Finite Group Databasebeta

← #13267922

Subgroups — #13267922

Order 1792.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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#11729143C211yesmaximal, characteristic, complemented
2#11729113C211yesmaximal, characteristic
3#11727046C211yesmaximal, characteristic
4#11726311C211yesmaximal, characteristic, complemented
5C7×#1364C211yesmaximal, characteristic, complemented
6#11725360C211yesmaximal, characteristic, complemented
7#11725279C211yesmaximal, characteristic
8C7×#500C2211yescharacteristic
9#89860—21no—
10#89857—21no—
11C2×#6374—21no—
12#89602—21no—
13C4×Dic28C2211yescharacteristic
14#89467C2211yescharacteristic
15C7×C4⋊C16C2211yescharacteristic
16C7×#335C2211yescharacteristic
17#89304—21no—
18#89287—21no—
19#89285—21no—
20#89279—21no—
21C4⋊(C7⋊C16)C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12417—71nomaximal, Sylow
24C7×C4⋊Q8—21no—
25C2×#1143—41no—
26C2×#1143—41no—
27C2×Dic28D411yescharacteristic
28C2×D56—21no—
29C2×D56—21no—
30#6437—21no—
31C4×Dic14—21no—
32C2×C7×C16—21no—
33C7×#107—21no—
34C7×#107—21no—
35C7×C4⋊C8—21no—
36C4×C7×C8C2311yescharacteristic
37#6392D411yescharacteristic
38#6390—21no—
39Dic7⋊C8—41no—
40C8×Dic7—41no—
41C2×C7⋊C16—21no—
42#6383—42no—
43C4×C7⋊C8D411yescharacteristic
44C4×C7⋊C8D411yescharacteristic
45#6374—42no—
46#2225—71no—
47#2207—71no—
48#1706—71no—
49C8×Q16—71no—
50#1367—71no—
51#1364D711yescharacteristic, complemented
52#1361—71no—
53C2×C7×Q8—41no—
54C2×D28—41no—
55C2×D28—41no—
56C2×D28—41no—
57C2×Dic14—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#1143—82no—
66C4⋊Dic7—21no—
67Dic7⋊C4—41no—
68C4×Dic7—41no—
69C2×C7⋊C8—21no—
70C2×C7⋊C8D811yescharacteristic
71C2×C7⋊C8D811yescharacteristic
72Dic28—21no—
73Dic28SD1612yescomplemented
74D56—41no—
75D56—41no—
76D56—81no—
77C7×C16—41no—
78C7⋊C16—41no—
79C2×SD32—141no—
80#500C2×D711yescharacteristic
81#493—71no—
82#492—141no—
83#486—141no—
84C8×Q8—141no—
85C4×Q16—71no—
86#367—141no—
87C4⋊C16—71no—
88C4⋊C16C2×D711yescharacteristic
89#358—141no—
90#357—141no—
91#335C2×D711yescharacteristic
92Q8⋊C8—141no—
93C82—71no—
94C22×D7—81no—
95C7×Q8—81no—
96C7×Q8—81no—
97C2×C4×C7—81no—
98C2×C4×C7#12111yescharacteristic
99C2×C4×C7C2×D811yescharacteristic
100C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
101C2×Dic7—41no—
102C2×Dic7—41no—
103D28—82no—
104D28—81no—
105D28—81no—
106D28—81no—
107D28—81no—
108Dic14—22no—
109Dic14—41no—
110C7×C8—21no—
111C7×C8—81no—
112C7×C8C2×SD1611yescharacteristic
113C7×C8#13511yescharacteristic
114C7⋊C8—21no—
115C7⋊C8—41no—
116C7⋊C8SD3212yescomplemented
117C2×Q16—71no—
118C2×SD16—281no—
119C2×SD16—281no—
120C2×D8—141no—
121C2×D8—141no—
122C4⋊Q8—21no—
123#127—141no—
124C4×Q8—141no—
125SD32—282no—
126C2×C16—21no—
127C2×C16—141no—
128#107—21no—
129#107—21no—
130#107—71no—
131C4⋊C8—21no—
132C4⋊C8—281no—
133#103—141no—
134#102—282no—
135C4×C8—71no—
136C4×C8—71no—
137C4×C8—281no—
138C4×C8C22×D711yescharacteristic
139C22×C7#45911yescharacteristic
140C2×D7—161no—
141C2×D7—161no—
142C2×D7—161no—
143C4×C7—21no—
144C4×C7—161no—
145C4×C7#46511yescharacteristic
146C4×C7#47111yescharacteristic
147C4×C7C2×SD3211yescharacteristic
148C4×C7#50911yescharacteristic
149Dic7—42no—
150Dic7—81no—
151C2×Q8—41no—
152C2×Q8—141no—
153C2×D4—281no—
154C2×D4—281no—
155C2×D4—281no—
156Q16—72no—
157Q16—141no—
158SD16—562no—
159SD16—562no—
160D8—281no—
161D8—281no—
162D8—561no—
163C2×C8—41no—
164C2×C8—71no—
165C2×C8—71no—
166C2×C8—141no—
167C2×C8#115711yescharacteristic
168C2×C8D4×D711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—141no—
172C4⋊C4—281no—
173C42—281no—
174C42D4×D711yescharacteristic
175C16—41no—
176C16—281no—
177C2×C7#166511yescharacteristic
178C2×C7#220711yescharacteristic
179C2×C7#221211yescharacteristic
180D7—321no—
181C23—561no—
182Q8—81no—
183Q8—81no—
184Q8—142no—
185Q8—281no—
186D4—562no—
187D4—561no—
188D4—561no—
189D4—561no—
190D4—561no—
191C2×C4—81no—
192C2×C4—281no—
193C2×C4—281no—
194C2×C4#644711yescharacteristic
195C2×C4D7×D811yescharacteristic
196C2×C4#647711yescharacteristic
197C8—21no—
198C8—72no—
199C8—81no—
200C8—141no—
201C8—281no—
202C8D7×SD1611yescharacteristic
203C8#648211yescharacteristic
204C7#1241711yescharacteristic, complemented, Sylow
205C22—1121no—
206C22—1121no—
207C22—1121no—
208C22#8955811yescharacteristic
209C4—21no—
210C4—161no—
211C4—282no—
212C4—561no—
213C4#8952611yescharacteristic
214C4#8957911yescharacteristic
215C4#8968311yescharacteristic
216C4D7×SD3211yescharacteristic
217C2—2241no—
218C2#1172619411yescharacteristic
219C2#1172695811yescharacteristic
220C2#1172702611yescharacteristic
subgroupG/H
centerC22#89558
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1364C2
socleC22×C7#459
prime radical#89467C22
hypercenter#1364D7
hypocenterC7#12417
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. #11729143C2semidirect
4. #11726311C2semidirect
5. C7×#1364C2semidirect
6. #11725360C2semidirect
51. #1364D7semidirect
73. Dic28SD16semidirect
116. C7⋊C8SD32semidirect
204. C7#12417semidirect

Maximal subgroups

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

subgroupindex
#117291432
#117291132
#117270462
#117263112
C7×#13642
#117253602
#117252792
#124177