Finite Group Databasebeta

← #13265915

Subgroups — #13265915

Order 1792.

424 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 71 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×#3189C211yesmaximal, characteristic
2#11728841C212yesmaximal, complemented
3#11727718C211yesmaximal, characteristic
4#11725582C212yesmaximal, complemented
5#11725479C211yesmaximal, characteristic
6C2×C7×C22⋊Q8—21no—
7C2×#6487—21no—
8C22×Dic28C2211yescharacteristic
9C7×#484—22no—
10C7×#467—21no—
11C7×#467C2212yes—
12C7×#451—22no—
13C2×C7×C22⋊C8C2211yescharacteristic
14C2×#6403—22no—
15C2×#6390—21no—
16#89346C2211yescharacteristic
17C22⋊(C7⋊C16)C2212yes—
18#89287—22no—
19#12106—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C23×C4×C7D411yescharacteristic
22C22×Dic14—21no—
23C2×C7×Q16—21no—
24C2×C7×Q16D412yes—
25C22×C7×C8D411yescharacteristic
26C22×C7×C8C2311yescharacteristic
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C7×C22⋊Q8—41no—
30C7×C22⋊Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C22⋊C4—41no—
34C2×C22⋊Dic7—41no—
35C2×C4⋊Dic7—21no—
36#6487—42no—
37#6487—42no—
38C2×Dic7⋊C4—41no—
39C2×Dic28—22no—
40C2×Dic28—42no—
41C7×#107—21no—
42C7×#107D412yes—
43C7×#103—22no—
44C7×#103—22no—
45C7×C22⋊C8—22no—
46#6403—44no—
47#6390—42no—
48C2×C7⋊C16—22no—
49#3189D711yescharacteristic
50#2188—72no—
51#2001—71no—
52#1348—72no—
53#1314—71no—
54C24×C7D811yescharacteristic
55C2×C7×Q8—22no—
56C2×C7×Q8—41no—
57C2×C7×Q8—81no—
58C2×C7×Q8—81no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7D811yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×Dic7—41no—
68C22×Dic7—41no—
69C2×Dic14—22no—
70C2×Dic14—41no—
71C2×Dic14—41no—
72C2×Dic14—41no—
73C2×Dic14—41no—
74C2×Dic14—41no—
75C7×Q16—42no—
76C7×Q16—81no—
77C2×C7×C8—21no—
78C2×C7×C8—41no—
79C2×C7×C8—41no—
80C2×C7×C8C2×D412yes—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—42no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C22⋊Dic7—44no—
88C4⋊Dic7—22no—
89C4⋊Dic7—41no—
90Dic7⋊C4—44no—
91Dic28—24no—
92Dic28—42no—
93C7⋊C16—42no—
94C22×Q16—71no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—141no—
97C2×Q32—142no—
98#484—22no—
99#467—21no—
100#467C2×D712yes—
101#451—22no—
102C2×#103—141no—
103C2×C22⋊C8C2×D711yescharacteristic
104#358—142no—
105C22⋊C16—72no—
106#340—71no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7Q3212yescomplemented
112C23×C7C2×D811yescharacteristic
113C7×Q8—42no—
114C7×Q8—42no—
115C7×Q8—82no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—81no—
125C2×C4×C7—82no—
126C2×C4×C7—82no—
127C2×C4×C7#12011yescharacteristic
128C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
129C2×Dic7—42no—
130C2×Dic7—42no—
131C2×Dic7—81no—
132C2×Dic7—81no—
133C2×Dic7—81no—
134C2×Dic7—81no—
135Dic14—42no—
136Dic14—42no—
137Dic14—81no—
138Dic14—81no—
139C7×C8—22no—
140C7×C8—41no—
141C22×Q8—21no—
142C22×Q8—141no—
143C23×C4C22⋊D711yescharacteristic
144C2×Q16—21no—
145C2×Q16—142no—
146C2×Q16—282no—
147C2×Q16C22⋊D712yes—
148C22×C8C22⋊D711yescharacteristic
149C22×C8C22×D711yescharacteristic
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—282no—
155C22⋊Q8—282no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—141no—
159C2×C4⋊C4—281no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—281no—
162Q32—284no—
163C2×C16—142no—
164#107—21no—
165#107C22⋊D712yes—
166#103—22no—
167#103—22no—
168#103—282no—
169C22⋊C8—22no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—22no—
173C22×C7—42no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—81no—
177C22×C7#44711yescharacteristic
178C22×C7C2×Q3211yescharacteristic
179C22×C7#51011yescharacteristic
180C4×C7—21no—
181C4×C7—41no—
182C4×C7—82no—
183C4×C7—82no—
184C4×C7#45311yescharacteristic
185C4×C7B(4,𝔽2)11yescharacteristic
186Dic7—82no—
187Dic7—82no—
188C24#114111yescharacteristic
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8—142no—
194C2×Q8—281no—
195C2×Q8—281no—
196C2×Q8—281no—
197C2×Q8—281no—
198C2×Q8—281no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—281no—
206C22×C4—281no—
207C22×C4#114111yescharacteristic
208C22×C4C2×C22⋊D711yescharacteristic
209Q16—42no—
210Q16—81no—
211Q16—144no—
212Q16—282no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8C2×C22⋊D712yes—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—142no—
222C4⋊C4—281no—
223C4⋊C4—284no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—284no—
227C16—282no—
228C2×C7—21no—
229C2×C7—21no—
230C2×C7—42no—
231C2×C7#159511yescharacteristic
232C2×C7#218811yescharacteristic
233C2×C7#219111yescharacteristic
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23#640312yescomplemented
239C23C2×#114111yescharacteristic
240Q8—42no—
241Q8—42no—
242Q8—82no—
243Q8—282no—
244Q8—282no—
245Q8—561no—
246Q8—561no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—282no—
259C2×C4—282no—
260C2×C4—561no—
261C2×C4—561no—
262C2×C4—561no—
263C2×C4—561no—
264C2×C4#649511yescharacteristic
265C2×C4#651611yescharacteristic
266C8—22no—
267C8—41no—
268C7#1210611yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22#8995111yescharacteristic
277C22C2×#640311yescharacteristic
278C22#8998911yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—562no—
284C4—562no—
285C4#9001311yescharacteristic
286C4#9001611yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2#1172959911yescharacteristic
291C2#1172981811yescharacteristic
292C2#1172986711yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#3189C2
socleC22×C7#447
prime radicalC24×C7D8
hypercenter#3189D7
hypocenterC7#12106
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
2. #11728841C2semidirect
4. #11725582C2semidirect
111. C23×C7Q32semidirect
238. C23#6403semidirect
268. C7#12106semidirect

Maximal subgroups

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

subgroupindex
C7×#31892
#117288412
#117277182
#117255822
#117254792
#121067