Finite Group Databasebeta

← #13268574

Subgroups — #13268574

Order 1792.

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

normal subgroups only

This group has 284 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×#2247C211yesmaximal, characteristic, complemented
2#11729067C211yesmaximal, characteristic
3#11729064C211yesmaximal, characteristic, complemented
4#11729051C211yesmaximal, characteristic
5#11729050C211yesmaximal, characteristic, complemented
6#11726812C211yesmaximal, characteristic
7#11725305C211yesmaximal, characteristic, complemented
8C2×C7×Q32—22no—
9C7×#494C2211yescharacteristic
10C7×#494C2211yescharacteristic
11C2×#6402—22no—
12#89865—21no—
13#89860—21no—
14#89860—21no—
15#89851—21no—
16#89467C2211yescharacteristic
17#89454C2211yescharacteristic
18C4×C7×C16C2211yescharacteristic
19#89359—22no—
20#89304—22no—
21C4×C7⋊C16C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12530—71nomaximal, Sylow
24C2×C7×Q16—21no—
25C2×C7×Q16—21no—
26C2×C7×Q16—21no—
27C2×C7×Q16—21no—
28C7×C4⋊Q8—21no—
29C7×C4⋊Q8—21no—
30C2×D56—21no—
31C2×D56—21no—
32#6437—21no—
33#6432—21no—
34C7×Q32—44no—
35C2×C7×C16D412yes—
36C4×C7×C8C2311yescharacteristic
37#6402—44no—
38#6392—21no—
39#6392—21no—
40C2×C7⋊C16D412yes—
41#6384—42no—
42#6383—42no—
43#6383—42no—
44#6382—42no—
45C4×C7⋊C8D411yescharacteristic
46C4×C7⋊C8D411yescharacteristic
47#2248—71no—
48#2247D711yescharacteristic, complemented
49#2241—71no—
50#2240—71no—
51#1709—71no—
52#1707—71no—
53#1312—71no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×D28—41no—
57C2×D28—41no—
58C2×D28—41no—
59C2×Dic14—41no—
60C7×Q16—42no—
61C7×Q16—42no—
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
71C4⋊Dic7—41no—
72C4⋊Dic7—41no—
73C2×C7⋊C8D812yes—
74C2×C7⋊C8D812yes—
75D56—42no—
76D56—42no—
77C7×C16—22no—
78C7⋊C16—22no—
79C2×Q32—22no—
80C2×SD32—142no—
81#500—71no—
82#498—141no—
83#494C2×D711yescharacteristic
84#494C2×D711yescharacteristic
85#493—71no—
86#487—141no—
87#486—141no—
88#486—141no—
89#358—142no—
90#357—142no—
91C4×C16—71no—
92C4×C16C2×D711yescharacteristic
93C82—71no—
94C22×D7—81no—
95C7×Q8—81no—
96C7×Q8—81no—
97C7×Q8—81no—
98C7×Q8—81no—
99C2×C4×C7—81no—
100C2×C4×C7—81no—
101C2×C4×C7#12711yescharacteristic
102C2×C4×C7C2×D811yescharacteristic
103C2×C4×C7C2×D811yescharacteristic
104C2×Dic7—81no—
105D28—81no—
106D28—81no—
107D28—82no—
108D28—82no—
109Dic14—82no—
110C7×C8AGL(1,ℤ/8)12yes—
111C7×C8AGL(1,ℤ/8)12yes—
112C7⋊C8—22no—
113C7⋊C8—22no—
114C2×Q16—21no—
115C2×Q16—21no—
116C2×Q16—21no—
117C2×Q16—21no—
118C2×D8—141no—
119C2×D8—141no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—141no—
123#127—141no—
124Q32—44no—
125SD32—284no—
126C2×C16—72no—
127C2×C16C22⋊D712yes—
128#107—141no—
129#107—141no—
130#106—282no—
131#103—282no—
132#102—282no—
133#102—282no—
134C4×C8—71no—
135C4×C8—71no—
136C4×C8C22×D711yescharacteristic
137C22×C7#49311yescharacteristic
138C2×D7—161no—
139C2×D7—161no—
140C2×D7—161no—
141C4×C7—161no—
142C4×C7—161no—
143C4×C7#49611yescharacteristic
144C4×C7#49611yescharacteristic
145C4×C7#50812yes—
146C4×C7#51012yes—
147Dic7—161no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—281no—
151C2×D4—281no—
152C2×D4—281no—
153C2×D4—281no—
154Q16—42no—
155Q16—42no—
156Q16—42no—
157Q16—42no—
158D8—282no—
159D8—282no—
160C2×C8—72no—
161C2×C8—72no—
162C2×C8D4×D711yescharacteristic
163C2×C8C2×C22⋊D711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—281no—
169C4⋊C4—281no—
170C42D4×D711yescharacteristic
171C16—22no—
172C16—142no—
173C2×C7#172111yescharacteristic
174C2×C7#224911yescharacteristic
175C2×C7#225111yescharacteristic
176D7—321no—
177C23—561no—
178Q8—81no—
179Q8—81no—
180Q8—81no—
181Q8—81no—
182Q8—562no—
183D4—561no—
184D4—562no—
185D4—562no—
186D4—561no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—561no—
190C2×C4D7×D811yescharacteristic
191C2×C4D7×D811yescharacteristic
192C2×C4#650311yescharacteristic
193C8—142no—
194C8—142no—
195C8#647412yes—
196C8#649512yes—
197C7#1253011yescharacteristic, complemented, Sylow
198C22—1121no—
199C22—1121no—
200C22—1121no—
201C22#8992311yescharacteristic
202C4—161no—
203C4—161no—
204C4—1121no—
205C4#8969012yes—
206C4#8969112yes—
207C4#8984611yescharacteristic
208C4#8992611yescharacteristic
209C2—2241no—
210C2#1172903311yescharacteristic
211C2#1172925411yescharacteristic
212C2#1172925511yescharacteristic
subgroupG/H
centerC22#89923
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#2247C2
socleC22×C7#493
prime radical#89467C22
hypercenter#2247D7
hypocenterC7#12530
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2247

Complements

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

NH ≅ G/Ntype
1. C7×#2247C2semidirect
3. #11729064C2semidirect
5. #11729050C2semidirect
7. #11725305C2semidirect
48. #2247D7semidirect
197. C7#12530semidirect

Maximal subgroups

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

subgroupindex
C7×#22472
#117290672
#117290642
#117290512
#117290502
#117268122
#117253052
#125307