Finite Group Databasebeta

← #13268216

Subgroups — #13268216

Order 1792.

240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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#11729135C211yesmaximal, characteristic, complemented
2#11729125C211yesmaximal, characteristic, complemented
3#11727031C211yesmaximal, characteristic, complemented
4#11726330C211yesmaximal, characteristic
5C7×#1364C211yesmaximal, characteristic, complemented
6#11725366C211yesmaximal, characteristic, complemented
7#11725275C211yesmaximal, characteristic, complemented
8C7×#500C2211yescharacteristic
9#89855—21no—
10C2×D7×C16—21no—
11#89615—21no—
12#89467C2211yescharacteristic
13C8×D28C2211yescharacteristic
14C7×C4⋊C16C2211yescharacteristic
15C7×#335C2211yescharacteristic
16#89342—21no—
17(C2×D7)⋊C16—21no—
18#89286—21no—
19#89286—21no—
20#89280—21no—
21C4×C7⋊C16C2211yescharacteristic
22#89253C2211yescharacteristic
23#12467—71nomaximal, Sylow
24C7×C4⋊Q8—21no—
25C2×#1143—41no—
26C2×D56—21no—
27C2×D56—21no—
28C2×D56—41no—
29C2×C8×D7—21no—
30#6437—21no—
31C4×D28D411yescharacteristic
32C2×C7×C16—21no—
33C7×#107—21no—
34C7×#107—21no—
35C7×C4⋊C8—21no—
36C4×C7×C8C2311yescharacteristic
37#6409—41no—
38#6395—41no—
39(C2×D7)⋊C8—21no—
40C2×C7⋊C16D411yescharacteristic
41C2×C7⋊C16D411yescharacteristic
42C4⋊(C7⋊C8)—21no—
43C4⋊(C7⋊C8)D411yescharacteristic
44D7×C16—42no—
45#2240—71no—
46D4×C16—71no—
47#1673—71no—
48#1667—71no—
49#1371—71no—
50#1364D711yescharacteristic, complemented
51#1361—71no—
52C2×C7×Q8—41no—
53C2×D28—41no—
54C2×D28—41no—
55C2×D28—41no—
56C2×D28D811yescharacteristic
57C2×C4×D7—21no—
58C2×C7×C8—41no—
59C2×C7×C8C2×D411yescharacteristic
60C2×C7×C8#4111yescharacteristic
61C7×C4⋊C4—41no—
62C7×C4⋊C4—41no—
63C42×C7C2×D411yescharacteristic
64#1143—82no—
65(C2×D7)⋊C4—21no—
66C4⋊Dic7D811yescharacteristic
67C2×C7⋊C8—21no—
68C2×C7⋊C8—41no—
69D56—41no—
70D56—41no—
71D56—81no—
72D56—82no—
73C8×D7—41no—
74C8×D7—41no—
75C7×C16—41no—
76C7⋊C16—21no—
77C7⋊C16—21no—
78C22×C16—141no—
79#500C2×D711yescharacteristic
80#493—71no—
81#461—141no—
82#459—141no—
83C8×D4—71no—
84C4⋊C16C2×D711yescharacteristic
85#357—141no—
86#357—141no—
87C22⋊C16—141no—
88C4×C16—71no—
89#335—71no—
90#335C2×D711yescharacteristic
91#332—141no—
92#331—141no—
93C22×D7—21no—
94C22×D7—81no—
95C7×Q8—81no—
96C7×Q8—81no—
97C2×C4×C7—81no—
98C2×C4×C7#12111yescharacteristic
99C2×C4×C7C2×D811yescharacteristic
100C2×C4×C7#13511yescharacteristic
101C2×Dic7—21no—
102D28—21no—
103D28—81no—
104D28—81no—
105D28—81no—
106D28—81no—
107D28—161no—
108D28D1612yes—
109C4×D7—41no—
110C4×D7—41no—
111C7×C8—21no—
112C7×C8—81no—
113C7×C8#13511yescharacteristic
114C7×C8AGL(1,ℤ/8)11yescharacteristic
115C7⋊C8—41no—
116C7⋊C8—81no—
