Finite Group Databasebeta

← #11735536

Subgroups — #11735536

Order 896.

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

normal subgroups only

This group has 464 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×#6453C211yesmaximal, characteristic, complemented
2C2×#6448C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic, complemented
4#89766C212yesmaximal, complemented
5#89764C211yesmaximal, characteristic, complemented
6#89751C212yesmaximal, complemented
7#89731C212yesmaximal
8#89725C211yesmaximal, characteristic, complemented
9#89720C212yesmaximal, complemented
10#89446C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30#6453—24no—
31#6448—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2820—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7—22no—
45C2×C22⋊D7—42no—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—42no—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—42no—
55C7×C22⋊C4—22no—
56C7×C22⋊C4—42no—
57C22⋊Dic7—22no—
58C22⋊Dic7—42no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—42no—
62(C2×D7)⋊C4—42no—
63C4⋊Dic7—22no—
64C4⋊Dic7—24no—
65Dic7⋊C4—24no—
66Dic7⋊C4—42no—
67Dic7⋊C4—42no—
68C4×Dic7D414yes—
69C2×#125—71no—
70C2×#124—71no—
71#399—71no—
72#399D711yescharacteristic, complemented
73#397—71no—
74#397—71no—
75#396—72no—
76#396—72no—
77#395—72no—
78#394—72no—
79#382—71no—
80C23×C7—21no—
81C23×C7—21no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7C2411yescharacteristic
86C22×D7—21no—
87C22×D7—21no—
88C22×D7—22no—
89C22×D7—21no—
90C22×D7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7C2×D412yes—
101C22⋊D7—44no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7C2×D412yes—
114C2×Dic7C2×D412yes—
115Dic14—44no—
116C22×Q8—71no—
117C22×D4—71no—
118#125—144no—
119#124—144no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4C2×D711yescharacteristic
126C2×C4⋊C4C2×D712yes—
127C2×C22⋊C4—71no—
128C2×C22⋊C4—71no—
129C2×C22⋊C4—71no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4C2×D711yescharacteristic
134C2×C22⋊C4C2×D712yes—
135C2×C42—71no—
136#95—72no—
137#95—71no—
138#95—72no—
139#95—72no—
140#95C2×D711yescharacteristic
141C22×C7—41no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7—42no—
145C22×C7—42no—
146C22×C7C22×D411yescharacteristic
147C22×C7H(5,𝔽2)12yes—
148C22×C7H(5,𝔽2)11yescharacteristic
149C22×C72−1+411yescharacteristic
150C22×C72−1+412yes—
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—42no—
155C2×D7—42no—
156C4×C7—22no—
157C4×C7—41no—
158C4×C7—42no—
159C4×C7—42no—
160Dic7—22no—
161Dic7—22no—
162Dic7—41no—
163Dic7—42no—
164Dic7—42no—
165C24—71no—
166C24C22×D711yescharacteristic
167C2×Q8—142no—
168C2×Q8—282no—
169C2×D4—142no—
170C2×D4—282no—
171C22×C4—71no—
172C22×C4—71no—
173C22×C4—72no—
174C22×C4—72no—
175C22×C4—71no—
176C22×C4C22×D712yes—
177C22×C4C22×D711yescharacteristic
178C22×C4C22×D712yes—
179C22×C4C22×D711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—144no—
183C4⋊C4—144no—
184C4⋊C4—142no—
185C4⋊C4—282no—
186C4⋊C4—282no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C22⋊C4—142no—
190C22⋊C4—142no—
191C22⋊C4—142no—
192C22⋊C4—282no—
193C22⋊C4—282no—
194C22⋊C4—282no—
195C42—74no—
196C2×C7—81no—
197C2×C7#53511yescharacteristic
198C2×C7#53611yescharacteristic
199C2×C7#53711yescharacteristic
200C2×C7#56212yes—
201C2×C7#56312yes—
202D7—81no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—141no—
209C23—142no—
210C23—141no—
211C23—141no—
212C23—142no—
213C23C23×D711yescharacteristic
214Q8—284no—
215D4—284no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—72no—
226C2×C4—72no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4D4×D712yes—
239C7#282011yescharacteristic, complemented, Sylow
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—281no—
246C22—281no—
247C22—282no—
248C22—281no—
249C22—282no—
250C22C2×D4×D711yescharacteristic
251C22#654812yes—
252C22#655111yescharacteristic
253C22#655311yescharacteristic
254C22#655412yes—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—142no—
260C4—142no—
261C4—281no—
262C4—282no—
263C4—282no—
264C2—81no—
265C2—561no—
266C2#9032011yescharacteristic
267C2#9033712yes—
268C2#9033911yescharacteristic
269C2#9035812yes—
270C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2820
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6453C2semidirect
3. C7×#399C2semidirect
4. #89766C2semidirect
5. #89764C2semidirect
6. #89751C2semidirect
8. #89725C2semidirect
9. #89720C2semidirect
10. #89446C2semidirect
11. #89422C2semidirect
72. #399D7semidirect
239. C7#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64482
C7×#3992
#897662
#897642
#897512
#897312
#897252
#897202
#894462
#894222
#28207