Finite Group Databasebeta

← #13243099

Subgroups — #13243099

Order 1792.

494 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 83 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#11734018C211yesmaximal, characteristic
2C7×#2095C211yesmaximal, characteristic, complemented
3#11727341C212yesmaximal, complemented
4#11725448C211yesmaximal, characteristic, complemented
5#11725285C212yesmaximal, complemented
6C2×#6510—21no—
7C2×C7×#107—21no—
8C2×C7×#103—21no—
9C7×#391C2211yescharacteristic
10C7×#384C2211yescharacteristic
11#89763—21no—
12C2×#6383—22no—
13C2×#6382—22no—
14#89711C2212yes—
15#89443—22no—
16#89442—22no—
17C7×#336C2211yescharacteristic
18#89319C2212yes—
19#7830—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×D28—21no—
22C22×D28—21no—
23C22×D28D412yes—
24C22×C7×C8—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C42×C7C2311yescharacteristic
29#6510—42no—
30#6510—42no—
31C2×(C2×D7)⋊C4—22no—
32C2×(C2×D7)⋊C4—22no—
33C2×(C2×D7)⋊C4—22no—
34C2×(C2×D7)⋊C4—22no—
35C2×C4⋊Dic7D412yes—
36C2×C4×Dic7—21no—
37C22×C7⋊C8—22no—
38C7×#107—42no—
39C7×#103—42no—
40C7×#95—22no—
41C7⋊#95—21no—
42#6383—44no—
43#6382—44no—
44#2733—71no—
45#2095D711yescharacteristic, complemented
46#2094—72no—
47#1272—71no—
48#1270—72no—
49C23×D7—22no—
50C2×C7×Q8—21no—
51C2×C7×Q8—41no—
52C2×C7×Q8—41no—
53C2×C7×Q8SD1612yes—
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—41no—
57C22×C4×C7C2×D411yescharacteristic
58C22×C4×C7C2×D412yes—
59C22×Dic7—22no—
60C2×D28—22no—
61C2×D28—22no—
62C2×D28—42no—
63C2×D28—42no—
64C2×D28—42no—
65C2×D28—81no—
66C2×D28—81no—
67C2×D28—82no—
68C2×D28—81no—
69C2×C7×C8—22no—
70C2×C7×C8—41no—
71C2×C7×C8—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C42×C7—41no—
78(C2×D7)⋊C4—44no—
79(C2×D7)⋊C4—42no—
80(C2×D7)⋊C4—42no—
81(C2×D7)⋊C4—42no—
82(C2×D7)⋊C4—42no—
83(C2×D7)⋊C4—82no—
84C4⋊Dic7—42no—
85C4×Dic7—42no—
86C2×C7⋊C8—24no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C2×#124—141no—
90C2×#107—21no—
91C2×#106—142no—
92C2×#103—21no—
93C2×#102—142no—
94#394—141no—
95#392—142no—
96#391C2×D711yescharacteristic
97#390—72no—
98#388—142no—
99#384C2×D711yescharacteristic
100#336—72no—
101#336C2×D711yescharacteristic
102C23×C7#12011yescharacteristic
103C22×D7—42no—
104C22×D7—42no—
105C22×D7—42no—
106C22×D7—42no—
107C22×D7—42no—
108C22×D7—42no—
109C22×D7—42no—
110C7×Q8—42no—
111C7×Q8—81no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—41no—
121C2×C4×C7—82no—
122C2×C4×C7—81no—
123C2×C4×C7C2×SD1611yescharacteristic
124C2×C4×C7#13512yes—
125C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C7#13712yes—
127C2×Dic7—42no—
128C2×Dic7—42no—
129C2×Dic7—42no—
130D28—44no—
131D28—82no—
132D28—84no—
133D28—84no—
134C7×C8—42no—
135C7⋊C8—44no—
136C22×Q8C22⋊D711yescharacteristic
137C22×D4—72no—
138C22×D4—141no—
139C22×D4—141no—
140C22×C8—21no—
141C22×C8—142no—
142#124—282no—
143#124—282no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4C22⋊D711yescharacteristic
148C2×C22⋊C4—142no—
149C2×C22⋊C4—142no—
150C2×C22⋊C4—142no—
151C2×C22⋊C4—142no—
152C2×C42—141no—
153C2×C42C22×D711yescharacteristic
154#107—42no—
155#106—284no—
156#103—42no—
157#102—284no—
158#95—22no—
159#95—141no—
160C22×C7#45011yescharacteristic
161C22×C7#45212yes—
162C22×C7#45312yes—
163C22×C7#45311yescharacteristic
164C22×C7#45811yescharacteristic
165C2×D7—82no—
166C2×D7—82no—
167C2×D7—82no—
168C2×D7—82no—
169C2×D7—82no—
170C2×D7—82no—
171C2×D7—82no—
172C4×C7—21no—
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—42no—
177C4×C7—81no—
178C4×C7—161no—
179Dic7—82no—
180C24—142no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#114312yes—
185C2×D4—142no—
186C2×D4—142no—
187C2×D4—282no—
188C2×D4—282no—
189C2×D4—282no—
190C2×D4—561no—
191C2×D4—561no—
192C2×D4—562no—
193C2×D4—561no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—142no—
198C22×C4D4×D712yes—
199C22×C4C2×C22⋊D711yescharacteristic
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—144no—
204C2×C8—282no—
205C2×C8—282no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—282no—
212C22⋊C4—284no—
213C22⋊C4—282no—
214C22⋊C4—282no—
215C22⋊C4—282no—
216C22⋊C4—282no—
217C22⋊C4—562no—
218C42—41no—
219C42—282no—
220C2×C7#161311yescharacteristic
221C2×C7#161812yes—
222C2×C7#163112yes—
223C2×C7#163411yescharacteristic
224C2×C7#165511yescharacteristic
225D7—162no—
226C23—282no—
227C23—282no—
228C23—282no—
229C23—282no—
230C23—282no—
231C23—282no—
232C23—282no—
233C23#650011yescharacteristic
234Q8—42no—
235Q8—81no—
236D4—284no—
237D4—562no—
238D4—564no—
239D4—564no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—282no—
252C2×C4—282no—
253C2×C4—282no—
254C2×C4#648112yes—
255C2×C4#648212yes—
256C2×C4C2×#114311yescharacteristic
257C2×C4#651211yescharacteristic
258C8—42no—
259C8—284no—
260C7#783011yescharacteristic, complemented, Sylow
261C22—562no—
262C22—562no—
263C22—562no—
264C22—562no—
265C22—562no—
266C22—562no—
267C22—562no—
268C22#8981811yescharacteristic
269C22#8982511yescharacteristic
270C22#8984911yescharacteristic
271C22#8995912yes—
272C22#8997012yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—161no—
280C4—562no—
281C2—1122no—
282C2#1172746811yescharacteristic
283C2#1172891412yes—
284C2#1172893112yes—
285C2#1172905811yescharacteristic
286C2#1172908011yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2095C2
socleC23×C7#120
prime radical#11734018C2
hypercenter#2095D7
hypocenterC7#7830
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
2. C7×#2095C2semidirect
3. #11727341C2semidirect
4. #11725448C2semidirect
5. #11725285C2semidirect
45. #2095D7semidirect
260. C7#7830semidirect

Maximal subgroups

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

subgroupindex
#117340182
C7×#20952
#117273412
#117254482
#117252852
#78307