117C2×SD16—281no—
118C2×D8—141no—
119C2×D8—141no—
120C2×D8—281no—
121C22×C8—141no—
122C4⋊Q8—21no—
123#127—141no—
124C4×D4—71no—
125C2×C16—21no—
126C2×C16—71no—
127C2×C16—71no—
128C2×C16—282no—
129#107—21no—
130#107—21no—
131C4⋊C8—21no—
132C4⋊C8—71no—
133C4⋊C8—141no—
134#103—281no—
135#102—281no—
136C22⋊C8—141no—
137C4×C8C22×D711yescharacteristic
138C22×C7#46611yescharacteristic
139C2×D7—22no—
140C2×D7—41no—
141C2×D7—41no—
142C2×D7—161no—
143C2×D7—161no—
144C2×D7—161no—
145C4×C7—21no—
146C4×C7—161no—
147C4×C7#46311yescharacteristic
148C4×C7#47111yescharacteristic
149C4×C7C2×D1611yescharacteristic
150C4×C7#50811yescharacteristic
151Dic7—41no—
152C2×Q8—41no—
153C2×D4—71no—
154C2×D4—281no—
155C2×D4—281no—
156C2×D4—281no—
157C22×C4—141no—
158SD16—562no—
159D8—281no—
160D8—281no—
161D8—561no—
162D8—562no—
163C2×C8—41no—
164C2×C8—141no—
165C2×C8—281no—
166C2×C8—281no—
167C2×C8—281no—
168C2×C8#115711yescharacteristic
169C2×C8D4×D711yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—71no—
173C22⋊C4—141no—
174C42D4×D711yescharacteristic
175C16—41no—
176C16—141no—
177C16—141no—
178C2×C7#168111yescharacteristic
179C2×C7#221511yescharacteristic
180C2×C7#222311yescharacteristic
181D7—42no—
182D7—321no—
183C23—141no—
184C23—561no—
185Q8—81no—
186Q8—81no—
187D4—72no—
188D4—141no—
189D4—561no—
190D4—561no—
191D4—561no—
192D4—561no—
193D4—1121no—
194C2×C4—81no—
195C2×C4—141no—
196C2×C4—281no—
197C2×C4—281no—
198C2×C4#644711yescharacteristic
199C2×C4D7×D811yescharacteristic
200C2×C4#647911yescharacteristic
201C8—21no—
202C8—81no—
203C8—281no—
204C8—561no—
205C8#647711yescharacteristic
206C8#648211yescharacteristic
207C7#1246711yescharacteristic, complemented, Sylow
208C22—142no—
209C22—281no—
210C22—281no—
211C22—1121no—
212C22—1121no—
213C22—1121no—
214C22#8954711yescharacteristic
215C4—21no—
216C4—161no—
217C4—281no—
218C4#8952611yescharacteristic
219C4#8959111yescharacteristic
220C4D7×D1611yescharacteristic
221C4#8968811yescharacteristic
222C2—282no—
223C2—2241no—
224C2#1172620011yescharacteristic
225C2#1172694911yescharacteristic
226C2#1172702511yescharacteristic
subgroupG/H
centerC22#89547
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1364C2
socleC22×C7#466
prime radical#11726330C2
hypercenter#1364D7
hypocenterC7#12467
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #11729135C2semidirect
2. #11729125C2semidirect
3. #11727031C2semidirect
5. C7×#1364C2semidirect
6. #11725366C2semidirect
7. #11725275C2semidirect
50. #1364D7semidirect
207. C7#12467semidirect

Maximal subgroups

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

subgroupindex
#117291352
#117291252
#117270312
#117263302
C7×#13642
#117253662
#117252752
#124677