Finite Group Databasebeta

← #13263017

Subgroups — #13263017

Order 1792.

454 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×(C22⋊D7)⋊C8C211yesmaximal, characteristic
2D7×C22⋊C16C212yesmaximal, complemented
3C7×#1314C211yesmaximal, characteristic, complemented
4#11725479C211yesmaximal, characteristic, complemented
5#11725330C212yesmaximal, complemented
6C2×C4×C22⋊D7C411yescharacteristic
7C23×C7⋊C8C411yescharacteristic
8C22×C8×D7C2211yescharacteristic
9C2×C7×C22⋊C8C2211yescharacteristic
10C2×(C2×D7)⋊C8C411yescharacteristic
11C2×Dic7⋊C8C2211yescharacteristic
12C2×C8×Dic7C411yescharacteristic
13C2×D7×C16—22no—
14(C22⋊D7)⋊C8—22no—
15(C22⋊D7)⋊C8—22no—
16C7×C22⋊C16C2212yes—
17C22⋊(C7⋊C16)C2212yes—
18(C2×D7)⋊C16—22no—
19#11465—71nomaximal, Sylow
20C23×C4×C7C2×C411yescharacteristic
21C22×C22⋊D7C811yescharacteristic
22C22×C4×D7C2×C411yescharacteristic
23C22×C7×C8C2×C411yescharacteristic
24C22×C7×C8C2311yescharacteristic
25C2×C22⋊Dic7C811yescharacteristic
26C4×C22⋊D7—22no—
27C4×C22⋊D7—22no—
28C2×(C2×D7)⋊C4C811yescharacteristic
29C2×Dic7⋊C4C811yescharacteristic
30C2×C4×Dic7C2×C411yescharacteristic
31C22×C7⋊C8—21no—
32C22×C7⋊C8—21no—
33C22×C7⋊C8—42no—
34C22×C7⋊C8C2×C411yescharacteristic
35C22×C7⋊C8C2×C411yescharacteristic
36C2×C8×D7—42no—
37C2×C8×D7D412yes—
38C2×C8×D7D412yes—
39C2×C7×C16—22no—
40C7×C22⋊C8—22no—
41(C2×D7)⋊C8—22no—
42Dic7⋊C8—22no—
43C8×Dic7—21no—
44C8×Dic7—21no—
45C2×C7⋊C16—22no—
46D7×C16—44no—
47C2×C8×D4—71no—
48C2×C22⋊C16—72no—
49#1322—72no—
50#1314—71no—
51#1314D711yescharacteristic, complemented
52C24×C7C2×C811yescharacteristic
53C23×D7C2×C811yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—41no—
57C22×C4×C7—41no—
58C22×C4×C7C2×C811yescharacteristic
59C22×C4×C7C22×C411yescharacteristic
60C2×C22⋊D7—22no—
61C2×C22⋊D7—22no—
62C2×C22⋊D7C1614yescomplemented
63C22×Dic7C2×C811yescharacteristic
64C22×Dic7C2×C811yescharacteristic
65C22×Dic7C2×C811yescharacteristic
66C2×C4×D7—41no—
67C2×C4×D7—41no—
68C2×C4×D7C22⋊C412yes—
69C2×C4×D7M1612yes—
70C2×C7×C8—21no—
71C2×C7×C8—41no—
72C2×C7×C8—41no—
73C2×C7×C8C2×D412yes—
74C22⋊Dic7—22no—
75(C2×D7)⋊C4—22no—
76Dic7⋊C4—22no—
77C4×Dic7—21no—
78C4×Dic7—21no—
79C2×C7⋊C8—22no—
80C2×C7⋊C8—21no—
81C2×C7⋊C8—42no—
82C2×C7⋊C8—42no—
83C2×C7⋊C8—41no—
84C2×C7⋊C8C22⋊C411yescharacteristic
85C2×C7⋊C8C22⋊C411yescharacteristic
86C8×D7—24no—
87C8×D7—42no—
88C7×C16—42no—
89C7⋊C16—42no—
90C23×C8—71no—
91C23×C8—71no—
92C2×C4×D4—71no—
93C22×C16—142no—
94C8×D4—142no—
95C8×D4—142no—
96C2×C4⋊C8—71no—
97C2×C22⋊C8—71no—
98C2×C22⋊C8C2×D711yescharacteristic
99C2×C4×C8—71no—
100C22⋊C16—72no—
101C22⋊C16—142no—
102C22⋊C16C2×D712yes—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—41no—
106C23×C7—41no—
107C23×C7C2×C1612yescomplemented
108C23×C7C22×C811yescharacteristic
109C22×D7—41no—
110C22×D7—41no—
111C22×D7C22⋊C812yes—
112C22×D7C2×C1612yes—
113C22×D7M3212yes—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7C2×C22⋊C411yescharacteristic
122C2×C4×C7C2×M1611yescharacteristic
123C22⋊D7—28no—
124C2×Dic7—21no—
125C2×Dic7—21no—
126C2×Dic7—21no—
127C2×Dic7—41no—
128C2×Dic7—41no—
129C2×Dic7C22⋊C812yes—
130C2×Dic7C2×C1612yescomplemented
131C4×D7—22no—
132C4×D7—22no—
133C4×D7—42no—
134C7×C8—22no—
135C7×C8—41no—
136C7⋊C8—21no—
137C7⋊C8—41no—
138C7⋊C8#9912yes—
139C22×D4—71no—
140C23×C4—71no—
