Finite Group Databasebeta

← #13242056

Subgroups — #13242056

Order 1792.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#11733925C211yesmaximal, characteristic
2#11726357C212yesmaximal, complemented
3C7×#1272C211yesmaximal, characteristic, complemented
4#11725448C211yesmaximal, characteristic, complemented
5#11725285C212yesmaximal
6C2×#6510C411yescharacteristic
7C2×Q8×Dic7C411yescharacteristic
8C2×#6456C411yescharacteristic
9C7×#391C2211yescharacteristic
10C2×#6389—22no—
11C2×Dic7⋊C8—22no—
12#89749C411yescharacteristic
13#89711C2211yescharacteristic
14C42×Dic7C2211yescharacteristic
15#89442—22no—
16#89438—22no—
17C7×#336C2212yes—
18#89319C2212yes—
19#7635—71nomaximal, Sylow
20C22×C7×Q8C2×C411yescharacteristic
21C22×D28C2×C411yescharacteristic
22C22×C4×D7—21no—
23C22×C7×C8—22no—
24C2×C7×C4⋊C4C2×C411yescharacteristic
25C2×C42×C7C2311yescharacteristic
26#6510—22no—
27#6510—22no—
28Q8×Dic7—42no—
29C2×(C2×D7)⋊C4—22no—
30C2×(C2×D7)⋊C4—22no—
31C2×(C2×D7)⋊C4—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7C2×C411yescharacteristic
34C2×C4×Dic7—21no—
35C2×C4×Dic7C2×C411yescharacteristic
36C2×C4×Dic7C2×C411yescharacteristic
37C2×C4×Dic7D412yes—
38C2×C4×Dic7D412yes—
39C22×C7⋊C8—22no—
40#6456—42no—
41C7×#95—22no—
42C7⋊#95—22no—
43#6389—44no—
44Dic7⋊C8—28no—
45C4×#124—71no—
46#1767—72no—
47#1272—71no—
48#1272D711yescharacteristic, complemented
49#1270—72no—
50C23×D7—21no—
51C2×C7×Q8—21no—
52C2×C7×Q8—41no—
53C2×C7×Q8—41no—
54C2×C7×Q8M1612yes—
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7C22×C411yescharacteristic
58C22×C4×C7C2×D412yes—
59C22×Dic7—21no—
60C22×Dic7C22⋊C412yes—
61C22×Dic7C22⋊C411yescharacteristic
62C22×Dic7C22⋊C411yescharacteristic
63C2×D28—21no—
64C2×D28—41no—
65C2×D28—41no—
66C2×D28M1612yes—
67C2×C4×D7—42no—
68C2×C4×D7—44no—
69C2×C7×C8—24no—
70C2×C7×C8—42no—
71C2×C7×C8—42no—
72C7×C4⋊C4—41no—
73C42×C7—41no—
74(C2×D7)⋊C4—28no—
75(C2×D7)⋊C4—42no—
76(C2×D7)⋊C4—42no—
77(C2×D7)⋊C4—42no—
78C4⋊Dic7—42no—
79C4⋊Dic7—41no—
80C4×Dic7—21no—
81C4×Dic7—21no—
82C4×Dic7—22no—
83C4×Dic7—22no—
84C4×Dic7—22no—
85C4×Dic7—24no—
86C4×Dic7—42no—
87C2×C7⋊C8—24no—
88C2×C7⋊C8—42no—
89C2×C7⋊C8—42no—
90C2×#124—71no—
91C2×C4×Q8—71no—
92C2×C4×D4—71no—
93C2×C4⋊C8—142no—
94C2×#97—142no—
95#391C2×D711yescharacteristic
96#390—71no—
97#388—142no—
98#382—71no—
99C4×C22⋊C4—142no—
100C43—71no—
101#336—72no—
102#336C2×D712yes—
103C23×C7C2×C22⋊C411yescharacteristic
104C22×D7—22no—
105C22×D7—41no—
106C22×D7—41no—
107C22×D7—42no—
108C22×D7—42no—
109C7×Q8—42no—
110C7×Q8—42no—
111C2×C4×C7—22no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7C2×M1611yescharacteristic
121C2×C4×C7C2×M1611yescharacteristic
122C2×C4×C7AGL(1,ℤ/8)12yes—
123C2×C4×C7#13712yes—
124C2×Dic7—22no—
125C2×Dic7—21no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—41no—
130C2×Dic7—42no—
131C2×Dic7#10012yes—
132C2×Dic7#10414yes—
133D28—42no—
134D28—42no—
135C4×D7—84no—
136C7×C8—44no—
137C7⋊C8—44no—
138C22×Q8C4×D711yescharacteristic
139C22×D4—71no—
140C23×C4—141no—
141C22×C8—22no—
142C22×C8—142no—
143#124—142no—
