C22⋊D711yescharacteristic
124#127—141no—
125#125—141no—
126#123—282no—
127#121—142no—
128#121—282no—
129C4×D4—142no—
130#117—21no—
131#117C22⋊D711yescharacteristic
132M32—142no—
133#107—141no—
134#106—72no—
135#102—142no—
136#102—142no—
137#97—22no—
138#97C4×D711yescharacteristic
139#97D2811yescharacteristic
140C4×C8—22no—
141C4×C8C4×D711yescharacteristic
142C4×C8D2811yescharacteristic
143C22×C7—21no—
144C22×C7—41no—
145C22×C7#41811yescharacteristic
146C2×D7—82no—
147C2×D7—82no—
148C2×D7—82no—
149C4×C7—21no—
150C4×C7—41no—
151C4×C7—41no—
152C4×C7—82no—
153C4×C7#41011yescharacteristic
154C4×C7C2×#10411yescharacteristic
155Dic7—82no—
156C2×D4—142no—
157C2×D4—282no—
158C2×D4—282no—
159C22×C4—282no—
160C22×C4C2×C22⋊D711yescharacteristic
161D8—282no—
162D8—281no—
163D8—281no—
164M16—41no—
165M16—41no—
166M16—41no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8(C2×D7)⋊C411yescharacteristic
171C2×C8(C2×D7)⋊C411yescharacteristic
172C2×C8C2×C4×D711yescharacteristic
173C2×C8C2×D2811yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—142no—
177C4⋊C4—282no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—282no—
181C22⋊C4—282no—
182C42—22no—
183C42(C2×D7)⋊C411yescharacteristic
184C42(C2×D7)⋊C411yescharacteristic
185C16—282no—
186C2×C7—21no—
187C2×C7—41no—
188C2×C7#150511yescharacteristic
189D7—162no—
190C23—282no—
191C23#651211yescharacteristic
192D4—282no—
193D4—282no—
194D4—1122no—
195D4—1122no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—282no—
202C2×C4—1122no—
203C2×C4C2×(C2×D7)⋊C411yescharacteristic
204C2×C4#651211yescharacteristic
205C8—21no—
206C8—21no—
207C8—21no—
208C8—42no—
209C8#637911yescharacteristic
210C8#637911yescharacteristic
211C7#1177011yescharacteristic, complemented, Sylow
212C22—21no—
213C22—41no—
214C22—562no—
215C22—562no—
216C22—562no—
217C22#8977311yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—562no—
223C4C2×#637911yescharacteristic
224C4#8977811yescharacteristic
225C2—21no—
226C2—41no—
227C2—1122no—
228C2#1172718611yescharacteristic
subgroupG/H
centerC2#11727186
commutatorC2×C7×C8C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1747C2
socleC2×C7#1505
prime radical#11736014C2
cosocleC2×C7×M16C23
hypercenter#1747D7
hypocenterC7#11770
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
2. C7×#1747C2semidirect
3. #11727640C2semidirect
4. #11725289C2semidirect
5. #11725287C2semidirect
12. #89483C4semidirect
13. #89467C4semidirect
15. #89455C4semidirect
49. #1747D7semidirect
211. C7#11770semidirect

Maximal subgroups

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

subgroupindex
#117360142
C7×#17472
#117276402
#117252892
#117252872
#117707