Finite Group Databasebeta

← #13266813

Subgroups — #13266813

Order 1792.

454 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2819C211yesmaximal, characteristic
2#11727458C212yesmaximal, complemented
3#11727297C211yesmaximal, characteristic, complemented
4#11725453C212yesmaximal, complemented
5#11725438C211yesmaximal, characteristic, complemented
6C2×C7×#124—21no—
7#90191C2211yescharacteristic
8C7×#394—21no—
9C7×#394C2212yes—
10C7×#393—22no—
11C7×#392—22no—
12C7×#382C2211yescharacteristic
13#89990—22no—
14#89736—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12250—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4—21no—
22C22×C7×D4—21no—
23C22×C7×D4D412yes—
24C7×#124—42no—
25C7×#124—42no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C42×C7D411yescharacteristic
33C2×#1142—42no—
34#6489—21no—
35#6487—22no—
36C2×#1136—21no—
37#6453—41no—
38#6451—41no—
39C7×#95—21no—
40C7×#95—22no—
41C7×#95D412yes—
42#6406—42no—
43C22⋊(C7⋊C8)—22no—
44#6382—41no—
45#6381—41no—
46#2819D711yescharacteristic
47#2061—71no—
48#1599—72no—
49#1345—72no—
50#1324—71no—
51C24×C7—22no—
52C2×C7×Q8—42no—
53C2×C7×Q8—81no—
54C2×C7×Q8—81no—
55C2×C7×D4—42no—
56C2×C7×D4—42no—
57C2×C7×D4—82no—
58C2×C7×D4—81no—
59C2×C7×D4—81no—
60C2×C7×D4—82no—
61C2×C7×D4—81no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7C2×D412yes—
67C2×Dic14—22no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—41no—
70C7×C22⋊C4—44no—
71C7×C22⋊C4—42no—
72C7×C22⋊C4—42no—
73C7×C22⋊C4—82no—
74C7×C22⋊C4—82no—
75C42×C7—21no—
76C42×C7SD1612yescomplemented
77C22⋊Dic7—41no—
78#1142—84no—
79C4⋊Dic7—22no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82Dic7⋊C4—42no—
83C4×Dic7—41no—
84#1136—22no—
85#1136—41no—
86C2×C7⋊C8—41no—
87C2×C7⋊C8—42no—
88#544—71no—
89C2×#124—21no—
90#450—142no—
91#428—141no—
92#394—21no—
93#394C2×D712yes—
94#393—22no—
95#392—22no—
96#382C2×D711yescharacteristic
97#343—71no—
98#339—141no—
99#328—142no—
100#324—72no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7#12011yescharacteristic
109C7×Q8—84no—
110C7×D4—44no—
111C7×D4—82no—
112C7×D4—84no—
113C7×D4—84no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7C2×SD1611yescharacteristic
127C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
128C2×Dic7—41no—
129C2×Dic7—41no—
130Dic14—82no—
131C7⋊C8—42no—
132C7⋊C8—82no—
133C22×Q8—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22⋊D712yes—
137C2×SD16—282no—
138C2×M16—141no—
139C4⋊Q8—281no—
140#125—281no—
141#124—42no—
142#124—42no—
143C22⋊Q8—142no—
144#117—141no—
145C2×C4⋊C4C22⋊D711yescharacteristic
146C2×C4⋊C4C22×D711yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42C22⋊D711yescharacteristic
152#107—281no—
153#106—281no—
154#102—282no—
155C22⋊C8—142no—
156#95—21no—
157#95—22no—
158#95C22⋊D712yes—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—82no—
162C22×C7—82no—
163C22×C7—82no—
164C22×C7—82no—
165C22×C7—82no—
166C22×C7—82no—
167C22×C7—82no—
168C22×C7#45011yescharacteristic
169C22×C7#45611yescharacteristic
170C22×C7B(4,𝔽2)11yescharacteristic
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—42no—
174C4×C7—42no—
175C4×C7—82no—
176C4×C7—81no—
177Dic7—81no—
178Dic7—81no—
179C24—22no—
180C2×Q8—42no—
181C2×Q8—81no—
182C2×Q8—81no—
183C2×Q8—142no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—82no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—82no—
190C2×D4—81no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4C2×C22⋊D711yescharacteristic
195C22×C4C2×C22⋊D712yes—
196SD16—564no—
197M16—142no—
198M16—281no—
199C2×C8—281no—
200C2×C8—282no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—142no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—282no—
207C22⋊C4—44no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—281no—
213C42—21no—
214C42—281no—
215C42#114212yescomplemented
216C2×C7—21no—
217C2×C7—21no—
218C2×C7—162no—
219C2×C7#160411yescharacteristic
220C2×C7#219911yescharacteristic
221C2×C7#220211yescharacteristic
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23#651611yescharacteristic
230Q8—84no—
231Q8—562no—
232D4—44no—
233D4—82no—
234D4—84no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—281no—
249C2×C4—281no—
250C2×C4#649511yescharacteristic
251C2×C4C2×#114211yescharacteristic
252C8—282no—
253C8—562no—
254C7#1225011yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#8999011yescharacteristic
265C22#9001611yescharacteristic
266C22#9001811yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—561no—
274C4—561no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2#1172963411yescharacteristic
279C2#1172970411yescharacteristic
280C2#1172970511yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
Fitting, prime radicalC7×#2819C2
socleC22×C7B(4,𝔽2)
hypercenter#2819D7
hypocenterC7#12250
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2819

Complements

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

NH ≅ G/Ntype
2. #11727458C2semidirect
3. #11727297C2semidirect
4. #11725453C2semidirect
5. #11725438C2semidirect
76. C42×C7SD16semidirect
215. C42#1142semidirect
254. C7#12250semidirect

Maximal subgroups

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

subgroupindex
C7×#28192
#117274582
#117272972
#117254532
#117254382
#122507