Finite Group Databasebeta

← #13266638

Subgroups — #13266638

Order 1792.

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

normal subgroups only

This group has 380 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#11734049C211yesmaximal, characteristic
2C7×#2028C211yesmaximal, characteristic
3#11729658C212yesmaximal
4#11725449C211yesmaximal, characteristic
5#11725300C212yesmaximal
6C7×#544C2211yescharacteristic
7C7×#419—21no—
8C7×#391C2211yescharacteristic
9#90002—21no—
10#89822—22no—
11#89751C2212yes—
12#89701—21no—
13#89428—22no—
14#89418—22no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17#89333—22no—
18#89323C2212yes—
19#12224—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×Dic14—21no—
22C22×Dic14D412yes—
23C2×C7×M16—21no—
24C7×C4⋊Q8—41no—
25C7×#125—41no—
26C7×C22⋊Q8—22no—
27C7×#117—21no—
28C2×C7×C4⋊C4C2311yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×#1144—42no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×Dic7⋊C4—22no—
34C2×C4×Dic7—21no—
35C7×#103—41no—
36C7×#103—41no—
37C7×#95—22no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40C7⋊#95—22no—
41C7⋊#95D412yes—
42C22⋊(C7⋊C8)—22no—
43#6384—42no—
44#2811—71no—
45#2028D711yescharacteristic
46#1605—72no—
47#1354—72no—
48#1319—71no—
49C2×C7×Q8—21no—
50C2×C7×Q8—22no—
51C2×C7×Q8—41no—
52C2×C7×Q8—41no—
53C2×C7×Q8Q1612yescomplemented
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D412yes—
58C22×Dic7—22no—
59C22×Dic7—22no—
60C2×Dic14—24no—
61C2×Dic14—42no—
62C2×Dic14—82no—
63C2×Dic14—81no—
64C2×Dic14—81no—
65C7×M16—41no—
66C7×M16—41no—
67C2×C7×C8—41no—
68C7×C4⋊C4—21no—
69C7×C4⋊C4—21no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—42no—
73C7×C22⋊C4—41no—
74C42×C7—41no—
75C42×C7—41no—
76#1144—84no—
77C4⋊Dic7—81no—
78C4⋊Dic7—81no—
79Dic7⋊C4—82no—
80C4×Dic7—81no—
81C2×C7⋊C8—42no—
82#544C2×D711yescharacteristic
83#451—142no—
84#419—21no—
85#395—72no—
86#395—142no—
87#391—142no—
88#391—141no—
89#391—141no—
90#391C2×D711yescharacteristic
91#343C2×D711yescharacteristic
92#339—21no—
93#328—142no—
94#324—72no—
95C23×C7#12011yescharacteristic
96C7×Q8—42no—
97C7×Q8—81no—
98C7×Q8—82no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—41no—
105C2×C4×C7—41no—
106C2×C4×C7—41no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7—41no—
110C2×C4×C7—41no—
111C2×C4×C7C2×Q1611yescharacteristic
112C2×C4×C7#13711yescharacteristic
113C2×Dic7—24no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—42no—
117C2×Dic7—42no—
118C2×Dic7—42no—
119Dic14—44no—
120Dic14—84no—
121Dic14—82no—
122C7×C8—81no—
123C7⋊C8—82no—
124C22×Q8—72no—
125C22×Q8—141no—
126C22×Q8C22⋊D711yescharacteristic
127C2×Q16—282no—
128C2×M16—21no—
129C4⋊Q8—41no—
130#125—41no—
131C22⋊Q8—22no—
132#117—21no—
133C2×C4⋊C4—141no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4—142no—
136C2×C4⋊C4C22×D711yescharacteristic
137C2×C42—141no—
138C2×C42C22⋊D711yescharacteristic
139#103—41no—
140#103—41no—
141#103—282no—
142C22⋊C8—142no—
143#95—22no—
144#95—72no—
145#95—142no—
146#95—142no—
147#95—142no—
148C22×C7—21no—
149C22×C7—21no—
150C22×C7#45111yescharacteristic
151C22×C7#45511yescharacteristic
152C22×C7B(4,𝔽2)11yescharacteristic
153C4×C7—21no—
154C4×C7—21no—
155C4×C7—42no—
156C4×C7—42no—
157C4×C7—42no—
158C4×C7—81no—
159C4×C7—81no—
160Dic7—44no—
161Dic7—82no—
162C2×Q8—21no—
163C2×Q8—22no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—144no—
167C2×Q8—282no—
168C2×Q8—562no—
169C2×Q8—561no—
170C2×Q8—561no—
171C2×Q8#114412yescomplemented
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—142no—
175C22×C4—142no—
176C22×C4D4×D712yes—
177C22×C4C2×C22⋊D711yescharacteristic
178Q16—564no—
179M16—41no—
180M16—41no—
181C2×C8—41no—
182C2×C8—282no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—562no—
189C4⋊C4—561no—
190C4⋊C4—561no—
191C22⋊C4—41no—
192C42—41no—
193C42—41no—
194C42—561no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7#160211yescharacteristic
198C2×C7#219911yescharacteristic
199C2×C7#220511yescharacteristic
200C23#650011yescharacteristic
201Q8—42no—
202Q8—81no—
203Q8—82no—
204Q8—284no—
205Q8—564no—
206Q8—562no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—144no—
220C2×C4—282no—
221C2×C4—282no—
222C2×C4—282no—
223C2×C4—282no—
224C2×C4—282no—
225C2×C4C2×#114411yescharacteristic
226C2×C4#651411yescharacteristic
227C8—81no—
228C8—562no—
229C7#1222411yescharacteristic, complemented, Sylow
230C22—21no—
231C22—21no—
232C22#8980411yescharacteristic
233C22#8982211yescharacteristic
234C22#8983311yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—284no—
243C4—562no—
244C2—21no—
245C2—21no—
246C2#1172745511yescharacteristic
247C2#1172874511yescharacteristic
248C2#1172874911yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2028C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2028D7
hypocenterC7#12224
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
53. C2×C7×Q8Q16semidirect
171. C2×Q8#1144semidirect
229. C7#12224semidirect

Maximal subgroups

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

subgroupindex
#117340492
C7×#20282
#117296582
#117254492
#117253002
#122247