Finite Group Databasebeta

← #11731753

Subgroups — #11731753

Order 896.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6435C211yesmaximal, characteristic
2C2×C4×Dic14C211yesmaximal, characteristic, complemented
3C7×#383C211yesmaximal, characteristic, complemented
4#89446C211yesmaximal, characteristic
5#89445C211yesmaximal, characteristic
6#89442C212yesmaximal
7#89440C212yesmaximal
8#89433C211yesmaximal, characteristic, complemented
9#89428C211yesmaximal, characteristic, complemented
10#89425C212yesmaximal, complemented
11#89425C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×(C2×D7)⋊C4C2211yescharacteristic
16C2×(C2×D7)⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×C4×Dic7C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6435—24no—
28C4×Dic14—24no—
29C7×#95C2212yes—
30C7×#95C2212yes—
31C7×#95C2211yescharacteristic
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2599—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—22no—
51C2×C4×D7—22no—
52C2×Dic14—22no—
53C2×Dic14—24no—
54C42×C7D414yes—
55(C2×D7)⋊C4—24no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—24no—
59C4⋊Dic7—22no—
60C4⋊Dic7—24no—
61C4⋊Dic7—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65C4×Dic7—24no—
66C4×Dic7—22no—
67C2×C4×Q8—71no—
68C2×#117—71no—
69#398—71no—
70#397—71no—
71#395—71no—
72#393—71no—
73#388—72no—
74#387—72no—
75#387—72no—
76#385—72no—
77#383D711yescharacteristic, complemented
78C23×C7C2411yescharacteristic
79C22×D7—21no—
80C22×D7—22no—
81C22×D7—21no—
82C22×D7—22no—
83C22×D7#4112yes—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×C4×C7C2×D412yes—
96C2×C4×C7C2×D412yes—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7#4112yes—
109C2×Dic7#4114yes—
110C4×D7—28no—
111Dic14—28no—
112C22×Q8—71no—
113C23×C4—71no—
114C4×Q8—144no—
115#117—144no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—71no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C42—71no—
127C2×C42—72no—
128C2×C42C2×D711yescharacteristic
129#95—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95C2×D712yes—
134#95C2×D712yes—
135#95C2×D711yescharacteristic
136#95C2×D711yescharacteristic
137C22×C7C22×D411yescharacteristic
138C22×C7C2×#4111yescharacteristic
139C22×C7C2×#4112yes—
140C22×C7C2×#4111yescharacteristic
141C22×C72−1+412yes—
142C2×D7—22no—
143C2×D7—24no—
144C2×D7—41no—
145C2×D7—41no—
146C2×D7—42no—
147C4×C7—22no—
148C4×C7—22no—
149C4×C7—22no—
150C4×C7—42no—
151C4×C7—42no—
152Dic7—24no—
153Dic7—22no—
154Dic7—41no—
155Dic7—42no—
156Dic7—41no—
157C24—71no—
158C2×Q8—144no—
159C2×Q8—142no—
160C22×C4—72no—
161C22×C4—71no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4—142no—
166C22×C4—142no—
167C22×C4—142no—
168C22×C4C22×D712yes—
169C22×C4C22×D712yes—
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D711yescharacteristic
172C22×C4C22×D711yescharacteristic
173C4⋊C4—144no—
174C4⋊C4—144no—
175C4⋊C4—142no—
176C4⋊C4—142no—
177C4⋊C4—144no—
178C4⋊C4—142no—
179C22⋊C4—144no—
180C22⋊C4—144no—
181C22⋊C4—142no—
182C22⋊C4—142no—
183C42—142no—
184C42—144no—
185C42D2814yes—
186C2×C7#52511yescharacteristic
187C2×C7#52511yescharacteristic
188C2×C7#53611yescharacteristic
189C2×C7#55212yes—
190C2×C7#55612yes—
191D7—42no—
192C23—72no—
193C23—141no—
194C23—142no—
195C23—141no—
196C23—142no—
197C23C23×D711yescharacteristic
198Q8—148no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—74no—
210C2×C4—72no—
211C2×C4—148no—
212C2×C4—142no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4C2×D2812yes—
224C2×C4C2×D2812yes—
225C2×C4C2×D2812yes—
226C7#259911yescharacteristic, complemented, Sylow
227C22—144no—
228C22—142no—
229C22—282no—
230C22—281no—
231C22—281no—
232C22C22×D2811yescharacteristic
233C22D7×#4111yescharacteristic
234C22D7×#4112yes—
235C22D7×#4111yescharacteristic
236C22#655412yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—142no—
243C4—144no—
244C4—281no—
245C4—282no—
246C4—281no—
247C2—282no—
248C2#9021511yescharacteristic
249C2#9021511yescharacteristic
250C2#9021711yescharacteristic
251C2#9039312yes—
252C2#9039912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#383C2
prime radicalC23×D7C23
hypercenter#383D7
hypocenterC7#2599
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic14C2semidirect
3. C7×#383C2semidirect
8. #89433C2semidirect
9. #89428C2semidirect
10. #89425C2semidirect
11. #89425C2semidirect
77. #383D7semidirect
226. C7#2599semidirect

Maximal subgroups

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

subgroupindex
C2×#64352
C2×C4×Dic142
C7×#3832
#894462
#894452
#894422
#894402
#894332
#894282
#894252
#25997