Finite Group Databasebeta

← #11725932

Subgroups — #11725932

Order 896.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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#90248C211yesmaximal, characteristic
2#89609C211yesmaximal, characteristic, complemented
3#89457C211yesmaximal, characteristic, complemented
4C7×D4⋊C8C211yesmaximal, characteristic, complemented
5D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#89279C211yesmaximal, characteristic, complemented
7#89251C211yesmaximal, characteristic, complemented
8C4×C7×D4C2211yescharacteristic
9#6501C411yescharacteristic
10#6499C411yescharacteristic
11C4×C22⋊D7—21no—
12#6489—21no—
13C2×#1131—21no—
14#6460C411yescharacteristic
15#6453C411yescharacteristic
16#6446—21no—
17#6442—21no—
18C42×D7C2211yescharacteristic
19C4×Dic14C2211yescharacteristic
20C7×C4⋊C8C2211yescharacteristic
21C4×C7×C8C2211yescharacteristic
22(C2×D7)⋊C8—21no—
23Dic7⋊C8—21no—
24Dic7⋊C8—21no—
25C4⋊(C7⋊C8)C2211yescharacteristic
26C4×C7⋊C8C2211yescharacteristic
27#1495—71nomaximal, Sylow
28C2×C7×D4C2×C411yescharacteristic
29C22×C4×C7—21no—
30C2×C22⋊D7—21no—
31C2×C4×D7C2×C411yescharacteristic
32C2×C4×D7D411yescharacteristic
33C2×C4×D7D411yescharacteristic
34C2×Dic14C2×C411yescharacteristic
35C2×C7×C8—21no—
36C2×C7×C8—21no—
37C7×C4⋊C4C2×C411yescharacteristic
38C7×C22⋊C4—21no—
39C42×C7C2311yescharacteristic
40C22⋊Dic7—21no—
41C22⋊Dic7—21no—
42C22⋊Dic7—21no—
43(C2×D7)⋊C4—21no—
44C4⋊Dic7—21no—
45C4⋊Dic7—21no—
46C4⋊Dic7C2×C411yescharacteristic
47Dic7⋊C4—21no—
48Dic7⋊C4—21no—
49C4×Dic7—21no—
50C4×Dic7C2×C411yescharacteristic
51C4×Dic7D411yescharacteristic
52C4×Dic7D411yescharacteristic
53C2×C7⋊C8—21no—
54C2×C7⋊C8—21no—
55#1131—42no—
56#529—71no—
57#431—71no—
58#423—71no—
59Q8⋊C8—71no—
60Q8⋊C8—71no—
61D4⋊C8—71no—
62D4⋊C8D711yescharacteristic, complemented
63C23×C7—21no—
64C22×D7C22⋊C411yescharacteristic
65C7×D4—21no—
66C7×D4—21no—
67C2×C4×C7—21no—
68C2×C4×C7—41no—
69C2×C4×C7—41no—
70C2×C4×C7C22×C411yescharacteristic
71C2×C4×C7C2×D411yescharacteristic
72C2×C4×C7C2×D411yescharacteristic
73C22⋊D7—42no—
74C2×Dic7—21no—
75C2×Dic7—21no—
76C2×Dic7C22⋊C411yescharacteristic
77C2×Dic7C22⋊C411yescharacteristic
78C2×Dic7C22⋊C411yescharacteristic
79C4×D7—21no—
80C4×D7—21no—
81C4×D7—21no—
82C4×D7—21no—
83C4×D7—41no—
84Dic14—21no—
85Dic14—21no—
86C7×C8—41no—
87C7×C8—41no—
88C7⋊C8—41no—
89C7⋊C8—41no—
90C2×M16—141no—
91#126—141no—
92#125—71no—
93#124—71no—
94#123—141no—
95C22⋊Q8—71no—
96#121—71no—
97C4×Q8—71no—
98C4×D4—141no—
99C4×D4C2×D711yescharacteristic
100#117—141no—
101C2×C42—71no—
102C4⋊C8—71no—
103C4⋊C8—141no—
104C4⋊C8—141no—