141C23×C4C4×D711yescharacteristic
142C22×C8—71no—
143C22×C8—72no—
144C22×C8—72no—
145C22×C8—71no—
146C22×C8—141no—
147C22×C8—141no—
148C22×C8—282no—
149C22×C8—282no—
150C22×C8C4×D711yescharacteristic
151C22×C8C22×D711yescharacteristic
152C4×D4—142no—
153C4×D4—142no—
154C2×C4⋊C4—71no—
155C2×C22⋊C4—71no—
156C2×C22⋊C4—71no—
157C2×C42—71no—
158C2×C16—22no—
159C2×C16—142no—
160C2×C16—284no—
161C4⋊C8—142no—
162C22⋊C8—22no—
163C22⋊C8—142no—
164C4×C8—141no—
165C4×C8—141no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7—41no—
172C22×C7—41no—
173C22×C7C2×C22⋊C811yescharacteristic
174C22×C7C22×C1611yescharacteristic
175C22×C7C2×M3211yescharacteristic
176C2×D7—22no—
177C2×D7—22no—
178C2×D7—42no—
179C2×D7—42no—
180C2×D7C22⋊C1614yes—
181C4×C7—21no—
182C4×C7—41no—
183C4×C7C2×#9911yescharacteristic
184C4×C7#41311yescharacteristic
185Dic7—22no—
186Dic7—22no—
187Dic7—41no—
188C24—71no—
189C24C8×D711yescharacteristic
190C2×D4—74no—
191C2×D4—142no—
192C2×D4—142no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—72no—
200C22×C4—72no—
201C22×C4—71no—
202C22×C4—281no—
203C22×C4—281no—
204C22×C4C8×D711yescharacteristic
205C22×C4C2×C4×D711yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—71no—
210C2×C8—71no—
211C2×C8—144no—
212C2×C8—142no—
213C2×C8—141no—
214C2×C8—282no—
215C2×C8—282no—
216C2×C8—282no—
217C2×C8—281no—
218C2×C8D4×D712yes—
219C4⋊C4—142no—
220C22⋊C4—142no—
221C22⋊C4—142no—
222C42—141no—
223C42—141no—
224C16—42no—
225C16—282no—
226C2×C7—21no—
227C2×C7—21no—
228C2×C7—42no—
229C2×C7#145911yescharacteristic
230C2×C7C2×C22⋊C1611yescharacteristic
231C2×C7#211511yescharacteristic
232D7—24no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—72no—
238C23—72no—
239C23—72no—
240C23—281no—
241C23—281no—
242C23D7×C1612yescomplemented
243C23C2×C8×D711yescharacteristic
244D4—148no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—72no—
253C2×C4—72no—
254C2×C4—141no—
255C2×C4—141no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—141no—
259C2×C4—282no—
260C2×C4—281no—
261C2×C4—281no—
262C2×C4D7×C22⋊C411yescharacteristic
263C2×C4D7×M1611yescharacteristic
264C8—22no—
265C8—41no—
266C8—72no—
267C8—141no—
268C8—281no—
269C7#1146511yescharacteristic, complemented, Sylow
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—74no—
278C22—142no—
279C22—142no—
280C22—282no—
281C22—282no—
282C22D7×C22⋊C811yescharacteristic
283C22C2×D7×C1611yescharacteristic
284C22D7×M3211yescharacteristic
285C4—21no—
286C4—41no—
287C4—142no—
288C4—142no—
289C4—281no—
290C4D7×#9911yescharacteristic
291C4#8951711yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—144no—
296C2#1172587311yescharacteristic
297C2D7×C22⋊C1611yescharacteristic
298C2#1172687311yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC22×C7C22×C16
FrattiniC22×C8C22×D7
FittingC7×#1314C2
socleC22×C7C2×C22⋊C8
prime radicalC22×C22⋊D7C8
cosocleC22×C7×C8C23
hypercenter#1314D7
hypocenterC7#11465
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. D7×C22⋊C16C2semidirect
3. C7×#1314C2semidirect
4. #11725479C2semidirect
5. #11725330C2semidirect
51. #1314D7semidirect
62. C2×C22⋊D7C16semidirect
107. C23×C7C2×C16semidirect
130. C2×Dic7C2×C16semidirect
242. C23D7×C16semidirect
269. C7#11465semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D7)⋊C82
D7×C22⋊C162
C7×#13142
#117254792
#117253302
#114657