Finite Group Databasebeta

← #11734636

Subgroups — #11734636

Order 896.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×Q8×Dic7C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic
4#90001C211yesmaximal, characteristic
5#89758C211yesmaximal, characteristic
6#89753C212yesmaximal
7#89744C212yesmaximal
8#89431C211yesmaximal, characteristic
9#89427C212yesmaximal
10#89422C211yesmaximal, characteristic
11#89418C212yesmaximal
12C22×C7×Q8C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16Q8×Dic7—24no—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2211yescharacteristic
27#6450—24no—
28C7×#95C2212yes—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2581—71nomaximal, Sylow
38C2×C7×Q8—22no—
39C2×C7×Q8—24no—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14Q814yescomplemented
54C7×C4⋊C4—22no—
55C7×C4⋊C4—24no—
56C4⋊Dic7—22no—
57C4⋊Dic7—24no—
58Dic7⋊C4—22no—
59Dic7⋊C4—24no—
60Dic7⋊C4—22no—
61Dic7⋊C4—24no—
62C4×Dic7—22no—
63C4×Dic7—22no—
64C4×Dic7—24no—
65C4×Dic7D414yes—
66C2×C4×Q8—71no—
67C2×C4×Q8—71no—
68#400—71no—
69#400—71no—
70#395—71no—
71#395D711yescharacteristic
72#391—72no—
73#389—72no—
74#387—72no—
75#387—72no—
76#382—71no—
77C23×C7C2411yescharacteristic
78C7×Q8—28no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—22no—
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×C7C2×D412yes—
90C2×C4×C7C2×Q812yes—
91C2×C4×C7#4114yes—
92C2×Dic7—21no—
93C2×Dic7—21no—
94C2×Dic7—22no—
95C2×Dic7—21no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—24no—
101C2×Dic7—22no—
102C2×Dic7C2×D412yes—
103C2×Dic7C2×D412yes—
104C2×Dic7C2×Q812yes—
105C2×Dic7C2×Q812yes—
106C2×Dic7#4112yes—
107Dic14—28no—
108C22×Q8—71no—
109C22×Q8C2×D711yescharacteristic
110C4×Q8—144no—
111C4×Q8—144no—
112C2×C4⋊C4—71no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D712yes—
120C2×C42—71no—
121C2×C42—71no—
122C2×C42—71no—
123C2×C42—72no—
124#95—72no—
125#95—71no—
126#95—72no—
127#95—72no—
128#95—72no—
129#95—72no—
130#95—72no—
131#95C2×D711yescharacteristic
132#95C2×D712yes—
133C22×C7C22×D411yescharacteristic
134C22×C7C22×Q811yescharacteristic
135C22×C7C2×#4112yes—
136C22×C7C2×#4111yescharacteristic
137C22×C72−1+412yes—
138C4×C7—22no—
139C4×C7—22no—
140C4×C7—24no—
141C4×C7—41no—
142C4×C7—42no—
143Dic7—22no—
144Dic7—22no—
145Dic7—41no—
146Dic7—42no—
147Dic7—42no—
148Dic7C22⋊Q814yes—
149C2×Q8—22no—
150C2×Q8—24no—
151C2×Q8—74no—
152C2×Q8—142no—
153C2×Q8—142no—
154C22×C4—71no—
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—72no—
158C22×C4—72no—
159C22×C4—71no—
160C22×C4C22×D712yes—
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D711yescharacteristic
163C22×C4C22×D712yes—
164C22×C4C22×D711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—144no—
168C4⋊C4—142no—
169C4⋊C4—144no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—144no—
173C42—74no—
174C42—142no—
175C42—144no—
176C42—142no—
177C2×C7C2×C22⋊Q811yescharacteristic
178C2×C7#52511yescharacteristic
179C2×C7#53611yescharacteristic
180C2×C7#55612yes—
181C2×C7#55712yes—
182C23C23×D711yescharacteristic
183Q8—28no—
184Q8—148no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—72no—
196C2×C4—72no—
197C2×C4—72no—
198C2×C4—72no—
199C2×C4—72no—
200C2×C4—141no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4—142no—
204C2×C4—141no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—144no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4D4×D712yes—
211C2×C4#115914yes—
212C2×C4Q8×D712yes—
213C7#258111yescharacteristic, complemented, Sylow
214C22C2×D4×D711yescharacteristic
215C22C2×#115912yes—
216C22C2×Q8×D711yescharacteristic
217C22#655112yes—
218C22D7×#4111yescharacteristic
219C4—22no—
220C4—24no—
221C4—22no—
222C4—41no—
223C4—42no—
224C4—74no—
225C4—142no—
226C4—142no—
227C4—282no—
228C4—281no—
229C4—282no—
230C2#9027711yescharacteristic
231C2D7×C22⋊Q811yescharacteristic
232C2#9033212yes—
233C2#9041212yes—
234C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2581
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
53. C2×Dic14Q8semidirect
213. C7#2581semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic72
C2×#64502
C7×#3952
#900012
#897582
#897532
#897442
#894312
#894272
#894222
#894182
#25817