Finite Group Databasebeta

← #13268561

Subgroups — #13268561

Order 1792.

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

normal subgroups only

This group has 306 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
1C7×#2241C211yesmaximal, characteristic, complemented
2#11729135C211yesmaximal, characteristic, complemented
3#11729113C211yesmaximal, characteristic, complemented
4#11729067C211yesmaximal, characteristic, complemented
5#11729051C211yesmaximal, characteristic, complemented
6#11726814C211yesmaximal, characteristic
7#11725305C211yesmaximal, characteristic, complemented
8C7×#500C2211yescharacteristic
9C7×#494C2211yescharacteristic
10C2×#6402—22no—
11#89860—21no—
12#89860—21no—
13#89860—21no—
14#89857—21no—
15C2×D112—22no—
16#89467C2211yescharacteristic
17#89467C2211yescharacteristic
18C7×#358—22no—
19C4×C7×C16C2211yescharacteristic
20#89286—22no—
21C4×C7⋊C16C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12527—71nomaximal, Sylow
24C2×C7×Q16—21no—
25C2×C7×Q16—21no—
26C7×C4⋊Q8—21no—
27C7×C4⋊Q8—21no—
28C2×#1143—42no—
29C2×D56—21no—
30C2×D56—21no—
31C2×D56—21no—
32C2×D56—21no—
33#6437—21no—
34#6437—21no—
35C2×C7×C16D412yes—
36C7×#107—21no—
37C7×#107—21no—
38C4×C7×C8C2311yescharacteristic
39#6402—44no—
40C2×C7⋊C16D412yes—
41#6383—42no—
42#6383—42no—
43#6383—42no—
44C4×C7⋊C8D411yescharacteristic
45C4×C7⋊C8D411yescharacteristic
46D112—44no—
47#2248—71no—
48#2246—71no—
49#2241D711yescharacteristic, complemented
50#2240—71no—
51#1709—71no—
52#1706—71no—
53#1312—71no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×D28—41no—
57C2×D28—41no—
58C2×D28—41no—
59C2×D28—41no—
60C2×D28—41no—
61C2×D28—41no—
62C7×Q16—42no—
63C7×Q16—42no—
64C2×C7×C8C2×D411yescharacteristic
65C2×C7×C8C2×D411yescharacteristic
66C7×C4⋊C4—41no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—41no—
70C42×C7C2×D411yescharacteristic
71#1143—84no—
72C2×C7⋊C8D812yes—
73C2×C7⋊C8D812yes—
74D56—42no—
75D56—42no—
76D56—42no—
77D56—42no—
78C7×C16—22no—
79C7⋊C16—22no—
80C2×SD32—142no—
81C2×D16—142no—
82#500C2×D711yescharacteristic
83#494C2×D711yescharacteristic
84#493—71no—
85#493—71no—
86#492—141no—
87#486—141no—
88#486—141no—
89#486—141no—
90#358—22no—
91#357—142no—
92C4×C16—71no—
93C4×C16C2×D711yescharacteristic
94C82—71no—
95C22×D7—81no—
96C22×D7—81no—
97C7×Q8—81no—
98C7×Q8—81no—
99C7×Q8—82no—
100C2×C4×C7—81no—
101C2×C4×C7—81no—
102C2×C4×C7#12711yescharacteristic
103C2×C4×C7C2×D811yescharacteristic
104C2×C4×C7C2×D811yescharacteristic
105D28—81no—
106D28—81no—
107D28—81no—
108D28—81no—
109D28—82no—
110D28—82no—
111D28—82no—
112D28—82no—
113C7×C8AGL(1,ℤ/8)12yes—
114C7×C8AGL(1,ℤ/8)12yes—
115C7⋊C8—22no—
116C7⋊C8—22no—
117C2×Q16—21no—
118C2×Q16—21no—
119C2×SD16—282no—
120C2×D8—141no—
121C2×D8—141no—
122C2×D8—141no—
123C2×D8—141no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#127—141no—
127#127—141no—
128SD32—284no—
129D16—284no—
130C2×C16—72no—
131C2×C16C22⋊D712yes—
132#107—21no—
133#107—21no—
134#102—282no—
135#102—282no—
136#102—282no—
137C4×C8—71no—
138C4×C8—71no—
139C4×C8C22×D711yescharacteristic
140C22×C7#49311yescharacteristic
141C2×D7—161no—
142C2×D7—161no—
143C2×D7—161no—
144C2×D7—161no—
145C2×D7—161no—
146C2×D7—161no—
147C4×C7—161no—
148C4×C7—161no—
149C4×C7#49611yescharacteristic
150C4×C7#49611yescharacteristic
151C4×C7#50812yes—
152C4×C7#50912yes—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—281no—
156C2×D4—281no—
157C2×D4—281no—
158C2×D4—281no—
159C2×D4—281no—
160C2×D4—281no—
161Q16—42no—
162Q16—42no—
163SD16—564no—
164D8—282no—
165D8—282no—
166D8—282no—
167D8—282no—
168C2×C8—72no—
169C2×C8—72no—
170C2×C8D4×D711yescharacteristic
171C2×C8C2×C22⋊D711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C42D4×D711yescharacteristic
177C16—22no—
178C16—142no—
179C2×C7#172111yescharacteristic
180C2×C7#224911yescharacteristic
181C2×C7#225011yescharacteristic
182D7—321no—
183D7—321no—
184C23—561no—
185C23—561no—
186Q8—82no—
187Q8—81no—
188Q8—81no—
189D4—562no—
190D4—562no—
191D4—562no—
192D4—562no—
193D4—561no—
194D4—561no—
195D4—561no—
196D4—561no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4D7×D811yescharacteristic
200C2×C4D7×D811yescharacteristic
201C2×C4#650311yescharacteristic
202C8—142no—
203C8—142no—
204C8#647712yes—
205C8#651212yes—
206C7#1252711yescharacteristic, complemented, Sylow
207C22—1121no—
208C22—1121no—
209C22—1121no—
210C22—1121no—
211C22—1121no—
212C22—1121no—
213C22#8992311yescharacteristic
214C4—161no—
215C4—161no—
216C4#8968612yes—
217C4#8969112yes—
218C4#8983011yescharacteristic
219C4#8994811yescharacteristic
220C2—2241no—
221C2—2241no—
222C2#1172894811yescharacteristic
223C2#1172923511yescharacteristic
224C2#1172925411yescharacteristic
subgroupG/H
centerC22#89923
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#2241C2
socleC22×C7#493
prime radical#11726814C2
hypercenter#2241D7
hypocenterC7#12527
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
1. C7×#2241C2semidirect
2. #11729135C2semidirect
3. #11729113C2semidirect
4. #11729067C2semidirect
5. #11729051C2semidirect
7. #11725305C2semidirect
49. #2241D7semidirect
206. C7#12527semidirect

Maximal subgroups

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

subgroupindex
C7×#22412
#117291352
#117291132
#117290672
#117290512
#117268142
#117253052
#125277