Finite Group Databasebeta

← #13266874

Subgroups — #13266874

Order 1792.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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#11731663C211yesmaximal, characteristic
2C7×#2061C211yesmaximal, characteristic
3#11729658C212yesmaximal
4#11725438C211yesmaximal, characteristic
5#11725300C212yesmaximal
6C7×#544C2211yescharacteristic
7C7×#428—21no—
8C7×#382C2211yescharacteristic
9#89822—22no—
10#89758—21no—
11#89751C2212yes—
12#89708—21no—
13#89432—22no—
14#89428—22no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17#89333—22no—
18#89323C2212yes—
19#12260—71nomaximal, Sylow
20C22×Dic14D412yes—
21C2×C7×M16—21no—
22C7×C4⋊Q8—41no—
23C7×#125—41no—
24C7×C22⋊Q8—22no—
25C7×#117—21no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×#1144—42no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×Dic7⋊C4—22no—
34C7×#107—41no—
35C7×#106—41no—
36C7×#95—22no—
37C7⋊#95—22no—
38C7⋊#95—22no—
39C7⋊#95—21no—
40C7⋊#95—22no—
41C7⋊#95D412yes—
42C22⋊(C7⋊C8)—22no—
43#6384—42no—
44#2844—71no—
45#2061D711yescharacteristic
46#1605—72no—
47#1354—72no—
48#1324—71no—
49C2×C7×Q8—22no—
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7C2×D411yescharacteristic
53C22×C4×C7C2×D412yes—
54C22×Dic7—22no—
55C22×Dic7—22no—
56C2×Dic14—42no—
57C2×Dic14—82no—
58C7×M16—22no—
59C7×M16—41no—
60C2×C7×C8—41no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—42no—
67C7×C22⋊C4—41no—
68C42×C7—21no—
69C42×C7—41no—
70C42×C7Q1612yescomplemented
71#1144—84no—
72C4⋊Dic7—81no—
73C4⋊Dic7—81no—
74C4⋊Dic7—81no—
75Dic7⋊C4—82no—
76C2×C7⋊C8—42no—
77#544C2×D711yescharacteristic
78#451—142no—
79#428—21no—
80#400—142no—
81#400—141no—
82#395—72no—
83#395—142no—
84#382—141no—
85#382C2×D711yescharacteristic
86#343C2×D711yescharacteristic
87#339—21no—
88#328—142no—
89#324—72no—
90C23×C7#12011yescharacteristic
91C7×Q8—82no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—41no—
97C2×C4×C7—41no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7C2×Q1611yescharacteristic
104C2×C4×C7#13711yescharacteristic
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111Dic14—44no—
112Dic14—82no—
113C7×C8—42no—
114C7⋊C8—82no—
115C22×Q8—72no—
116C2×Q16—282no—
117C2×M16—21no—
118C4⋊Q8—41no—
119#125—41no—
120C22⋊Q8—22no—
121#117—21no—
122C2×C4⋊C4—141no—
123C2×C4⋊C4—141no—
124C2×C4⋊C4—141no—
125C2×C4⋊C4—142no—
126C2×C4⋊C4C22⋊D711yescharacteristic
127C2×C4⋊C4C22×D711yescharacteristic
128C2×C42C22⋊D711yescharacteristic
129#107—41no—
130#106—41no—
131#103—282no—
132C22⋊C8—142no—
133#95—22no—
134#95—72no—
135#95—142no—
136#95—142no—
137#95—141no—
138#95—142no—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7#45111yescharacteristic
142C22×C7#45611yescharacteristic
143C22×C7B(4,𝔽2)11yescharacteristic
144C4×C7—21no—
145C4×C7—21no—
146C4×C7—42no—
147C4×C7—42no—
148C4×C7—81no—
149C4×C7—81no—
150C4×C7—81no—
151Dic7—82no—
152Dic7—82no—
153C2×Q8—22no—
154C2×Q8—282no—
155C2×Q8—562no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—142no—
159C22×C4—142no—
160C22×C4D4×D712yes—
161C22×C4C2×C22⋊D711yescharacteristic
162Q16—564no—
163M16—22no—
164M16—41no—
165C2×C8—41no—
166C2×C8—282no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—562no—
174C4⋊C4—561no—
175C4⋊C4—561no—
176C4⋊C4—561no—
177C22⋊C4—41no—
178C42—21no—
179C42—41no—
180C42#114412yescomplemented
181C2×C7—21no—
182C2×C7—21no—
183C2×C7#160511yescharacteristic
184C2×C7#219911yescharacteristic
185C2×C7#220311yescharacteristic
186C23#650011yescharacteristic
187Q8—82no—
188Q8—284no—
189Q8—562no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4—282no—
207C2×C4C2×#114411yescharacteristic
208C2×C4#651411yescharacteristic
209C8—42no—
210C8—562no—
211C7#1226011yescharacteristic, complemented, Sylow
212C22—21no—
213C22—21no—
214C22#8980411yescharacteristic
215C22#8982211yescharacteristic
216C22#8986711yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—562no—
225C4—562no—
226C2—21no—
227C2—21no—
228C2#1172746111yescharacteristic
229C2#1172874311yescharacteristic
230C2#1172874511yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2061C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2061D7
hypocenterC7#12260
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
70. C42×C7Q16semidirect
180. C42#1144semidirect
211. C7#12260semidirect

Maximal subgroups

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

subgroupindex
#117316632
C7×#20612
#117296582
#117254382
#117253002
#122607