Finite Group Databasebeta

← #11726508

Subgroups — #11726508

Order 896.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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
1D7×#117C211yesmaximal, characteristic
2#89898C211yesmaximal, characteristic
3C4×(C2×D7)⋊C4C211yesmaximal, characteristic
4C7×#339C211yesmaximal, characteristic, complemented
5#89350C211yesmaximal, characteristic, complemented
6#89327C211yesmaximal, characteristic, complemented
7#89318C211yesmaximal, characteristic, complemented
8C22×C4×D7C2211yescharacteristic
9C2×C7×M16C2211yescharacteristic
10C7×#117C2211yescharacteristic
11C2×C42×C7C2211yescharacteristic
12C2×(C2×D7)⋊C4—21no—
13#6489C2211yescharacteristic
14C2×C4×Dic7C2211yescharacteristic
15C2×#1136C2211yescharacteristic
16#6455C411yescharacteristic, complemented
17#6455C411yescharacteristic, complemented
18D7×C4⋊C4C411yescharacteristic, complemented
19D7×C4⋊C4C411yescharacteristic, complemented
20D7×C22⋊C4—21no—
21#6440—21no—
22#6435—21no—
23C42×D7—21no—
24C7⋊#95—21no—
25(C2×D7)⋊C8—21no—
26(C2×D7)⋊C8—21no—
27#1826—71nomaximal, Sylow
28C23×D7D411yescharacteristic
29C22×C4×C7—21no—
30C22×C4×C7C2311yescharacteristic
31C22×Dic7—21no—
32C22×Dic7D411yescharacteristic
33C2×C4×D7—21no—
34C2×C4×D7—21no—
35C2×C4×D7—21no—
36C2×C4×D7—21no—
37C2×C4×D7—21no—
38C2×C4×D7—21no—
39C2×C4×D7C2×C411yescharacteristic
40C2×C4×D7C2×C411yescharacteristic
41C2×C4×D7D411yescharacteristic
42C2×C4×D7D411yescharacteristic
43C7×M16—21no—
44C7×M16—21no—
45C2×C7×C8—21no—
46C7×C4⋊C4C2×C411yescharacteristic
47C7×C4⋊C4C2×C411yescharacteristic
48C7×C22⋊C4—21no—
49C42×C7—21no—
50C42×C7—21no—
51C42×C7—21no—
52C22⋊Dic7—21no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—21no—
55(C2×D7)⋊C4—21no—
56C4⋊Dic7C2×C411yescharacteristic
57C4⋊Dic7C2×C411yescharacteristic
58Dic7⋊C4—21no—
59Dic7⋊C4—21no—
60C4×Dic7—21no—
61C4×Dic7—21no—
62C4×Dic7—21no—
63C4×Dic7—21no—
64C4×Dic7—21no—
65#1136—21no—
66#1136—21no—
67C2×C7⋊C8—21no—
68C2×#117—71no—
69#407—71no—
70C4×C22⋊C4—71no—
71#339—71no—
72#339—71no—
73#339—71no—
74#339D711yescharacteristic, complemented
75C23×C7C2×D411yescharacteristic
76C22×D7—21no—
77C22×D7—21no—
78C22×D7—22no—
79C22×D7—41no—
80C22×D7C22⋊C411yescharacteristic
81C22×D7C22⋊C411yescharacteristic
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—41no—
86C2×C4×C7—41no—
87C2×C4×C7C22×C411yescharacteristic
88C2×C4×C7C2×D411yescharacteristic
89C2×C4×C7#4111yescharacteristic
90C2×C4×C7#4111yescharacteristic
91C2×C4×C7#4111yescharacteristic
92C2×C4×C7#4111yescharacteristic
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—41no—
99C2×Dic7—41no—
100C2×Dic7C22⋊C411yescharacteristic
101C2×Dic7C22⋊C411yescharacteristic
102C4×D7—21no—
103C4×D7—21no—
104C4×D7—21no—
105C4×D7—21no—
106C4×D7—21no—
107C4×D7—21no—
108C4×D7—21no—
109C4×D7—21no—
110C4×D7—42no—
111C4×D7—42no—
112C7×C8—41no—
113C7⋊C8—41no—
114C23×C4—71no—
115C2×M16—71no—
116C2×M16C2×D711yescharacteristic
117#117—71no—
118#117—71no—
119#117—71no—
120#117—141no—
121#117—141no—
122#117C2×D711yescharacteristic
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C22⋊C4—141no—
126C2×C22⋊C4—141no—
127C2×C42—71no—
128C2×C42—141no—
129C2×C42C2×D711yescharacteristic
130C22⋊C8—141no—
131C22⋊C8—141no—
132#95—141no—
133C22×C7—21no—
134C22×C7—21no—
