Finite Group Databasebeta

← #11735762

Subgroups — #11735762

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×#6461C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C7×#400C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic
5#89758C211yesmaximal, characteristic, complemented
6#89449C211yesmaximal, characteristic
7#89447C212yesmaximal
8#89445C212yesmaximal
9#89429C212yesmaximal, complemented
10#89428C212yesmaximal, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×Dic14C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×(C2×D7)⋊C4C2211yescharacteristic
16C2×(C2×D7)⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6461—24no—
28#6451—24no—
29C7×#95C2211yescharacteristic
30C7×#95C2212yes—
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2821—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×Dic14—24no—
50C2×Dic14—42no—
51C2×Dic14—42no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—22no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—42no—
58(C2×D7)⋊C4—42no—
59C4⋊Dic7—22no—
60C4⋊Dic7—22no—
61C4⋊Dic7—42no—
62Dic7⋊C4—24no—
63Dic7⋊C4—22no—
64Dic7⋊C4—42no—
65C4×Dic7D414yes—
66C2×C4⋊Q8—71no—
67C2×#126—71no—
68#400—71no—
69#400D711yescharacteristic, complemented
70#399—71no—
71#399—71no—
72#398—72no—
73#397—72no—
74#395—72no—
75#393—72no—
76#383—71no—
77C23×C7C2411yescharacteristic
78C22×D7—21no—
79C22×D7—22no—
80C22×D7—21no—
81C22×D7—22no—
82C22×D7—21no—
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7C2×D412yes—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
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—72no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C22⋊C4—71no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C42—71no—
127#95—72no—
128#95—72no—
129#95—71no—
130#95—72no—
131#95—72no—
132#95C2×D711yescharacteristic
133#95C2×D712yes—
134#95C2×D712yes—
135C22×C7C22×D411yescharacteristic
136C22×C7H(5,𝔽2)12yes—
137C22×C72−1+411yescharacteristic
138C22×C72−1+412yes—
139C22×C72−1+411yescharacteristic
140C2×D7—41no—
141C2×D7—41no—
142C2×D7—41no—
143C2×D7—42no—
144C2×D7—42no—
145C4×C7—22no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7—42no—
150Dic7—22no—
151Dic7—22no—
152Dic7—42no—
153Dic7—41no—
154Dic7—42no—
155C24—71no—
156C2×Q8—144no—
157C2×Q8—282no—
158C2×Q8—282no—
159C22×C4—71no—
160C22×C4—71no—
161C22×C4—72no—
162C22×C4—72no—
163C22×C4—71no—
164C22×C4C22×D712yes—
165C22×C4C22×D711yescharacteristic
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D712yes—
168C22×C4C22×D711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—142no—
172C4⋊C4—144no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—282no—
176C4⋊C4—282no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—142no—
180C22⋊C4—282no—
181C22⋊C4—282no—
182C42—74no—
183C2×C7#53511yescharacteristic
184C2×C7#53611yescharacteristic
185C2×C7#53611yescharacteristic
186C2×C7#56312yes—
187C2×C7#56312yes—
188D7—81no—
189C23—141no—
190C23—141no—
191C23—142no—
192C23—142no—
193C23—141no—
194C23C23×D711yescharacteristic
195Q8—288no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—72no—
209C2×C4—72no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4D4×D712yes—
222C7#282111yescharacteristic, complemented, Sylow
223C22—282no—
224C22—282no—
225C22—281no—
226C22—281no—
227C22—281no—
228C22C2×D4×D711yescharacteristic
229C22#655111yescharacteristic
230C22#655312yes—
231C22#655412yes—
232C22#655411yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—142no—
239C4—142no—
240C4—282no—
241C4—282no—
242C4—281no—
243C2—561no—
244C2#9031911yescharacteristic
245C2#9033911yescharacteristic
246C2#9036511yescharacteristic
247C2#9039012yes—
248C2#9040312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2821
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
3. C7×#400C2semidirect
5. #89758C2semidirect
9. #89429C2semidirect
10. #89428C2semidirect
11. #89423C2semidirect
69. #400D7semidirect
222. C7#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#64612
C2×#64512
C7×#4002
#897642
#897582
#894492
#894472
#894452
#894292
#894282
#894232
#28217