Finite Group Databasebeta

← #11737819

Subgroups — #11737819

Order 896.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#90404C211yesmaximal, characteristic
2Q8×Dic14C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic
4#89938C212yesmaximal, complemented
5#89936C212yesmaximal, complemented
6#89842C211yesmaximal, characteristic
7C4×#1144C211yesmaximal, characteristic, complemented
8#89622C211yesmaximal, characteristic, complemented
9#89603C211yesmaximal, characteristic, complemented
10#89537C212yesmaximal, complemented
11#89531C212yesmaximal, complemented
12C2×C7×SD16C2212yes—
13C7×#127C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15Q8×Dic7C2212yes—
16C2×#1144C2211yescharacteristic
17#6499—21no—
18#6499C2212yes—
19#6498—22no—
20D4×Dic7—21no—
21D4×Dic7C2212yes—
22#6451—22no—
23#6451C2212yes—
24#6432—21no—
25#6432C2211yescharacteristic
26C4×Dic14—21no—
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2211yescharacteristic
30C7×#102C2212yes—
31#6409C2211yescharacteristic
32#6406C2212yes—
33#6406C2212yes—
34#6392C2212yes—
35#6390C2212yes—
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3375—71nomaximal, Sylow
42C2×C7×Q8C2311yescharacteristic
43C2×C7×D4—21no—
44C2×C7×D4—21no—
45C2×C7×D4C2312yes—
46C22×Dic7—22no—
47C2×Dic14—21no—
48C2×Dic14—22no—
49C2×Dic14D411yescharacteristic
50C2×Dic14C2311yescharacteristic
51C7×SD16—24no—
52C2×C7×C8C2312yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4C2311yescharacteristic
55C42×C7—21no—
56C42×C7C2311yescharacteristic
57C22⋊Dic7—22no—
58C22⋊Dic7—22no—
59C22⋊Dic7—22no—
60#1144—22no—
61C4⋊Dic7—21no—
62C4⋊Dic7—22no—
63C4⋊Dic7D411yescharacteristic
64C4⋊Dic7C2312yes—
65C4⋊Dic7C2311yescharacteristic
66Dic7⋊C4—21no—
67Dic7⋊C4—22no—
68Dic7⋊C4D412yes—
69C4×Dic7—21no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74#559—71no—
75Q82—71no—
76#489—72no—
77#486—71no—
78#477—71no—
79#474—72no—
80#464—72no—
81#460D711yescharacteristic
82#446—71no—
83#440—72no—
84C4×Q16—71no—
85C23×C7—22no—
86C7×Q8—21no—
87C7×Q8#4112yes—
88C7×D4—22no—
89C7×D4—22no—
90C7×D4—42no—
91C7×D4—81no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2411yescharacteristic
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—82no—
101C2×Dic7C2×D412yes—
102C2×Dic7C2×D412yes—
103Dic14—22no—
104Dic14—21no—
105Dic14—21no—
106Dic14—82no—
107Dic14—81no—
108C7×C8—22no—
109C7⋊C8—21no—
110C7⋊C8—21no—
111C2×Q16—71no—
112C2×SD16C2×D712yes—
113C4⋊Q8—72no—
114C4⋊Q8—71no—
115C4⋊Q8—141no—
116C4⋊Q8—142no—
117#127C2×D711yescharacteristic
118#124—72no—
119#124—141no—
120#123—142no—
121C4×Q8—71no—
122C4×Q8—72no—
123C4×Q8—71no—
124C4×Q8—141no—
125C4×Q8C2×D711yescharacteristic
126C4×D4—72no—
127C4×D4—141no—
128#107—72no—
129#107—71no—
130C4⋊C8—72no—
131C4⋊C8C2×D711yescharacteristic
132#103—71no—
133#103—72no—
134#103—71no—
135#102—72no—
136#102—72no—
137#102C2×D712yes—
138#97—72no—
139C4×C8—71no—
140C22×C7—42no—
141C22×C7—42no—
142C22×C7—42no—
143C22×C7C22×D411yescharacteristic
144C4×C7—22no—
145C4×C7—21no—
146C4×C7—41no—
147C4×C7#13712yes—
148C4×C7C2×#4111yescharacteristic
149C4×C72−1+411yescharacteristic
150Dic7—22no—
151Dic7—22no—
152Dic7—41no—
153Dic7—42no—
154Dic7—41no—
155C2×Q8—71no—
156C2×Q8—71no—
157C2×Q8—142no—
158C2×Q8—141no—
159C2×Q8C22×D711yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4C22×D712yes—
163C22×C4—142no—
164Q16—142no—
165SD16—24no—
166C2×C8—71no—
167C2×C8—71no—
168C2×C8C22×D712yes—
169C4⋊C4—21no—
170C4⋊C4—72no—
171C4⋊C4—71no—
172C4⋊C4—72no—
173C4⋊C4—71no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—141no—
177C4⋊C4—141no—
178C4⋊C4C22×D711yescharacteristic
179C22⋊C4—142no—
180C22⋊C4—142no—
181C22⋊C4—142no—
182C42—21no—
183C42—72no—
184C42—142no—
185C42—141no—
186C42C22×D711yescharacteristic
187C2×C7—82no—
188C2×C7#54811yescharacteristic
189C2×C7C2×#13711yescharacteristic
190C2×C7#57611yescharacteristic
191C23—22no—
192Q8—21no—
193Q8—142no—
194Q8—141no—
195Q8—141no—
196Q8—561no—
197Q8—562no—
198Q8#115912yes—
199D4—22no—
200D4—22no—
201D4—42no—
202D4—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—141no—
208C2×C4—141no—
209C2×C4—142no—
210C2×C4—562no—
211C2×C4D4×D711yescharacteristic
212C2×C4D4×D711yescharacteristic
213C2×C4C23×D711yescharacteristic
214C8—22no—
215C8—141no—
216C8—141no—
217C7#337511yescharacteristic, complemented, Sylow
218C22—42no—
219C22—42no—
220C22—42no—
221C22C2×D4×D711yescharacteristic
222C4—21no—
223C4—22no—
224C4—41no—
225C4—142no—
226C4—142no—
227C4—281no—
228C4—282no—
229C4—281no—
230C4#647812yes—
231C4C2×#115911yescharacteristic
232C4#655411yescharacteristic
233C2—82no—
234C2#9028911yescharacteristic
235C2C2×#647811yescharacteristic
236C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
prime radicalC7×#127C22
hypercenter#460D7
hypocenterC7#3375
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. Q8×Dic14C2semidirect
4. #89938C2semidirect
5. #89936C2semidirect
7. C4×#1144C2semidirect
8. #89622C2semidirect
9. #89603C2semidirect
10. #89537C2semidirect
11. #89531C2semidirect
217. C7#3375semidirect

Maximal subgroups

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

subgroupindex
#904042
Q8×Dic142
C7×#4602
#899382
#899362
#898422
C4×#11442
#896222
#896032
#895372
#895312
#33757