Finite Group Databasebeta

← #11737862

Subgroups — #11737862

Order 896.

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

normal subgroups only

This group has 340 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#90412C211yesmaximal, characteristic, complemented
2#90234C211yesmaximal, characteristic
3C7×#461C211yesmaximal, characteristic
4#89936C212yesmaximal
5#89934C212yesmaximal
6#89853C211yesmaximal, characteristic, complemented
7C4×#1142C211yesmaximal, characteristic
8#89622C211yesmaximal, characteristic, complemented
9#89603C211yesmaximal, characteristic, complemented
10#89569C212yesmaximal, complemented
11#89568C212yesmaximal, complemented
12C2×C7×SD16C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15Q8×Dic7C2212yes—
16#6507C2212yes—
17D4×Dic7C2212yes—
18C2×#1142C2211yescharacteristic
19C2×C4⋊Dic7—21no—
20#6487—21no—
21#6453—21no—
22#6453C2212yes—
23#6452—22no—
24#6450—21no—
25#6441—22no—
26#6432C2211yescharacteristic
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2211yescharacteristic
30C7×#103C2212yes—
31#6409C2212yes—
32#6409C2212yes—
33#6406C2211yescharacteristic
34#6392C2212yes—
35#6390C2212yes—
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2211yescharacteristic
39#6382C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3372—71nomaximal, Sylow
42C2×C7×Q8C2312yes—
43C2×C7×D4C2311yescharacteristic
44C22×C4×C7—21no—
45C22×Dic7—22no—
46C2×Dic14—21no—
47C2×Dic14D411yescharacteristic
48C2×Dic14C2311yescharacteristic
49C7×SD16—24no—
50C2×C7×C8C2312yes—
51C7×C4⋊C4—21no—
52C7×C4⋊C4—21no—
53C7×C4⋊C4C2311yescharacteristic
54C7×C22⋊C4—21no—
55C42×C7C2311yescharacteristic
56C22⋊Dic7—22no—
57#1142—22no—
58C4⋊Dic7—21no—
59C4⋊Dic7—22no—
60C4⋊Dic7—22no—
61C4⋊Dic7D411yescharacteristic
62C4⋊Dic7C2311yescharacteristic
63C4⋊Dic7C2312yes—
64Dic7⋊C4—21no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4D412yes—
69C4×Dic7—21no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74#557—71no—
75#554—71no—
76#491—72no—
77#487—71no—
78#479—72no—
79#477—71no—
80#461—72no—
81#461D711yescharacteristic
82#446—71no—
83#440—72no—
84C4×SD16—71no—
85C23×C7—21no—
86C7×Q8—22no—
87C7×Q8—22no—
88C7×D4—21no—
89C7×D4#4112yes—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—42no—
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2411yescharacteristic
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—82no—
100C2×Dic7C2×D412yes—
101C2×Dic7C2×D412yes—
102Dic14—22no—
103Dic14—21no—
104Dic14—21no—
105Dic14—81no—
106C7×C8—22no—
107C7⋊C8—21no—
108C7⋊C8—21no—
109C2×SD16—71no—
110C2×SD16C2×D712yes—
111C4⋊Q8—72no—
112C4⋊Q8—71no—
113C4⋊Q8C2×D711yescharacteristic
114#125—72no—
115#125—142no—
116#125—141no—
117C22⋊Q8—142no—
118C22⋊Q8—141no—
119C4×Q8—71no—
120C4×Q8—72no—
121C4×Q8—71no—
122C4×Q8—141no—
123C4×D4—72no—
124C4×D4C2×D711yescharacteristic
125C2×C4⋊C4—141no—
126#107—72no—
127#106—71no—
128C4⋊C8—72no—
129C4⋊C8C2×D711yescharacteristic
130#103—72no—
131#103—72no—
132#103—72no—
133#103—71no—
134#103C2×D712yes—
135#102—71no—
136#97—72no—
137C4×C8—71no—
138C22×C7—22no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7C22×D411yescharacteristic
142C4×C7—21no—
143C4×C7—42no—
144C4×C7—41no—
145C4×C7#13712yes—
146C4×C7C2×#4111yescharacteristic
147C4×C72−1+411yescharacteristic
148Dic7—22no—
149Dic7—22no—
150Dic7—41no—
151Dic7—42no—
152Dic7—41no—
153C2×Q8—71no—
154C2×Q8—71no—
155C2×Q8—141no—
156C2×Q8C22×D712yes—
157C2×D4C22×D711yescharacteristic
158C22×C4—21no—
159C22×C4—142no—
160SD16—24no—
161SD16—142no—
162C2×C8—71no—
163C2×C8—71no—
164C2×C8C22×D712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—72no—
168C4⋊C4—71no—
169C4⋊C4—71no—
170C4⋊C4—72no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—141no—
177C4⋊C4—141no—
178C4⋊C4C22×D711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—142no—
181C42—72no—
182C42—142no—
183C42—141no—
184C42C22×D711yescharacteristic
185C2×C7—42no—
186C2×C7#54811yescharacteristic
187C2×C7C2×#13711yescharacteristic
188C2×C7#57711yescharacteristic
189C23—21no—
190Q8—22no—
191Q8—22no—
192Q8—142no—
193Q8—141no—
194Q8—141no—
195Q8—561no—
196D4—21no—
197D4#115912yes—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—72no—
202C2×C4—72no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—142no—
206C2×C4—562no—
207C2×C4D4×D711yescharacteristic
208C2×C4D4×D711yescharacteristic
209C2×C4C23×D711yescharacteristic
210C8—22no—
211C8—141no—
212C8—141no—
213C7#337211yescharacteristic, complemented, Sylow
214C22—22no—
215C22—41no—
216C22—41no—
217C22C2×D4×D711yescharacteristic
218C4—21no—
219C4—41no—
220C4—42no—
221C4—142no—
222C4—142no—
223C4—282no—
224C4—281no—
225C4—281no—
226C4#647812yes—
227C4C2×#115911yescharacteristic
228C4#655411yescharacteristic
229C2—42no—
230C2#9028911yescharacteristic
231C2C2×#647811yescharacteristic
232C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#461D7
hypocenterC7#3372
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #90412C2semidirect
6. #89853C2semidirect
8. #89622C2semidirect
9. #89603C2semidirect
10. #89569C2semidirect
11. #89568C2semidirect
213. C7#3372semidirect

Maximal subgroups

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

subgroupindex
#904122
#902342
C7×#4612
#899362
#899342
#898532
C4×#11422
#896222
#896032
#895692
#895682
#33727