Finite Group Databasebeta

← #11735801

Subgroups — #11735801

Order 896.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#6456C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic, complemented
5#89751C211yesmaximal, characteristic, complemented
6#89445C211yesmaximal, characteristic, complemented
7#89440C212yesmaximal, complemented
8#89439C212yesmaximal, complemented
9#89437C212yesmaximal
10D7×#95C211yesmaximal, characteristic, complemented
11#89427C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C22×Dic14C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28#6460—24no—
29#6456—24no—
30C7×#95C2212yes—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2564—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C2×D28—22no—
50C2×D28—22no—
51C2×D28Q814yescomplemented
52C2×C4×D7—24no—
53C2×C4×D7—22no—
54C2×C4×D7—24no—
55C2×C4×D7—42no—
56C2×C4×D7—42no—
57C2×C4×D7—42no—
58C2×C4×D7D414yes—
59C2×Dic14—22no—
60C2×Dic14—42no—
61C7×C4⋊C4—22no—
62C7×C4⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—42no—
68C4⋊Dic7—24no—
69Dic7⋊C4—24no—
70Dic7⋊C4—42no—
71C4×Dic7—22no—
72C2×C22⋊Q8—71no—
73C2×C4×D4—71no—
74#400D711yescharacteristic, complemented
75#399—71no—
76#395—71no—
77#393—71no—
78#387—72no—
79#386—72no—
80#385—72no—
81#380—72no—
82C2×#95—71no—
83C23×C7C2411yescharacteristic
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—21no—
87C22×D7—22no—
88C22×D7—21no—
89C22×D7—22no—
90C22×D7C2×D412yes—
91C22×D7C2×Q812yes—
92C22×D7C2×Q812yes—
93C22×D7#4112yes—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7C2×D412yes—
106C2×C4×C7C2×Q812yes—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—21no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7C2×D412yes—
116C2×Dic7#4114yes—
117D28—28no—
118C4×D7—44no—
119C4×D7—48no—
120C4×D7—44no—
121Dic14—44no—
122C22×Q8—71no—
123C22×D4—71no—
124C23×C4—71no—
125C23×C4—72no—
126C23×C4—71no—
127C22⋊Q8—144no—
128C4×D4—144no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C22⋊C4—71no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C42—71no—
140#95—72no—
141#95—71no—
142#95—72no—
143#95—72no—
144#95C2×D711yescharacteristic
145#95C2×D712yes—
146#95C2×D712yes—
147C22×C7C22×D411yescharacteristic
148C22×C7C22×Q811yescharacteristic
149C22×C7C2×#4111yescharacteristic
150C22×C7C2×#4112yes—
151C22×C7H(5,𝔽2)12yes—
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—24no—
155C2×D7—24no—
156C2×D7—22no—
157C2×D7—24no—
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—42no—
161C2×D7C22⋊Q814yes—
162C4×C7—22no—
163C4×C7—22no—
164C4×C7—42no—
165C4×C7—41no—
166C4×C7—42no—
167Dic7—22no—
168Dic7—24no—
169Dic7—41no—
170Dic7—42no—
171C24—71no—
172C24—71no—
173C2×Q8—142no—
174C2×Q8—282no—
175C2×D4—74no—
176C2×D4—142no—
177C2×D4—142no—
178C22×C4—72no—
179C22×C4—71no—
180C22×C4—74no—
181C22×C4—71no—
182C22×C4—72no—
183C22×C4—142no—
184C22×C4—144no—
185C22×C4—144no—
186C22×C4—282no—
187C22×C4—282no—
188C22×C4—282no—
189C22×C4C22×D712yes—
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D711yescharacteristic
193C22×C4C22×D711yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—144no—
197C4⋊C4—144no—
198C4⋊C4—282no—
199C22⋊C4—144no—
200C22⋊C4—142no—
201C22⋊C4—144no—
202C22⋊C4—142no—
203C22⋊C4—282no—
204C42—142no—
205C2×C7C2×C22⋊Q811yescharacteristic
206C2×C7#53211yescharacteristic
207C2×C7#53411yescharacteristic
208C2×C7#55112yes—
209C2×C7#55412yes—
210D7—24no—
211D7—42no—
212C23—72no—
213C23—72no—
214C23—72no—
215C23—72no—
216C23—142no—
217C23—142no—
218C23—141no—
219C23—142no—
220C23—141no—
221C23—142no—
222C23C23×D711yescharacteristic
223Q8—284no—
224D4—148no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—72no—
237C2×C4—74no—
238C2×C4—142no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—141no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—284no—
247C2×C4—288no—
248C2×C4—284no—
249C2×C4D4×D712yes—
250C2×C4Q8×D712yes—
251C7#256411yescharacteristic, complemented, Sylow
252C22—74no—
253C22—144no—
254C22—144no—
255C22—142no—
256C22—142no—
257C22—144no—
258C22—142no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22C2×D4×D711yescharacteristic
263C22C2×Q8×D711yescharacteristic
264C22D7×#4112yes—
265C22D7×#4111yescharacteristic
266C22#655312yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—41no—
271C4—42no—
272C4—144no—
273C4—142no—
274C4—282no—
275C4—281no—
276C2—144no—
277C2—282no—
278C2D7×C22⋊Q811yescharacteristic
279C2#9034411yescharacteristic
280C2#9035911yescharacteristic
281C2#9038112yes—
282C2#9039612yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC22×D28C22
hypercenter#400D7
hypocenterC7#2564
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
3. C7×#400C2semidirect
4. #89764C2semidirect
5. #89751C2semidirect
6. #89445C2semidirect
7. #89440C2semidirect
8. #89439C2semidirect
10. D7×#95C2semidirect
11. #89427C2semidirect
51. C2×D28Q8semidirect
74. #400D7semidirect
251. C7#2564semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64562
C7×#4002
#897642
#897512
#894452
#894402
#894392
#894372
D7×#952
#894272
#25647