135C22×C7C2×C22⋊C411yescharacteristic
136C22×C7#12311yescharacteristic
137C22×C7#12311yescharacteristic
138C2×D7—21no—
139C2×D7—21no—
140C2×D7—21no—
141C2×D7—21no—
142C2×D7—21no—
143C2×D7—42no—
144C2×D7—42no—
145C2×D7#10412yes—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7#11711yescharacteristic
150C4×C7#11711yescharacteristic
151C4×C7#12311yescharacteristic
152C4×C7#12311yescharacteristic
153Dic7—21no—
154Dic7—21no—
155Dic7—42no—
156Dic7—41no—
157Dic7—41no—
158C24—71no—
159C22×C4—21no—
160C22×C4—71no—
161C22×C4—71no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—71no—
165C22×C4—141no—
166C22×C4—141no—
167C22×C4—141no—
168C22×C4—141no—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4—141no—
172C22×C4C22×D711yescharacteristic
173M16—21no—
174M16—21no—
175M16—141no—
176M16—141no—
177C2×C8—21no—
178C2×C8—141no—
179C4⋊C4—71no—
180C4⋊C4—71no—
181C4⋊C4—141no—
182C4⋊C4—141no—
183C4⋊C4C4×D711yescharacteristic
184C4⋊C4C4×D711yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—141no—
187C22⋊C4—141no—
188C22⋊C4—144no—
189C22⋊C4—141no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—141no—
194C42—141no—
195C42—141no—
196C42—141no—
197C42—141no—
198C2×C7—21no—
199C2×C7—21no—
200C2×C7#38111yescharacteristic
201C2×C7C2×#10411yescharacteristic
202C2×C7#42111yescharacteristic
203D7—22no—
204D7—41no—
205C23—71no—
206C23—71no—
207C23—142no—
208C23—141no—
209C23—141no—
210C23—281no—
211C23D4×D711yescharacteristic
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—71no—
218C2×C4—71no—
219C2×C4—141no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—141no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4—141no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—141no—
232C2×C4—282no—
233C2×C4—282no—
234C2×C4—281no—
235C2×C4—281no—
236C2×C4C2×C4×D711yescharacteristic
237C2×C4#115711yescharacteristic
238C2×C4#115711yescharacteristic
239C2×C4D4×D711yescharacteristic
240C2×C4#115911yescharacteristic
241C2×C4#116111yescharacteristic
242C8—41no—
243C8—281no—
244C7#182611yescharacteristic, complemented, Sylow
245C22—21no—
246C22—21no—
247C22—72no—
248C22—141no—
249C22—141no—
250C22—141no—
251C22—141no—
252C22—141no—
253C22—282no—
254C22—282no—
255C22D7×C22⋊C411yescharacteristic
256C22#644611yescharacteristic
257C22#645711yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—141no—
262C4—141no—
263C4—282no—
264C4—281no—
265C4—281no—
266C4#643511yescharacteristic
267C4#644611yescharacteristic
268C4#645511yescharacteristic
269C4#645711yescharacteristic
270C2—21no—
271C2—21no—
272C2—142no—
273C2—281no—
274C2#8943611yescharacteristic
275C2D7×#10411yescharacteristic
276C2#8959311yescharacteristic
subgroupG/H
centerC2×C4D4×D7
commutatorC2×C4×C7C22×C4
FrattiniC22×C4C22×D7
FittingC7×#339C2
socleC22×C7C2×C22⋊C4
prime radicalC23×D7D4
cosocleC22×C4×C7C23
hypercenter#339D7
hypocenterC7#1826
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. C7×#339C2semidirect
5. #89350C2semidirect
6. #89327C2semidirect
7. #89318C2semidirect
16. #6455C4semidirect
17. #6455C4semidirect
18. D7×C4⋊C4C4semidirect
19. D7×C4⋊C4C4semidirect
74. #339D7semidirect
244. C7#1826semidirect

Maximal subgroups

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

subgroupindex
D7×#1172
#898982
C4×(C2×D7)⋊C42
C7×#3392
#893502
#893272
#893182
#18267