144#124—142no—
145C4×Q8—282no—
146C4×D4—282no—
147C2×C4⋊C4—71no—
148C2×C4⋊C4—141no—
149C2×C4⋊C4C4×D711yescharacteristic
150C2×C22⋊C4—142no—
151C2×C22⋊C4—141no—
152C2×C22⋊C4—142no—
153C2×C42—71no—
154C2×C42—71no—
155C2×C42—72no—
156C2×C42—72no—
157C2×C42—141no—
158C2×C42C22×D711yescharacteristic
159C4⋊C8—148no—
160#97—284no—
161#95—22no—
162#95—142no—
163C22×C7#40711yescharacteristic
164C22×C7#41011yescharacteristic
165C22×C7C2×#10011yescharacteristic
166C22×C7#41712yes—
167C22×C7C2×#10412yes—
168C2×D7—42no—
169C2×D7—44no—
170C2×D7—81no—
171C2×D7—82no—
172C2×D7—81no—
173C4×C7—21no—
174C4×C7—21no—
175C4×C7—22no—
176C4×C7—22no—
177C4×C7—42no—
178C4×C7—81no—
179Dic7—41no—
180Dic7—42no—
181Dic7—81no—
182Dic7#32914yes—
183C24—141no—
184C2×Q8—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8#113112yes—
188C2×D4—72no—
189C2×D4—141no—
190C2×D4—281no—
191C2×D4—281no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—71no—
195C22×C4—72no—
196C22×C4—71no—
197C22×C4—141no—
198C22×C4—282no—
199C22×C4—284no—
200C22×C4C2×C4×D711yescharacteristic
201C22×C4D4×D712yes—
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—144no—
206C2×C8—282no—
207C2×C8—282no—
208C4⋊C4—41no—
209C4⋊C4—281no—
210C4⋊C4—282no—
211C22⋊C4—148no—
212C22⋊C4—282no—
213C22⋊C4—282no—
214C22⋊C4—282no—
215C42—41no—
216C42—144no—
217C42—142no—
218C42—141no—
219C42—141no—
220C42—142no—
221C42—142no—
222C42—282no—
223C2×C7#146611yescharacteristic
224C2×C7#148912yes—
225C2×C7#150912yes—
226C2×C7C2×#32911yescharacteristic
227C2×C7#156011yescharacteristic
228D7—82no—
229C23—142no—
230C23—281no—
231C23—281no—
232C23—282no—
233C23—282no—
234C23D7×C22⋊C411yescharacteristic
235Q8—42no—
236Q8—42no—
237D4—282no—
238D4—282no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—72no—
249C2×C4—74no—
250C2×C4—142no—
251C2×C4—141no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—281no—
256C2×C4—282no—
257C2×C4—564no—
258C2×C4C2×#113111yescharacteristic
259C2×C4D7×M1611yescharacteristic
260C2×C4#647712yes—
261C2×C4#648112yes—
262C8—44no—
263C8—284no—
264C7#763511yescharacteristic, complemented, Sylow
265C22—282no—
266C22—284no—
267C22—561no—
268C22—561no—
269C22—562no—
270C22#8949811yescharacteristic
271C22#8951211yescharacteristic
272C22D7×#10011yescharacteristic
273C22#8957512yes—
274C22D7×#10412yes—
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—74no—
281C4—81no—
282C4—281no—
283C4—282no—
284C4—561no—
285C2—562no—
286C2#1172585711yescharacteristic
287C2#1172599212yes—
288C2#1172606512yes—
289C2D7×#32911yescharacteristic
290C2#1172624311yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1272C2
socleC23×C7C2×C22⋊C4
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1272D7
hypocenterC7#7635
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #11726357C2semidirect
3. C7×#1272C2semidirect
4. #11725448C2semidirect
48. #1272D7semidirect
264. C7#7635semidirect

Maximal subgroups

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

subgroupindex
#117339252
#117263572
C7×#12722
#117254482
#117252852
#76357