Finite Group Databasebeta

← #11727985

Subgroups — #11727985

Order 896.

290 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C2×C7×#103C211yesmaximal, characteristic
3#90002C211yesmaximal, characteristic
4C2×#6409C211yesmaximal, characteristic
5#89346C211yesmaximal, characteristic
6#89345C211yesmaximal, characteristic
7#89324C211yesmaximal, characteristic
8C22×C7×Q8C2211yescharacteristic
9C22×Dic14—21no—
10C22×C7×C8C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C4⋊Dic7C2211yescharacteristic
13C2×C4⋊Dic7C2211yescharacteristic
14C2×Dic7⋊C4—21no—
15C2×C4×Dic7C2211yescharacteristic
16C22×C7⋊C8C2211yescharacteristic
17#6451—22no—
18#6451—22no—
19C7×#103—22no—
20#6409—22no—
21C7⋊#95—21no—
22C7⋊#95—21no—
23#2024—71nomaximal, Sylow
24C2×C7×Q8—21no—
25C2×C7×Q8—21no—
26C2×C7×Q8—21no—
27C2×C7×Q8—21no—
28C2×C7×Q8—21no—
29C2×C7×Q8Q812yescomplemented
30C22×C4×C7—21no—
31C22×C4×C7—21no—
32C22×C4×C7C2311yescharacteristic
33C22×Dic7—21no—
34C22×Dic7—21no—
35C22×Dic7D411yescharacteristic
36C22×Dic7D411yescharacteristic
37C2×Dic14—24no—
38C2×Dic14—42no—
39C2×Dic14—42no—
40C2×C7×C8—21no—
41C2×C7×C8—21no—
42C2×C7×C8—21no—
43C7×C4⋊C4—21no—
44C7×C4⋊C4D412yes—
45C4⋊Dic7—21no—
46C4⋊Dic7—21no—
47C4⋊Dic7—21no—
48C4⋊Dic7D412yes—
49Dic7⋊C4—24no—
50C4×Dic7—21no—
51C4×Dic7—21no—
52C2×C7⋊C8—21no—
53C2×C7⋊C8—21no—
54C2×C7⋊C8—21no—
55C2×C4⋊Q8—71no—
56C2×#103—71no—
57C2×#103D711yescharacteristic
58#391—71no—
59#340—71no—
60#340—71no—
61#336—71no—
62C23×C7C2×D411yescharacteristic
63C7×Q8—22no—
64C7×Q8—22no—
65C7×Q8—41no—
66C7×Q8—41no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—41no—
70C2×C4×C7—41no—
71C2×C4×C7—41no—
72C2×C4×C7—41no—
73C2×C4×C7C2×D411yescharacteristic
74C2×C4×C7C2×D411yescharacteristic
75C2×C4×C7C2×Q811yescharacteristic
76C2×C4×C7#4111yescharacteristic
77C2×C4×C7#4111yescharacteristic
78C2×C4×C7#4111yescharacteristic
79C2×Dic7—21no—
80C2×Dic7—21no—
81C2×Dic7—21no—
82C2×Dic7—21no—
83C2×Dic7—22no—
84C2×Dic7—41no—
85C2×Dic7—41no—
86C2×Dic7—41no—
87C2×Dic7—41no—
88C2×Dic7—41no—
89C2×Dic7SD1612yes—
90C2×Dic7Q1612yes—
91Dic14—48no—
92C7×C8—41no—
93C7⋊C8—41no—
94C22×Q8—141no—
95C22×Q8C2×D711yescharacteristic
96C22×C8—71no—
97C22×C8C2×D711yescharacteristic
98C4⋊Q8—142no—
99C4⋊Q8—142no—
100C2×C4⋊C4—71no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—141no—
103C2×C4⋊C4C2×D711yescharacteristic
104C2×C42—71no—
105#103—22no—
106#103—142no—
107#95—141no—
108#95—141no—
109C22×C7#12011yescharacteristic
110C22×C7C22⋊Q811yescharacteristic
111C22×C7#12411yescharacteristic
112C22×C7C2×SD1611yescharacteristic
113C22×C7C2×Q1611yescharacteristic
114C22×C7AGL(1,ℤ/8)11yescharacteristic
115C22×C7#13711yescharacteristic
116C4×C7—42no—
117C4×C7—42no—
118C4×C7C22⋊Q811yescharacteristic
119C4×C7C22⋊Q811yescharacteristic
120C4×C7#12411yescharacteristic
121C4×C7#12411yescharacteristic
122Dic7—22no—
123Dic7—22no—
124Dic7—42no—
125Dic7—81no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—144no—
132C2×Q8—282no—
133C2×Q8—282no—
134C2×Q8Dic1412yescomplemented
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—71no—
138C22×C4—71no—
139C22×C4—141no—
140C22×C4—141no—
141C22×C4C22×D711yescharacteristic
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—141no—
146C2×C8—141no—
147C2×C8—141no—
148C4⋊C4—21no—
149C4⋊C4—72no—
150C4⋊C4—144no—
151C4⋊C4—141no—
152C4⋊C4—141no—
153C4⋊C4—141no—
154C4⋊C4C22⋊D712yes—
155C42—141no—
156C42—141no—
157C2×C7#39311yescharacteristic
158C2×C7#45011yescharacteristic
159C2×C7#45111yescharacteristic
160C2×C7#47511yescharacteristic
161C2×C7#47711yescharacteristic
162C2×C7#48711yescharacteristic
163C2×C7#48911yescharacteristic
164C23D4×D711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—288no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—72no—
177C2×C4—72no—
178C2×C4—141no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4—142no—
183C2×C4—281no—
184C2×C4—281no—
185C2×C4—281no—
186C2×C4—281no—
187C2×C4—281no—
188C2×C4C2×Dic1411yescharacteristic
189C2×C4#115711yescharacteristic
190C2×C4D4×D711yescharacteristic
191C2×C4#115911yescharacteristic
192C2×C4#115911yescharacteristic
193C2×C4C2×C22⋊D711yescharacteristic
194C8—41no—
195C8—281no—
196C7#202411yescharacteristic, complemented, Sylow
197C22#644111yescharacteristic
198C22#644811yescharacteristic
199C22D7×SD1611yescharacteristic
200C22#647711yescharacteristic
201C22D7×Q1611yescharacteristic
202C22#648111yescharacteristic
203C22#650011yescharacteristic
204C4—42no—
205C4—42no—
206C4—142no—
207C4—142no—
208C4—282no—
209C4—561no—
210C4#644111yescharacteristic
211C4#644811yescharacteristic
212C4#648711yescharacteristic
213C4#649911yescharacteristic
214C2#8956311yescharacteristic
215C2#8956411yescharacteristic
216C2#8956611yescharacteristic
217C2#8956711yescharacteristic
218C2#8971911yescharacteristic
219C2#8994111yescharacteristic
220C2#8995811yescharacteristic
subgroupG/H
centerC23D4×D7
commutator, cosocleC22×C4×C7C23
FrattiniC22×C4C22×D7
FittingC2×C7×#103C2
socle, prime radicalC23×C7C2×D4
hypercenterC2×#103D7
hypocenterC7#2024
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
29. C2×C7×Q8Q8semidirect
134. C2×Q8Dic14semidirect
196. C7#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×C7×#1032
#900022
C2×#64092
#893462
#893452
#893242
#20247