Finite Group Databasebeta

← #11735537

Subgroups — #11735537

Order 896.

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

normal subgroups only

This group has 430 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×#6451C211yesmaximal, characteristic, complemented
2C2×#6442C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic
4#89756C212yesmaximal, complemented
5#89754C211yesmaximal, characteristic, complemented
6#89751C212yesmaximal, complemented
7#89724C212yesmaximal
8#89721C211yesmaximal, characteristic
9#89719C212yesmaximal
10#89431C211yesmaximal, characteristic, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×Dic14C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6451—24no—
28#6442—24no—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2821—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×Dic14—24no—
50C2×Dic14—42no—
51C2×Dic14—42no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—42no—
54C7×C22⋊C4—22no—
55C7×C22⋊C4—42no—
56C22⋊Dic7—22no—
57C22⋊Dic7—22no—
58C22⋊Dic7—42no—
59C4⋊Dic7—22no—
60C4⋊Dic7—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63Dic7⋊C4—24no—
64Dic7⋊C4—42no—
65C4×Dic7D414yes—
66C2×C4⋊Q8—71no—
67C2×#126—71no—
68#400—71no—
69#400—71no—
70#399—71no—
71#399D711yescharacteristic
72#398—72no—
73#397—72no—
74#395—72no—
75#393—72no—
76#383—71no—
77C23×C7—21no—
78C23×C7—22no—
79C23×C7—21no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7C2×D412yes—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7C2×D412yes—
108C2×Dic7C2×D412yes—
109Dic14—48no—
110C22×Q8—72no—
111C4⋊Q8—144no—
112#126—144no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D712yes—
121C2×C22⋊C4—71no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127#95—72no—
128#95—72no—
129#95—71no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95C2×D711yescharacteristic
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7—42no—
139C22×C7—42no—
140C22×C7C22×D411yescharacteristic
141C22×C7H(5,𝔽2)12yes—
142C22×C72−1+411yescharacteristic
143C22×C72−1+411yescharacteristic
144C22×C72−1+412yes—
145C4×C7—22no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—42no—
149Dic7—22no—
150Dic7—22no—
151Dic7—42no—
152Dic7—41no—
153Dic7—41no—
154Dic7—42no—
155C24C22×D711yescharacteristic
156C2×Q8—144no—
157C2×Q8—282no—
158C2×Q8—282no—
159C22×C4—71no—
160C22×C4—71no—
161C22×C4—72no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4C22×D712yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—142no—
172C4⋊C4—144no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—282no—
177C22⋊C4—22no—
178C22⋊C4—42no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C22⋊C4—282no—
182C42—74no—
183C2×C7—81no—
184C2×C7#53511yescharacteristic
185C2×C7#53611yescharacteristic
186C2×C7#53611yescharacteristic
187C2×C7#56312yes—
188C2×C7#56312yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23C23×D711yescharacteristic
195Q8—288no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—142no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—141no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—141no—
217C2×C4—142no—
218C2×C4—141no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4D4×D712yes—
222C7#282111yescharacteristic, complemented, Sylow
223C22—41no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—42no—
228C22C2×D4×D711yescharacteristic
229C22#654812yes—
230C22#655111yescharacteristic
231C22#655411yescharacteristic
232C22#655412yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—142no—
238C4—142no—
239C4—282no—
240C4—281no—
241C4—282no—
242C4—281no—
243C2—81no—
244C2#9031911yescharacteristic
245C2#9033712yes—
246C2#9033911yescharacteristic
247C2#9034212yes—
248C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC24×C7C23
hypercenter#399D7
hypocenterC7#2821
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
4. #89756C2semidirect
5. #89754C2semidirect
6. #89751C2semidirect
10. #89431C2semidirect
11. #89423C2semidirect
222. C7#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64422
C7×#3992
#897562
#897542
#897512
#897242
#897212
#897192
#894312
#894232
#28217