Finite Group Databasebeta

← #11728895

Subgroups — #11728895

Order 896.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#90244C211yesmaximal, characteristic
2C7×#476C211yesmaximal, characteristic, complemented
3#89863C211yesmaximal, characteristic, complemented
4#89857C211yesmaximal, characteristic, complemented
5D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#89280C211yesmaximal, characteristic, complemented
7#89279C211yesmaximal, characteristic, complemented
8C2×#1157—21no—
9C7×C4⋊Q8C2211yescharacteristic
10C4×C7×D4C2211yescharacteristic
11C2×#1143—22no—
12#6503—21no—
13#6493—21no—
14#6456—21no—
15#6447—22no—
16#6437C2211yescharacteristic
17C4×Dic14C2211yescharacteristic
18C7×#106—21no—
19C7×C4⋊C8C2211yescharacteristic
20C7×#102—21no—
21#6384—21no—
22#6381—21no—
23C4⋊(C7⋊C8)C2211yescharacteristic
24C4×C7⋊C8C2211yescharacteristic
25#1631—71nomaximal, Sylow
26C2×C7×Q8—21no—
27C2×C7×D4D411yescharacteristic
28C22×C4×C7—21no—
29C2×C22⋊D7—22no—
30C2×C22⋊D7—22no—
31#1157—42no—
32#1157—42no—
33C2×D28—21no—
34C2×D28—21no—
35C2×D28D412yes—
36C2×C4×D7—22no—
37C2×Dic14D411yescharacteristic
38C2×C7×C8—21no—
39C7×C4⋊C4—21no—
40C7×C4⋊C4—21no—
41C7×C4⋊C4D411yescharacteristic
42C7×C22⋊C4—21no—
43C42×C7C2311yescharacteristic
44#1143—44no—
45(C2×D7)⋊C4—22no—
46C4⋊Dic7D411yescharacteristic
47Dic7⋊C4—21no—
48C4×Dic7—21no—
49C2×C7⋊C8—21no—
50C2×C7⋊C8—21no—
51#550—71no—
52#492—71no—
53#477—71no—
54#476D711yescharacteristic, complemented
55#331—71no—
56Q8⋊C8—71no—
57D4⋊C8—71no—
58C23×C7—21no—
59C22×D7—22no—
60C7×Q8—42no—
61C7×D4—21no—
62C7×D4—21no—
63C2×C4×C7—21no—
64C2×C4×C7—41no—
65C2×C4×C7—41no—
66C2×C4×C7—41no—
67C2×C4×C7C2×D411yescharacteristic
68C2×C4×C7C2×D411yescharacteristic
69C2×C4×C7C2×D411yescharacteristic
70C22⋊D7—44no—
71C22⋊D7—82no—
72C2×Dic7—21no—
73C2×Dic7—21no—
74D28—42no—
75D28—41no—
76D28—41no—
77D28—41no—
78D28—41no—
79C4×D7—44no—
80Dic14—21no—
81Dic14SD1612yescomplemented
82C7×C8—41no—
83C7⋊C8—22no—
84C7⋊C8—41no—
85C2×#41—141no—
86C2×SD16—142no—
87C4⋊Q8C2×D711yescharacteristic
88#127—71no—
89#127—141no—
90#121—142no—
91#121—141no—
92C4×Q8—71no—
93C4×D4—141no—
94C4×D4C2×D711yescharacteristic
95#107—141no—
96#106—21no—
97C4⋊C8—71no—
98C4⋊C8C2×D711yescharacteristic
99#103—141no—
100#102—21no—
101C4×C8—71no—
102C22×C7—41no—
103C22×C7—41no—
104C22×C7—41no—
105C22×C7#12011yescharacteristic
106C2×D7—42no—
107C2×D7—42no—
108C2×D7—42no—
109C4×C7—21no—
110C4×C7—41no—
111C4×C7—81no—
112C4×C7C2×SD1611yescharacteristic
113C4×C7#13511yescharacteristic
114C4×C7AGL(1,ℤ/8)11yescharacteristic
115C4×C7#13711yescharacteristic
116Dic7—22no—
117Dic7—41no—
118#41—282no—
119#41—282no—
120C2×Q8—21no—
121C2×Q8—71no—
122C2×D4—72no—
123C2×D4—142no—
124C2×D4—142no—
125C2×D4—141no—
126C2×D4—141no—
127C2×D4C22⋊D711yescharacteristic
128C22×C4—21no—
129C22×C4—142no—
130SD16—284no—
131C2×C8—21no—
132C2×C8—141no—
133C2×C8—141no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—71no—
137C4⋊C4—141no—
138C4⋊C4C22⋊D711yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—142no—
141C42—141no—
142C42C22×D711yescharacteristic
143C2×C7—81no—
144C2×C7#45011yescharacteristic
145C2×C7#45211yescharacteristic
146C2×C7#45311yescharacteristic
147D7—82no—
148C23—21no—
149C23—142no—
150Q8—42no—
151Q8—72no—
152Q8—141no—
153D4—21no—
154D4—21no—
155D4—282no—
156D4—284no—
157D4—281no—
158D4—281no—
159D4—281no—
160D4—281no—
161D4—562no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—141no—
167C2×C4—141no—
168C2×C4—284no—
169C2×C4D4×D711yescharacteristic
170C2×C4D4×D711yescharacteristic
171C2×C4C2×C22⋊D711yescharacteristic
172C8—41no—
173C8—142no—
174C8—281no—
175C7#163111yescharacteristic, complemented, Sylow
176C22—41no—
177C22—41no—
178C22—41no—
179C22—282no—
180C22—282no—
181C22—282no—
182C22#650011yescharacteristic
183C4—21no—
184C4—41no—
185C4—81no—
186C4—142no—
187C4—281no—
188C4D7×SD1611yescharacteristic
189C4#647711yescharacteristic
190C4#650511yescharacteristic
191C4#651311yescharacteristic
192C2—81no—
193C2—562no—
194C2#8981911yescharacteristic
195C2#8994111yescharacteristic
196C2#8997011yescharacteristic
subgroupG/H
centerC22#6500
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#476C2
socleC22×C7#120
prime radical#90244C2
hypercenter#476D7
hypocenterC7#1631
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. C7×#476C2semidirect
3. #89863C2semidirect
4. #89857C2semidirect
5. D4⋊(C7⋊C8)C2semidirect
6. #89280C2semidirect
7. #89279C2semidirect
54. #476D7semidirect
81. Dic14SD16semidirect
175. C7#1631semidirect

Maximal subgroups

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

subgroupindex
#902442
C7×#4762
#898632
#898572
D4⋊(C7⋊C8)2
#892802
#892792
#16317