Finite Group Databasebeta

← #13266639

Subgroups — #13266639

Order 1792.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2811C211yesmaximal, characteristic
2#11727461C212yesmaximal
3#11727427C211yesmaximal, characteristic
4#11725454C212yesmaximal
5#11725449C211yesmaximal, characteristic
6#90191C2211yescharacteristic
7C7×#395—22no—
8C7×#395C2212yes—
9C7×#391—21no—
10C7×#391—21no—
11C7×#391—22no—
12C7×#391C2211yescharacteristic
13#90000—22no—
14#89997—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12224—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D411yescharacteristic
22C22×C7×Q8D412yes—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7—21no—
28C2×C42×C7D411yescharacteristic
29C2×#1144—42no—
30#6489—21no—
31#6487—22no—
32C2×#1136—21no—
33#6453—41no—
34#6451—41no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38C7×#95—22no—
39C7×#95D412yes—
40#6409—41no—
41#6409—41no—
42#6409—42no—
43C22⋊(C7⋊C8)—22no—
44#2811D711yescharacteristic
45#2028—71no—
46#1605—72no—
47#1354—72no—
48#1319—71no—
49C2×C7×Q8—24no—
50C2×C7×Q8—21no—
51C2×C7×Q8—41no—
52C2×C7×Q8—42no—
53C2×C7×Q8—41no—
54C2×C7×Q8—82no—
55C2×C7×Q8—81no—
56C2×C7×Q8—81no—
57C2×C7×Q8Q1612yescomplemented
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D412yes—
64C2×Dic14—22no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—82no—
67C7×C4⋊C4—81no—
68C7×C4⋊C4—81no—
69C42×C7—41no—
70C42×C7—81no—
71C22⋊Dic7—41no—
72#1144—84no—
73C4⋊Dic7—21no—
74C4⋊Dic7—22no—
75C4⋊Dic7—21no—
76Dic7⋊C4—42no—
77C4×Dic7—41no—
78#1136—41no—
79#1136—41no—
80C2×C7⋊C8—42no—
81C2×C7⋊C8—41no—
82#544—71no—
83#451—142no—
84#419—141no—
85#395—22no—
86#395C2×D712yes—
87#391—21no—
88#391—21no—
89#391—22no—
90#391C2×D711yescharacteristic
91#343—71no—
92#339—141no—
93#328—142no—
94#324—72no—
95C23×C7#12011yescharacteristic
96C7×Q8—42no—
97C7×Q8—44no—
98C7×Q8—81no—
99C7×Q8—84no—
100C7×Q8—82no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—24no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7—41no—
110C2×C4×C7—41no—
111C2×C4×C7—42no—
112C2×C4×C7—41no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7C2×Q1611yescharacteristic
118C2×C4×C7#13711yescharacteristic
119C2×Dic7—41no—
120C2×Dic7—41no—
121Dic14—82no—
122C7⋊C8—81no—
123C7⋊C8—82no—
124C22×Q8—21no—
125C22×Q8C22⋊D711yescharacteristic
126C22×Q8C22⋊D712yes—
127C2×Q16—282no—
128C2×M16—141no—
129C4⋊Q8—281no—
130#125—281no—
131C22⋊Q8—142no—
132#117—141no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C22×D711yescharacteristic
137C2×C42—21no—
138C2×C42C22⋊D711yescharacteristic
139#103—281no—
140#103—281no—
141#103—282no—
142C22⋊C8—142no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95C22⋊D712yes—
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—44no—
157C4×C7—42no—
158C4×C7—42no—
159C4×C7—82no—
160Dic7—81no—
161Dic7—81no—
162C2×Q8—24no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—82no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8—142no—
171C2×Q8#114412yescomplemented
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4C2×C22⋊D711yescharacteristic
177C22×C4C2×C22⋊D712yes—
178Q16—564no—
179M16—281no—
180M16—281no—
181C2×C8—282no—
182C2×C8—281no—
183C4⋊C4—41no—
184C4⋊C4—82no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—142no—
190C4⋊C4—282no—
191C22⋊C4—281no—
192C42—41no—
193C42—81no—
194C42—281no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7#160211yescharacteristic
198C2×C7#219911yescharacteristic
199C2×C7#220511yescharacteristic
200C23#651611yescharacteristic
201Q8—42no—
202Q8—44no—
203Q8—81no—
204Q8—84no—
205Q8—82no—
206Q8—562no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—281no—
224C2×C4—281no—
225C2×C4#650511yescharacteristic
226C2×C4C2×#114411yescharacteristic
227C8—561no—
228C8—562no—
229C7#1222411yescharacteristic, complemented, Sylow
230C22—21no—
231C22—21no—
232C22#9000011yescharacteristic
233C22#9001411yescharacteristic
234C22#9001611yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—561no—
243C4—561no—
244C2—21no—
245C2—21no—
246C2#1172964711yescharacteristic
247C2#1172969611yescharacteristic
248C2#1172970411yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2811C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2811D7
hypocenterC7#12224
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
57. 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
C7×#28112
#117274612
#117274272
#117254542
#117254492
#122247