Finite Group Databasebeta

← #13254445

Subgroups — #13254445

Order 1792.

602 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2815C211yesmaximal, characteristic
2#11740061C212yesmaximal, complemented
3#11729428C212yesmaximal, complemented
4#11727288C211yesmaximal, characteristic, complemented
5#11725435C211yesmaximal, characteristic, complemented
6C2×C7×#121C2212yes—
7C2×#6489C2211yescharacteristic
8C7×#397—22no—
9C7×#392—22no—
10C7×#392—22no—
11C7×#381C2211yescharacteristic
12#89987—22no—
13#89983C2211yescharacteristic
14#89979C2212yes—
15#89807—24no—
16#89806—22no—
17#89805—22no—
18#89736—22no—
19#89332—24no—
20#89322—22no—
21#9983—71nomaximal, Sylow
22C22×C7×D4—22no—
23C22×C7×D4—22no—
24C22×C7×D4D412yes—
25C23×C4×C7C2311yescharacteristic
26C7×#121—22no—
27C7×#121—22no—
28C7×#121—22no—
29C7×#121—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—24no—
33C2×C7×C22⋊C4D412yes—
34C2×C22⋊Dic7—21no—
35C2×C22⋊Dic7—21no—
36#6489—22no—
37#6489—22no—
38C2×C4⋊Dic7D412yescomplemented
39C2×C4×Dic7—21no—
40C2×#1136—22no—
41C7×#95—22no—
42C7×#95—22no—
43#6406—24no—
44#6406—24no—
45C7⋊#95—22no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)D414yescomplemented
48#6382—24no—
49#6381—24no—
50#3094—72no—
51#2815D711yescharacteristic
52#1820—71no—
53#1510—72no—
54#1306—71no—
55C24×C7—22no—
56C24×C7C2×D411yescharacteristic
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—44no—
60C2×C7×D4—42no—
61C2×C7×D4—42no—
62C2×C7×D4—84no—
63C2×C7×D4—84no—
64C2×C7×D4—82no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7C2×D412yes—
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D412yes—
71C22×C4×C7#4111yescharacteristic
72C22×Dic7—22no—
73C22×Dic7—41no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C22⋊C4—42no—
77C7×C22⋊C4—42no—
78C7×C22⋊C4—42no—
79C7×C22⋊C4—84no—
80C22⋊Dic7—41no—
81C22⋊Dic7—41no—
82C22⋊Dic7—42no—
83C4⋊Dic7—22no—
84C4⋊Dic7—22no—
85C4×Dic7—42no—
86#1136—44no—
87C2×C7⋊C8—24no—
88C2×#121C2×D712yes—
89C2×#117—71no—
90#482—144no—
91#481—142no—
92#480—142no—
93#428—142no—
94#418—142no—
95#407—72no—
96#397—22no—
97#392—22no—
98#392—22no—
99#381—71no—
100#381C2×D711yescharacteristic
101#342—142no—
102#327—144no—
103C23×C7—22no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—44no—
108C23×C7—42no—
109C23×C7—44no—
110C23×C7#12011yescharacteristic
111C23×C7#12111yescharacteristic
112C23×C7#12711yescharacteristic
113C7×D4—44no—
114C7×D4—82no—
115C7×D4—84no—
116C7×D4—84no—
117C7×D4—88no—
118C2×C4×C7—24no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—41no—
122C2×C4×C7—44no—
123C2×C4×C7—44no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7#12012yes—
127C2×C4×C7#12112yes—
128C2×Dic7—42no—
129C2×Dic7—82no—
130C2×Dic7—81no—
131C2×Dic7—82no—
132C7⋊C8—44no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22⋊D712yes—
136C23×C4C22×D711yescharacteristic
137C2×M16—142no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—22no—
142#117—142no—
143#117—142no—
144C2×C4⋊C4—72no—
145C2×C4⋊C4C22⋊D712yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C22⋊C4C22⋊D712yes—
151C2×C42—141no—
152#107—144no—
153#106—144no—
154#102—144no—
155#102—144no—
156C22⋊C8—74no—
157C22⋊C8—142no—
158#95—22no—
159#95—22no—
160#95—142no—
161C22×C7—22no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7—82no—
167C22×C7—84no—
168C22×C7—84no—
169C22×C7—82no—
170C22×C7—82no—
171C22×C7—82no—
172C22×C7#39211yescharacteristic
173C22×C7#45812yes—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—82no—
177C4×C7—82no—
178Dic7—82no—
179Dic7—161no—
180C24—22no—
181C24C2×C22⋊D711yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—44no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—84no—
188C2×D4—84no—
189C2×D4—82no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—142no—
194C22×C4—281no—
195C22×C4D4×D711yescharacteristic
196C22×C4D4×D712yes—
197C22×C4#115911yescharacteristic
198C22×C4C2×C22⋊D712yes—
199M16—284no—
200C2×C8—144no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—142no—
204C4⋊C4—142no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—84no—
209C22⋊C4—281no—
210C22⋊C4—281no—
211C22⋊C4—282no—
212C42—282no—
213C2×C7—22no—
214C2×C7—41no—
215C2×C7—41no—
216C2×C7—162no—
217C2×C7#200812yes—
218C2×C7#202211yescharacteristic
219C23—22no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—44no—
224C23—42no—
225C23—44no—
226C23#650111yescharacteristic
227C23#650311yescharacteristic
228C23#651611yescharacteristic
229D4—44no—
230D4—82no—
231D4—84no—
232D4—84no—
233D4—88no—
234C2×C4—24no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—282no—
243C2×C4—562no—
244C2×C4—561no—
245C2×C4—562no—
246C2×C4#650012yes—
247C2×C4#650212yes—
248C8—284no—
249C7#998311yescharacteristic, complemented, Sylow
250C22—22no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—82no—
256C22—84no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22#8999511yescharacteristic
262C22#9001512yes—
263C4—22no—
264C4—22no—
265C4—82no—
266C4—82no—
267C4—562no—
268C4—1121no—
269C2—22no—
270C2—41no—
271C2—41no—
272C2—162no—
273C2#1172960311yescharacteristic
274C2#1172961212yes—
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2815C2
socleC22×C7#392
hypercenter#2815D7
hypocenterC7#9983
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
2. #11740061C2semidirect
3. #11729428C2semidirect
4. #11727288C2semidirect
5. #11725435C2semidirect
38. C2×C4⋊Dic7D4semidirect
47. C22⋊(C7⋊C8)D4semidirect
249. C7#9983semidirect

Maximal subgroups

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

subgroupindex
C7×#28152
#117400612
#117294282
#117272882
#117254352
#99837