Finite Group Databasebeta

← #13258247

Subgroups — #13258247

Order 1792.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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×#2571C211yesmaximal, characteristic
2#11727331C211yesmaximal, characteristic
3#11727221C212yesmaximal
4#11725440C211yesmaximal, characteristic
5#11725425C212yesmaximal
6C7×#400—22no—
7C7×#389—22no—
8C7×#384C2211yescharacteristic
9C7×#382—22no—
10C4×C7×C4⋊C4C2212yes—
11C2×#6382—22no—
12C2×#6381—22no—
13#89707C2211yescharacteristic
14C2×C4⋊(C7⋊C8)C2212yes—
15C2×C4×C7⋊C8C2211yescharacteristic
16#89324—22no—
17#10576—71nomaximal, Sylow
18C2×C7×C4⋊C4—22no—
19C2×C7×C4⋊C4—22no—
20C2×C7×C4⋊C4D412yes—
21C2×C7×C4⋊C4D412yes—
22C2×C7×C4⋊C4D412yes—
23C2×C42×C7—22no—
24C2×C42×C7C2311yescharacteristic
25C2×C4⋊Dic7—21no—
26C2×C4⋊Dic7—22no—
27C22×C7⋊C8—21no—
28C22×C7⋊C8—22no—
29C7×#95—22no—
30C7×#95—24no—
31C7×#95—22no—
32#6382—44no—
33#6381—44no—
34C4⋊(C7⋊C8)—22no—
35C4⋊(C7⋊C8)Q814yes—
36C4×C7⋊C8—21no—
37C4×C7⋊C8Q812yes—
38#2571D711yescharacteristic
39#1948—72no—
40#1845—71no—
41#1296—71no—
42#1279—72no—
43C22×C4×C7—22no—
44C22×C4×C7—22no—
45C22×C4×C7—22no—
46C22×C4×C7C2×D411yescharacteristic
47C22×C4×C7C2×D412yes—
48C22×Dic7—41no—
49C7×C4⋊C4—22no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—22no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—42no—
54C7×C4⋊C4—42no—
55C7×C4⋊C4—42no—
56C7×C4⋊C4—82no—
57C42×C7—42no—
58C42×C7—42no—
59C42×C7C2×Q811yescharacteristic
60C42×C7C2×Q812yes—
61C42×C7#4111yescharacteristic
62C4⋊Dic7—42no—
63C4⋊Dic7—42no—
64C4⋊Dic7—42no—
65C2×C7⋊C8—22no—
66C2×C7⋊C8—22no—
67C2×C7⋊C8—22no—
68C2×C7⋊C8—24no—
69C2×C7⋊C8—42no—
70C2×C7⋊C8—42no—
71C2×#107—142no—
72C2×#106—142no—
73C2×C4⋊C8—72no—
74C2×C4×C8—71no—
75#400—22no—
76#389—22no—
77#384—71no—
78#384C2×D711yescharacteristic
79#382—22no—
80C4×C4⋊C4C2×D712yes—
81#340—142no—
82C23×C7#12011yescharacteristic
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—42no—
86C2×C4×C7—44no—
87C2×C4×C7—42no—
88C2×C4×C7—42no—
89C2×C4×C7—42no—
90C2×C4×C7—42no—
91C2×C4×C7—42no—
92C2×C4×C7—42no—
93C2×C4×C7C22⋊Q811yescharacteristic
94C2×C4×C7C22⋊Q812yes—
95C2×C4×C7C4⋊Q811yescharacteristic
96C2×C4×C7C4⋊Q812yes—
97C2×C4×C7#13512yes—
98C2×C4×C7#13512yes—
99C2×C4×C7AGL(1,ℤ/8)12yes—
100C2×C4×C7#13712yes—
101C2×Dic7—82no—
102C2×Dic7—81no—
103C7⋊C8—24no—
104C7⋊C8—44no—
105C22×C8—141no—
106C22×C8—142no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—142no—
110C2×C4⋊C4—141no—
111C2×C4⋊C4C22⋊D712yes—
112C2×C4⋊C4C22⋊D712yes—
113C2×C4⋊C4C22⋊D712yes—
114C2×C42—22no—
115C2×C42C22×D711yescharacteristic
116#107—284no—
117#106—284no—
118C4⋊C8—74no—
119C4⋊C8—142no—
120C4×C8—72no—
121C4×C8—141no—
122#95—22no—
123#95—24no—
124#95—22no—
125C22×C7#39511yescharacteristic
126C22×C7#44912yes—
127C22×C7#45212yes—
128C22×C7#45311yescharacteristic
129C22×C7#45611yescharacteristic
130C4×C7—21no—
131C4×C7—21no—
132C4×C7—82no—
133C4×C7—82no—
134C4×C7—82no—
135C4×C7#47812yes—
136C4×C7#47912yes—
137C4×C7#49912yes—
138C4×C7#50112yes—
139Dic7—161no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—281no—
144C22×C4C2×C22⋊D711yescharacteristic
145C22×C4C2×C22⋊D712yes—
146C2×C8—142no—
147C2×C8—142no—
148C2×C8—142no—
149C2×C8—144no—
150C2×C8—282no—
151C2×C8—282no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—282no—
161C4⋊C4—282no—
162C4⋊C4—282no—
163C42—42no—
164C42—42no—
165C42Q8×D711yescharacteristic
166C42Q8×D712yes—
167C42#116111yescharacteristic
168C2×C7#162011yescharacteristic
169C2×C7#163411yescharacteristic
170C2×C7#165012yes—
171C2×C7#205912yes—
172C2×C7#206011yescharacteristic
173C23#651611yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—562no—
185C2×C4—561no—
186C2×C4#650711yescharacteristic
187C2×C4#650712yes—
188C2×C4#650911yescharacteristic
189C2×C4#650912yes—
190C2×C4#651212yes—
191C2×C4#651312yes—
192C2×C4#651312yes—
193C2×C4#651412yes—
194C8—144no—
195C8—284no—
196C7#1057611yescharacteristic, complemented, Sylow
197C22#9000111yescharacteristic
198C22#9001012yes—
199C22#9001112yes—
200C22#9001311yescharacteristic
201C22#9001811yescharacteristic
202C4—21no—
203C4—21no—
204C4—82no—
205C4—82no—
206C4—82no—
207C4—1121no—
208C4#8983512yes—
209C4#8983612yes—
210C4#8983712yes—
211C4#8983812yes—
212C2#1172949312yes—
213C2#1172965411yescharacteristic
214C2#1172968211yescharacteristic
215C2#1172968712yes—
216C2#1172969711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2571C2
socle, prime radicalC23×C7#120
hypercenter#2571D7
hypocenterC7#10576
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
196. C7#10576semidirect

Maximal subgroups

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

subgroupindex
C7×#25712
#117273312
#117272212
#117254402
#117254252
#105767