Finite Group Databasebeta

← #11729549

Subgroups — #11729549

Order 896.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×C7×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90507C211yesmaximal, characteristic
3C2×#6505C211yesmaximal, characteristic
4C2×#6410C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6C2×#6379C211yesmaximal, characteristic
7#89318C211yesmaximal, characteristic, complemented
8C22×C7×Q8C2211yescharacteristic
9C22×C7×Q8C2211yescharacteristic
10C2×#1157C2211yescharacteristic
11C7×C4⋊Q8—22no—
12C7×C4⋊Q8—22no—
13C2×C7×C4⋊C4—22no—
14C2×C42×C7C2211yescharacteristic
15#6510—21no—
16#6509—22no—
17#6507—21no—
18C2×#1144—21no—
19#6505—22no—
20#6505—22no—
21C2×#1143—21no—
22#6492—22no—
23#6489C2211yescharacteristic
24C2×#1136C2211yescharacteristic
25C2×#1136C2211yescharacteristic
26#6410—22no—
27#6409—22no—
28#6379—22no—
29#2005—71nomaximal, Sylow
30C2×C7×Q8—21no—
31C2×C7×Q8—21no—
32C2×C7×Q8—21no—
33C2×C7×Q8—21no—
34C2×C7×Q8—21no—
35C2×C7×Q8—21no—
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38C2×C7×Q8—21no—
39C2×C7×Q8—21no—
40C2×C7×Q8—21no—
41C2×C7×Q8D412yes—
42C22×C4×C7—21no—
43C22×C4×C7—21no—
44C22×C4×C7D412yes—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—21no—
47#1157—21no—
48#1157—21no—
49#1157—21no—
50#1157—21no—
51C2×D28D411yescharacteristic
52C2×C4×D7—21no—
53C2×Dic14D411yescharacteristic
54C7×C4⋊C4—28no—
55C42×C7—21no—
56C42×C7D412yes—
57C22⋊Dic7—21no—
58#1144—24no—
59#1143—24no—
60(C2×D7)⋊C4—24no—
61C4⋊Dic7D412yes—
62Dic7⋊C4—24no—
63C4×Dic7—21no—
64#1136—21no—
65#1136—21no—
66#1136—21no—
67#1136D412yes—
68C2×C7⋊C8—21no—
69C2×C7⋊C8—21no—
70C2×#137—71no—
71#536—71no—
72C2×C4⋊Q8D711yescharacteristic, complemented
73C2×#104—71no—
74#419—71no—
75C2×#101—71no—
76#339—71no—
77C23×C7C2×D411yescharacteristic
78C22×D7—21no—
79C7×Q8—22no—
80C7×Q8—22no—
81C7×Q8—41no—
82C7×Q8—41no—
83C7×Q8—41no—
84C7×Q8—41no—
85C7×Q8—41no—
86C7×Q8—41no—
87C2×C4×C7—24no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—41no—
94C2×C4×C7—41no—
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7#4111yescharacteristic
101C22⋊D7—41no—
102C22⋊D7—41no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105D28—21no—
106D28—21no—
107C4×D7—41no—
108C4×D7—41no—
109Dic14—21no—
110Dic14—21no—
111C7⋊C8—22no—
112C7⋊C8—41no—
113C2×#41—71no—
114C22×Q8C2×D711yescharacteristic
115C22×Q8C2×D711yescharacteristic
116#137—142no—
117#137—142no—
118C2×Q16—141no—
119C2×SD16—141no—
120C2×M16—71no—
121C2×M16—71no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—141no—
125#124—141no—
126#123—142no—
127C22⋊Q8—142no—
128#117—71no—
129C2×C4⋊C4—22no—
130C2×C42C2×D711yescharacteristic
131#104—142no—
132#103—142no—
133#101—142no—
134C22×C7—22no—
135C22×C7#12011yescharacteristic
136C22×C7#12011yescharacteristic
137C22×C7#12111yescharacteristic
138C2×D7—41no—
139C2×D7—42no—
140C4×C7—24no—
141C4×C7—42no—
142C4×C7—42no—
143C4×C7#12011yescharacteristic
144C4×C7#12111yescharacteristic
145C4×C7#12111yescharacteristic
146C4×C7#12711yescharacteristic
147Dic7—41no—
148Dic7—42no—
149#41—141no—
150#41—141no—
151#41—141no—
152#41—141no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—71no—
165C2×Q8C22⋊D712yes—
166C2×D4—71no—
167C2×D4—141no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—141no—
171C22×C4C22⋊D712yes—
172C22×C4C22×D711yescharacteristic
173Q16—144no—
174SD16—144no—
175M16—72no—
176M16—141no—
177M16—141no—
178M16—141no—
179C2×C8—141no—
180C2×C8—141no—
181C4⋊C4—28no—
182C4⋊C4—72no—
183C4⋊C4—144no—
184C22⋊C4—144no—
185C22⋊C4—141no—
186C42—21no—
187C42—141no—
188C42C22⋊D712yes—
189C2×C7—21no—
190C2×C7—21no—
191C2×C7#39211yescharacteristic
192C2×C7#45612yes—
193D7—81no—
194C23—141no—
195C23C2×C22⋊D711yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—41no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—141no—
205Q8—141no—
206D4—141no—
207D4—141no—
208D4—281no—
209D4—281no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—142no—
219C2×C4—141no—
220C2×C4—281no—
221C2×C4—281no—
222C2×C4D4×D711yescharacteristic
223C2×C4D4×D711yescharacteristic
224C2×C4D4×D711yescharacteristic
225C2×C4#115911yescharacteristic
226C2×C4C2×C22⋊D711yescharacteristic
227C2×C4C2×C22⋊D711yescharacteristic
228C8—142no—
229C8—281no—
230C7#200511yescharacteristic, complemented, Sylow
231C22—22no—
232C22—282no—
233C22—281no—
234C22#650011yescharacteristic
235C22#650211yescharacteristic
236C22#651611yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—282no—
241C4—281no—
242C4#650011yescharacteristic
243C4#650111yescharacteristic
244C4#650211yescharacteristic
245C4#650311yescharacteristic
246C2—21no—
247C2—21no—
248C2—561no—
249C2#8986712yes—
250C2#8999511yescharacteristic
subgroupG/H
centerC22#6500
commutator, cosocleC22×C4×C7C23
FrattiniC22×C4C22×D7
FittingC2×C7×C4⋊Q8C2
socleC22×C7#120
prime radicalC2×#1157C22
hypercenterC2×C4⋊Q8D7
hypocenterC7#2005
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C4⋊Q8C2semidirect
4. C2×#6410C2semidirect
5. #89997C2semidirect
7. #89318C2semidirect
72. C2×C4⋊Q8D7semidirect
230. C7#2005semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C4⋊Q82
#905072
C2×#65052
C2×#64102
#899972
C2×#63792
#893182
#20057