Finite Group Databasebeta

← #11738422

Subgroups — #11738422

Order 896.

392 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 113 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#90416C211yesmaximal, characteristic, complemented
2#90248C211yesmaximal, characteristic, complemented
3C7×#474C211yesmaximal, characteristic, complemented
4#89866C211yesmaximal, characteristic, complemented
5#89787C211yesmaximal, characteristic, complemented
6#89663C212yesmaximal, complemented
7#89658C212yesmaximal, complemented
8#89620C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89559C212yesmaximal, complemented
11#89554C212yesmaximal, complemented
12C2×#1161—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15#6510—21no—
16#6501C2211yescharacteristic
17#6499C2211yescharacteristic
18C2×#1141C2212yes—
19C4×C22⋊D7—21no—
20#6489—21no—
21C2×#1132C2212yes—
22C2×#1131C2212yes—
23#6460C2211yescharacteristic
24#6458—21no—
25#6458C2212yes—
26#6456—21no—
27#6456C2212yes—
28#6453C2211yescharacteristic
29#6446—21no—
30#6442—21no—
31#6437C2211yescharacteristic
32C42×D7C2211yescharacteristic
33C4×Dic14C2211yescharacteristic
34C7×#107C2212yes—
35C7×C4⋊C8C2211yescharacteristic
36C7×#102C2212yes—
37#6395C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3251—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48#1161—44no—
49C2×D28—22no—
50C2×D28—21no—
51C2×D28—21no—
52C2×D28C2312yes—
53C2×C4×D7—22no—
54C2×C4×D7D411yescharacteristic
55C2×C4×D7D411yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14C2311yescharacteristic
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2312yes—
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—21no—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—21no—
65C22⋊Dic7—21no—
66C22⋊Dic7—21no—
67#1141—42no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—22no—
71C4⋊Dic7—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76C4×Dic7—21no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81#1132—42no—
82#1131—42no—
83#532—71no—
84#529—71no—
85#478—72no—
86#475—71no—
87#474D711yescharacteristic, complemented
88#469—72no—
89#468—72no—
90#463—72no—
91#460—71no—
92#459—71no—
93#423—71no—
94C23×C7—21no—
95C22×D7—22no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—42no—
98C7×D4—21no—
99C7×D4—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—42no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2411yescharacteristic
106C22⋊D7—42no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112D28—22no—
113D28—22no—
114D28—42no—
115D28—44no—
116D28—42no—
117C4×D7—22no—
118C4×D7—21no—
119C4×D7—21no—
120C4×D7—41no—
121C4×D7—44no—
122Dic14—21no—
123Dic14—21no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—141no—
127C2×SD16—72no—
128C2×D8—72no—
129C2×M16—72no—
130C4⋊Q8C2×D711yescharacteristic
131#127—71no—
132#126—141no—
133#125—71no—
134#124—71no—
135#124—141no—
136#123—141no—
137C22⋊Q8—71no—
138#121—71no—
139#121—72no—
140#121—141no—
141C4×Q8—71no—
142C4×D4—72no—
143C4×D4—141no—
144C4×D4—141no—
145C4×D4C2×D711yescharacteristic
146#117—141no—
147C2×C42—71no—
148#107C2×D712yes—
149#106—72no—
150C4⋊C8—72no—
151C4⋊C8—71no—
152C4⋊C8C2×D711yescharacteristic
153#103—72no—
154#102—72no—
155#102—72no—
156#102C2×D712yes—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7C22×D411yescharacteristic
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—42no—
167C4×C7—21no—
168C4×C7—42no—
169C4×C7—41no—
170C4×C7AGL(1,ℤ/8)12yes—
171C4×C7H(5,𝔽2)11yescharacteristic
172C4×C72−1+411yescharacteristic
173Dic7—21no—
174Dic7—21no—
175Dic7—21no—
176Dic7—41no—
177Dic7—41no—
178#41—284no—
179C2×Q8—21no—
180C2×Q8—71no—
181C2×D4—72no—
182C2×D4—142no—
183C2×D4—141no—
184C2×D4—141no—
185C2×D4—141no—
186C2×D4C22×D711yescharacteristic
187C22×C4—21no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—71no—
191C22×C4—142no—
192SD16—282no—
193D8—282no—
194M16—282no—
195C2×C8—72no—
196C2×C8C22×D712yes—
197C4⋊C4—21no—
198C4⋊C4—71no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4C22×D711yescharacteristic
204C4⋊C4C22×D712yes—
205C22⋊C4—21no—
206C22⋊C4—142no—
207C22⋊C4—142no—
208C22⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C42—71no—
213C42—71no—
214C42—71no—
215C42—141no—
216C42C22×D711yescharacteristic
217C2×C7—81no—
218C2×C7#53711yescharacteristic
219C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C7#57511yescharacteristic
221D7—41no—
222D7—82no—
223C23—21no—
224C23—71no—
225C23—142no—
226Q8—42no—
227Q8—141no—
228Q8—141no—
229D4—21no—
230D4—21no—
231D4—142no—
232D4—142no—
233D4—282no—
234D4—284no—
235D4—282no—
236D4—282no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—71no—
241C2×C4—71no—
242C2×C4—71no—
243C2×C4—142no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—281no—
249C2×C4—284no—
250C2×C4D4×D711yescharacteristic
251C2×C4D4×D711yescharacteristic
252C2×C4C23×D711yescharacteristic
253C8—22no—
254C8—142no—
255C7#325111yescharacteristic, complemented, Sylow
256C22—41no—
257C22—41no—
258C22—41no—
259C22—141no—
260C22—141no—
261C22—141no—
262C22—282no—
263C22—282no—
264C22—282no—
265C22C2×D4×D711yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—141no—
270C4—141no—
271C4—141no—
272C4—281no—
273C4—281no—
274C4#647412yes—
275C4#655111yescharacteristic
276C4#655311yescharacteristic
277C2—81no—
278C2—281no—
279C2—562no—
280C2#9032011yescharacteristic
281C2C2×#647411yescharacteristic
282C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#474C2
socleC22×C7C22×D4
hypercenter#474D7
hypocenterC7#3251
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90248C2semidirect
3. C7×#474C2semidirect
4. #89866C2semidirect
5. #89787C2semidirect
6. #89663C2semidirect
7. #89658C2semidirect
8. #89620C2semidirect
9. #89609C2semidirect
10. #89559C2semidirect
11. #89554C2semidirect
87. #474D7semidirect
255. C7#3251semidirect

Maximal subgroups

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

subgroupindex
#904162
#902482
C7×#4742
#898662
#897872
#896632
#896582
#896202
#896092
#895592
#895542
#32517