105C4⋊C8C2×D711yescharacteristic
106C22⋊C8—141no—
107C4×C8—71no—
108C4×C8C2×D711yescharacteristic
109C22×C7—41no—
110C22×C7—41no—
111C22×C7—41no—
112C22×C7C2×C22⋊C411yescharacteristic
113C2×D7—21no—
114C2×D7—21no—
115C2×D7—21no—
116C4×C7—21no—
117C4×C7—41no—
118C4×C7#13111yescharacteristic
119C4×C7#13111yescharacteristic
120C4×C7AGL(1,ℤ/8)11yescharacteristic
121C4×C7#13711yescharacteristic
122Dic7—21no—
123Dic7—21no—
124Dic7—21no—
125Dic7—41no—
126Dic7—41no—
127C2×Q8—71no—
128C2×D4—141no—
129C2×D4C4×D711yescharacteristic
130C22×C4—21no—
131C22×C4—71no—
132C22×C4—71no—
133C22×C4—71no—
134M16—282no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—141no—
138C2×C8—141no—
139C4⋊C4—71no—
140C4⋊C4—141no—
141C4⋊C4—141no—
142C4⋊C4—141no—
143C4⋊C4—141no—
144C4⋊C4C4×D711yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—141no—
147C22⋊C4—141no—
148C22⋊C4—141no—
149C22⋊C4—141no—
150C42—71no—
151C42—71no—
152C42—71no—
153C42—141no—
154C42C22×D711yescharacteristic
155C2×C7—81no—
156C2×C7#40811yescharacteristic
157C2×C7#41711yescharacteristic
158C2×C7#42111yescharacteristic
159D7—41no—
160C23—21no—
161C23—71no—
162Q8—141no—
163Q8—141no—
164D4—21no—
165D4—21no—
166D4—282no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—71no—
171C2×C4—71no—
172C2×C4—71no—
173C2×C4—141no—
174C2×C4—141no—
175C2×C4—141no—
176C2×C4—141no—
177C2×C4—141no—
178C2×C4—141no—
179C2×C4—281no—
180C2×C4C2×C4×D711yescharacteristic
181C2×C4D4×D711yescharacteristic
182C2×C4D4×D711yescharacteristic
183C8—41no—
184C8—41no—
185C8—281no—
186C8—281no—
187C7#149511yescharacteristic, complemented, Sylow
188C22—41no—
189C22—41no—
190C22—41no—
191C22—141no—
192C22—141no—
193C22—141no—
194C22D7×C22⋊C411yescharacteristic
195C4—21no—
196C4—41no—
197C4—141no—
198C4—141no—
199C4—141no—
200C4—281no—
201C4—281no—
202C4#646411yescharacteristic
203C4#647011yescharacteristic
204C4#647411yescharacteristic
205C4#647811yescharacteristic
206C2—81no—
207C2—281no—
208C2#8949911yescharacteristic
209C2#8954311yescharacteristic
210C2#8959311yescharacteristic
subgroupG/H
centerC2×C4D4×D7
commutatorC2×C4×C7C22×C4
FrattiniC42C22×D7
FittingC7×D4⋊C8C2
socleC22×C7C2×C22⋊C4
prime radical#6501C4
cosocleC42×C7C23
hypercenterD4⋊C8D7
hypocenterC7#1495
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #89609C2semidirect
3. #89457C2semidirect
4. C7×D4⋊C8C2semidirect
5. D4⋊(C7⋊C8)C2semidirect
6. #89279C2semidirect
7. #89251C2semidirect
62. D4⋊C8D7semidirect
187. C7#1495semidirect

Maximal subgroups

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

subgroupindex
#902482
#896092
#894572
C7×D4⋊C82
D4⋊(C7⋊C8)2
#892792
#892512
#14957