Finite Group Databasebeta

← #11735535

Subgroups — #11735535

Order 896.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6460C211yesmaximal, characteristic, complemented
2C2×#6447C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic, complemented
5#89763C212yesmaximal, complemented
6#89756C212yesmaximal, complemented
7#89733C212yesmaximal, complemented
8#89731C212yesmaximal, complemented
9#89719C211yesmaximal, characteristic, complemented
10#89445C211yesmaximal, characteristic, complemented
11#89436C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C22×Dic14C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2212yes—
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2212yes—
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30#6460—24no—
31#6447—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2816—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—42no—
48C2×C22⋊D7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C2×D28—22no—
54C2×D28—42no—
55C2×C4×D7—42no—
56C2×C4×D7D414yes—
57C2×Dic14—22no—
58C2×Dic14—42no—
59C7×C4⋊C4—22no—
60C7×C4⋊C4—42no—
61C7×C22⋊C4—22no—
62C7×C22⋊C4—42no—
63C22⋊Dic7—42no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—42no—
67(C2×D7)⋊C4—42no—
68(C2×D7)⋊C4—42no—
69C4⋊Dic7—24no—
70Dic7⋊C4—22no—
71Dic7⋊C4—42no—
72C2×C22⋊Q8—71no—
73C2×#121—71no—
74#399—71no—
75#399D711yescharacteristic, complemented
76#398—72no—
77#394—72no—
78#394—72no—
79#394—72no—
80#393—71no—
81#393—71no—
82#381—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7C2411yescharacteristic
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—21no—
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7C2×D412yes—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7C2×D412yes—
109C22⋊D7—44no—
110C22⋊D7—44no—
111C2×Dic7—22no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7C2×D412yes—
121D28—44no—
122C4×D7—44no—
123Dic14—44no—
124C22×Q8—71no—
125C22×D4—71no—
126C22×D4—71no—
127C22×D4—71no—
128C23×C4—71no—
129C22⋊Q8—144no—
130#121—144no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C4⋊C4C2×D712yes—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—72no—
142C2×C22⋊C4C2×D711yescharacteristic
143C2×C22⋊C4C2×D712yes—
144#95—71no—
145#95—72no—
146#95—72no—
147#95—72no—
148#95C2×D711yescharacteristic
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C22×C7H(5,𝔽2)12yes—
156C22×C7H(5,𝔽2)12yes—
157C22×C72−1+411yescharacteristic
158C22×C72−1+411yescharacteristic
159C2×D7—22no—
160C2×D7—22no—
161C2×D7—22no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—42no—
168C2×D7—42no—
169C4×C7—22no—
170C4×C7—41no—
171C4×C7—42no—
172C4×C7—42no—
173Dic7—22no—
174Dic7—42no—
175Dic7—41no—
176Dic7—42no—
177C24—71no—
178C24—71no—
179C24C22×D711yescharacteristic
180C2×Q8—142no—
181C2×Q8—282no—
182C2×D4—142no—
183C2×D4—142no—
184C2×D4—142no—
185C2×D4—282no—
186C2×D4—282no—
187C2×D4—282no—
188C22×C4—71no—
189C22×C4—74no—
190C22×C4—72no—
191C22×C4—72no—
192C22×C4—71no—
193C22×C4—282no—
194C22×C4C22×D712yes—
195C22×C4C22×D711yescharacteristic
196C22×C4C22×D712yes—
197C22×C4C22×D711yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—142no—
201C4⋊C4—144no—
202C4⋊C4—282no—
203C22⋊C4—22no—
204C22⋊C4—42no—
205C22⋊C4—144no—
206C22⋊C4—142no—
207C22⋊C4—282no—
208C22⋊C4—282no—
209C22⋊C4—282no—
210C22⋊C4—282no—
211C2×C7—81no—
212C2×C7#53411yescharacteristic
213C2×C7#53511yescharacteristic
214C2×C7#53611yescharacteristic
215C2×C7#56212yes—
216C2×C7#56212yes—
217D7—42no—
218D7—81no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—72no—
225C23—142no—
226C23—141no—
227C23—142no—
228C23—141no—
229C23—141no—
230C23—142no—
231C23—141no—
232C23—142no—
233C23—141no—
234C23C23×D711yescharacteristic
235Q8—284no—
236D4—284no—
237D4—284no—
238D4—284no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—72no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—284no—
259C2×C4D4×D712yes—
260C7#281611yescharacteristic, complemented, Sylow
261C22—41no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—142no—
267C22—142no—
268C22—142no—
269C22—282no—
270C22—282no—
271C22—281no—
272C22—281no—
273C22—282no—
274C22—281no—
275C22—282no—
276C22C2×D4×D711yescharacteristic
277C22#654812yes—
278C22#655111yescharacteristic
279C22#655312yes—
280C22#655411yescharacteristic
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—142no—
286C4—281no—
287C4—282no—
288C4—282no—
289C2—81no—
290C2—282no—
291C2—561no—
292C2#9031911yescharacteristic
293C2#9033812yes—
294C2#9034411yescharacteristic
295C2#9035812yes—
296C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC2×#6447C2
hypercenter#399D7
hypocenterC7#2816
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
3. C7×#399C2semidirect
4. #89764C2semidirect
5. #89763C2semidirect
6. #89756C2semidirect
7. #89733C2semidirect
8. #89731C2semidirect
9. #89719C2semidirect
10. #89445C2semidirect
11. #89436C2semidirect
75. #399D7semidirect
260. C7#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64472
C7×#3992
#897642
#897632
#897562
#897332
#897312
#897192
#894452
#894362
#28167