Finite Group Databasebeta

← #11734637

Subgroups — #11734637

Order 896.

468 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 161 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×#6452C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic
4#90001C211yesmaximal, characteristic
5#89757C212yesmaximal
6#89755C211yesmaximal, characteristic
7#89744C212yesmaximal
8#89433C211yesmaximal, characteristic
9#89421C211yesmaximal, characteristic
10#89419C212yesmaximal
11#89419C212yesmaximal
12C22×C7×Q8C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15Q8×Dic7—24no—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2212yes—
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30#6452—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2584—71nomaximal, Sylow
38C2×C7×Q8—22no—
39C2×C7×Q8—24no—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C7×C4⋊C4—22no—
52C7×C4⋊C4—24no—
53C4⋊Dic7—22no—
54C4⋊Dic7—22no—
55C4⋊Dic7—24no—
56C4⋊Dic7—42no—
57Dic7⋊C4—22no—
58Dic7⋊C4—24no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4—42no—
63Dic7⋊C4Q818yescomplemented
64C4×Dic7—24no—
65C4×Dic7—42no—
66C4×Dic7D414yes—
67C2×#125—71no—
68C2×C4×Q8—71no—
69#398—71no—
70#398—71no—
71#395—71no—
72#395D711yescharacteristic
73#389—72no—
74#389—72no—
75#389—72no—
76#389—72no—
77C4×C4⋊C4—71no—
78C23×C7C2411yescharacteristic
79C7×Q8—28no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2×Q814yes—
92C2×C4×C7#4114yes—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7C2×D412yes—
106C2×Dic7C2×D412yes—
107C2×Dic7C2×Q814yes—
108C2×Dic7C2×Q814yes—
109C22×Q8C2×D711yescharacteristic
110#125—144no—
111C4×Q8—144no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4—71no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4C2×D711yescharacteristic
124C2×C4⋊C4C2×D712yes—
125C2×C42—71no—
126C2×C42—72no—
127C2×C42—72no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95C2×D711yescharacteristic
133#95C2×D712yes—
134C22×C7C22×D411yescharacteristic
135C22×C7C22×Q812yes—
136C22×C7C2×#4112yes—
137C22×C7H(5,𝔽2)11yescharacteristic
138C22×C72−1+411yescharacteristic
139C4×C7—22no—
140C4×C7—24no—
141C4×C7—24no—
142C4×C7—41no—
143C4×C7—41no—
144Dic7—22no—
145Dic7—24no—
146Dic7—41no—
147Dic7—42no—
148Dic7—41no—
149Dic7C4⋊Q814yes—
150C2×Q8—22no—
151C2×Q8—24no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—71no—
155C22×C4—72no—
156C22×C4—71no—
157C22×C4—71no—
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D711yescharacteristic
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D712yes—
162C22×C4C22×D712yes—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—78no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—144no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—144no—
174C4⋊C4—282no—
175C4⋊C4—282no—
176C42—74no—
177C42—144no—
178C42—282no—
179C2×C7#52511yescharacteristic
180C2×C7C2×C4⋊Q811yescharacteristic
181C2×C7#53711yescharacteristic
182C2×C7#55412yes—
183C2×C7#55712yes—
184C23C23×D711yescharacteristic
185Q8—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—72no—
197C2×C4—74no—
198C2×C4—72no—
199C2×C4—74no—
200C2×C4—142no—
201C2×C4—141no—
202C2×C4—141no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—141no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4D4×D712yes—
213C2×C4#115914yes—
214C2×C4Q8×D714yes—
215C7#258411yescharacteristic, complemented, Sylow
216C22C2×D4×D711yescharacteristic
217C22C2×#115912yes—
218C22C2×Q8×D712yes—
219C22#655111yescharacteristic
220C22#655311yescharacteristic
221C4—22no—
222C4—24no—
223C4—24no—
224C4—41no—
225C4—41no—
226C4—74no—
227C4—142no—
228C4—144no—
229C4—281no—
230C4—282no—
231C4—281no—
232C2#9027711yescharacteristic
233C2#9032011yescharacteristic
234C2#9032612yes—
235C2#9041212yes—
236C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2584
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
63. Dic7⋊C4Q8semidirect
215. C7#2584semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic72
C2×#64522
C7×#3952
#900012
#897572
#897552
#897442
#894332
#894212
#894192
#25847