Finite Group Databasebeta

← #11738488

Subgroups — #11738488

Order 896.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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
2Q8×Dic14C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic
4#89851C211yesmaximal, characteristic
5C4×#1144C211yesmaximal, characteristic
6#89628C212yesmaximal
7#89626C212yesmaximal
8#89622C211yesmaximal, characteristic
9#89603C211yesmaximal, characteristic
10#89568C212yesmaximal
11#89561C212yesmaximal
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14Q8×Dic7—21no—
15Q8×Dic7C2212yes—
16#6507—21no—
17C2×#1144C2211yescharacteristic
18#6453C2212yes—
19#6452—22no—
20#6451—22no—
21#6451C2212yes—
22#6450C2212yes—
23#6432—21no—
24#6432C2211yescharacteristic
25C4×Dic14—21no—
26C4×Dic14C2211yescharacteristic
27C4×Dic14C2211yescharacteristic
28C7×#106C2212yes—
29C7×C4⋊C8C2211yescharacteristic
30C7×#103C2212yes—
31#6409C2211yescharacteristic
32#6392C2212yes—
33#6390C2212yes—
34#6389C2212yes—
35Dic7⋊C8C2212yes—
36#6384C2211yescharacteristic
37#6382C2212yes—
38#6382C2212yes—
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3387—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44C2×Dic14—21no—
45C2×Dic14—22no—
46C2×Dic14D411yescharacteristic
47C2×Dic14C2311yescharacteristic
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4C2312yes—
52C7×C4⋊C4C2311yescharacteristic
53C42×C7—21no—
54C42×C7C2311yescharacteristic
55#1144Q814yescomplemented
56C4⋊Dic7—21no—
57C4⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7D411yescharacteristic
60C4⋊Dic7C2311yescharacteristic
61C4⋊Dic7C2312yes—
62Dic7⋊C4—21no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4D412yes—
67C4×Dic7—21no—
68C4×Dic7—22no—
69C4×Dic7—22no—
70C4×Dic7C2312yes—
71C2×C7⋊C8C2311yescharacteristic
72C2×C7⋊C8C2311yescharacteristic
73Q82—71no—
74#557—71no—
75#501—72no—
76#498—71no—
77#479—72no—
78#477—71no—
79#475—72no—
80#475D711yescharacteristic
81#446—71no—
82#441—72no—
83C4×Q16—71no—
84C7×Q8—21no—
85C7×Q8—42no—
86C7×Q8C2×Q812yes—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2411yescharacteristic
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7C2×D412yes—
97C2×Dic7C2×D412yes—
98Dic14—22no—
99Dic14—21no—
100Dic14—82no—
101Dic14—81no—
102Dic14C2×Q812yes—
103C7×C8—22no—
104C7⋊C8—21no—
105C7⋊C8C2×Q812yes—
106C2×Q16—71no—
107C4⋊Q8—72no—
108C4⋊Q8—71no—
109C4⋊Q8—141no—
110C4⋊Q8—142no—
111C4⋊Q8—141no—
112C4⋊Q8C2×D711yescharacteristic
113#125—72no—
114#125—142no—
115C4×Q8—71no—
116C4×Q8—72no—
117C4×Q8—72no—
118C4×Q8—71no—
119C4×Q8—141no—
120C4×Q8—141no—
121C4×Q8C2×D711yescharacteristic
122#107—72no—
123#107—71no—
124#106—72no—
125#106—72no—
126#106C2×D712yes—
127C4⋊C8—72no—
128C4⋊C8C2×D711yescharacteristic
129#103—71no—
130#103—72no—
131#103—71no—
132#103C2×D712yes—
133#97—72no—
134C4×C8—71no—
135C22×C7C22×D411yescharacteristic
136C4×C7—22no—
137C4×C7—21no—
138C4×C7—42no—
139C4×C7—41no—
140C4×C7#13712yes—
141C4×C7C22×Q811yescharacteristic
142C4×C72−1+411yescharacteristic
143Dic7—22no—
144Dic7—22no—
145Dic7—22no—
146Dic7—42no—
147Dic7—41no—
148C2×Q8—21no—
149C2×Q8—71no—
150C2×Q8—71no—
151C2×Q8—142no—
152C2×Q8—141no—
153C2×Q8C22×D711yescharacteristic
154Q16—74no—
155C2×C8—71no—
156C2×C8—71no—
157C2×C8C22×D712yes—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—72no—
161C4⋊C4—71no—
162C4⋊C4—72no—
163C4⋊C4—71no—
164C4⋊C4—142no—
165C4⋊C4—142no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—141no—
170C4⋊C4—141no—
171C4⋊C4C22×D711yescharacteristic
172C4⋊C4C22×D712yes—
173C42—21no—
174C42—72no—
175C42—142no—
176C42—142no—
177C42—141no—
178C42C22×D711yescharacteristic
179C2×C7D4×Q811yescharacteristic
180C2×C7C2×#13711yescharacteristic
181C2×C7#57711yescharacteristic
182Q8—21no—
183Q8—42no—
184Q8—72no—
185Q8—142no—
186Q8—141no—
187Q8—561no—
188Q8—562no—
189Q8Q8×D712yes—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—72no—
194C2×C4—72no—
195C2×C4—141no—
196C2×C4—141no—
197C2×C4—142no—
198C2×C4D4×D711yescharacteristic
199C2×C4D4×D711yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—22no—
202C8—72no—
203C8—141no—
204C7#338711yescharacteristic, complemented, Sylow
205C22C2×D4×D711yescharacteristic
206C4—21no—
207C4—22no—
208C4—42no—
209C4—41no—
210C4—142no—
211C4—142no—
212C4—142no—
213C4—282no—
214C4—281no—
215C4#647812yes—
216C4C2×Q8×D711yescharacteristic
217C4#655411yescharacteristic
218C2#9032311yescharacteristic
219C2C2×#647811yescharacteristic
220C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socle, prime radicalC22×C7C22×D4
hypercenter#475D7
hypocenterC7#3387
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
55. #1144Q8semidirect
204. C7#3387semidirect

Maximal subgroups

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

subgroupindex
#904122
Q8×Dic142
C7×#4752
#898512
C4×#11442
#896282
#896262
#896222
#896032
#895682
#895612
#33877