Finite Group Databasebeta

← #13266542

Subgroups — #13266542

Order 1792.

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

normal subgroups only

This group has 388 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×#2817C211yesmaximal, characteristic
2#11727464C212yesmaximal
3#11727290C211yesmaximal, characteristic
4#11725455C212yesmaximal
5#11725433C211yesmaximal, characteristic
6C2×C7×C22⋊Q8—21no—
7#90174C2211yescharacteristic
8C7×#400—22no—
9C7×#397—21no—
10C7×#397C2212yes—
11C7×#395—22no—
12C7×#381C2211yescharacteristic
13#89983—21no—
14#89743—22no—
15#89322—21no—
16#89321C2211yescharacteristic
17#89320C2212yes—
18#89247—22no—
19#12207—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C23×C4×C7D411yescharacteristic
22C7×C22⋊Q8—42no—
23C7×C22⋊Q8—42no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4D412yes—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4D411yescharacteristic
30C2×C7×C22⋊C4C2311yescharacteristic
31C2×C22⋊Dic7—21no—
32C2×C22⋊Dic7—21no—
33#6487—22no—
34#6441—42no—
35C7×#95—22no—
36C7×#95—21no—
37C7×#95—22no—
38C7×#95—22no—
39C7×#95D412yes—
40C7⋊#95—42no—
41C22⋊(C7⋊C8)—22no—
42#6384—42no—
43#6382—42no—
44#2817D711yescharacteristic
45#2030—71no—
46#1608—72no—
47#1344—72no—
48#1317—71no—
49C24×C7C2×D411yescharacteristic
50C2×C7×Q8—42no—
51C2×C7×Q8—81no—
52C2×C7×Q8—81no—
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7SD1612yes—
61C22×C4×C7C2×D412yes—
62C22×Dic7—22no—
63C22×Dic7—41no—
64C2×Dic14—22no—
65C7×C4⋊C4—42no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—44no—
68C7×C4⋊C4—81no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—42no—
71C7×C22⋊C4—42no—
72C7×C22⋊C4—41no—
73C22⋊Dic7—41no—
74C22⋊Dic7—41no—
75C22⋊Dic7—41no—
76C22⋊Dic7—41no—
77C4⋊Dic7—22no—
78Dic7⋊C4—42no—
79C2×C7⋊C8—42no—
80#543—71no—
81C2×C22⋊Q8—21no—
82#483—142no—
83#400—22no—
84#397—21no—
85#397C2×D712yes—
86#395—22no—
87#381—141no—
88#381C2×D711yescharacteristic
89#342—71no—
90#342—141no—
91#328—142no—
92#323—72no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—42no—
96C23×C7#12011yescharacteristic
97C23×C7C2×SD1611yescharacteristic
98C23×C7AGL(1,ℤ/8)11yescharacteristic
99C7×Q8—84no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—44no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—81no—
114C2×Dic7—42no—
115C2×Dic7—82no—
116C2×Dic7—81no—
117C2×Dic7—82no—
118Dic14—82no—
119C7⋊C8—82no—
120C22×Q8—21no—
121C23×C4C22⋊D711yescharacteristic
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—142no—
125C22⋊Q8—282no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C22⋊D712yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—141no—
132C2×C22⋊C4—141no—
133C2×C22⋊C4C22⋊D711yescharacteristic
134C2×C22⋊C4C22×D711yescharacteristic
135#106—282no—
136#103—282no—
137C22⋊C8—142no—
138#95—22no—
139#95—21no—
140#95—22no—
141#95—22no—
142#95—282no—
143#95C22⋊D712yes—
144C22×C7—22no—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—42no—
150C22×C7—81no—
151C22×C7#45011yescharacteristic
152C22×C7#45511yescharacteristic
153C22×C7B(4,𝔽2)11yescharacteristic
154C4×C7—42no—
155C4×C7—42no—
156C4×C7—82no—
157C4×C7—81no—
158C4×C7—82no—
159Dic7—82no—
160Dic7—161no—
161C24C2×C22⋊D711yescharacteristic
162C2×Q8—42no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8—142no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—142no—
174C22×C4—281no—
175C22×C4#114212yes—
176C22×C4C2×C22⋊D712yes—
177C2×C8—282no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—81no—
182C4⋊C4—142no—
183C4⋊C4—282no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—41no—
188C22⋊C4—281no—
189C22⋊C4—281no—
190C22⋊C4—281no—
191C22⋊C4—281no—
192C2×C7—21no—
193C2×C7—21no—
194C2×C7—41no—
195C2×C7—41no—
196C2×C7#160111yescharacteristic
197C2×C7#220112yes—
198C23—21no—
199C23—21no—
200C23—42no—
201C23#649511yescharacteristic
202C23C2×#114211yescharacteristic
203C23#651611yescharacteristic
204Q8—84no—
205Q8—562no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—282no—
221C2×C4—562no—
222C2×C4—561no—
223C2×C4—562no—
224C8—562no—
225C7#1220711yescharacteristic, complemented, Sylow
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#8999011yescharacteristic
234C22#9001411yescharacteristic
235C22#9001611yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—562no—
242C4—1121no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#1172960011yescharacteristic
248C2#1172971212yes—
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2817C2
socleC22×C7B(4,𝔽2)
prime radicalC24×C7C2×D4
hypercenter#2817D7
hypocenterC7#12207
LCSG ▷ C2×C7×C22⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
225. C7#12207semidirect

Maximal subgroups

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

subgroupindex
C7×#28172
#117274642
#117272902
#117254552
#117254332
#122077