Finite Group Databasebeta

← #11729613

Subgroups — #11729613

Order 896.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C7×#535C211yesmaximal, characteristic
2#90012C211yesmaximal, characteristic, complemented
3#90009C211yesmaximal, characteristic, complemented
4#89331C214yesmaximal, complemented
5C2×C7×#41C2211yescharacteristic
6C22×C7×D4C411yescharacteristic, complemented
7C7×#127—21no—
8C7×#124—21no—
9C7×#121—21no—
10C7×#121—21no—
11C7×#120C412yescomplemented
12C7×#120C2212yes—
13C7×#117C411yescharacteristic, complemented
14#6489—21no—
15C2×#1136—21no—
16#6410—21no—
17#6408—21no—
18#6408C2212yes—
19#6407C2212yes—
20#2122—71nomaximal, Sylow
21C24×C7C2×C412yes—
22C7×#41—41no—
23C7×#41—41no—
24C2×C7×Q8D411yescharacteristic
25C2×C7×D4—22no—
26C2×C7×D4—22no—
27C2×C7×D4—21no—
28C2×C7×D4—21no—
29C2×C7×D4—41no—
30C2×C7×D4—41no—
31C2×C7×D4C2×C411yescharacteristic
32C2×C7×D4D411yescharacteristic
33C2×C7×D4C2311yescharacteristic
34C22×C4×C7C2×C411yescharacteristic
35C22×C4×C7D411yescharacteristic
36C22×C4×C7D411yescharacteristic
37C7×C4⋊C4—21no—
38C7×C22⋊C4—22no—
39C7×C22⋊C4—22no—
40C7×C22⋊C4C2×C412yes—
41C42×C7—21no—
42C22⋊Dic7—22no—
43C4⋊Dic7—41no—
44C4×Dic7—41no—
45#1136—22no—
46#1136—41no—
47C2×C7⋊C8—41no—
48#535D711yescharacteristic
49#413—71no—
50#410—71no—
51AΣL(1,𝔽16)—74no—
52C23×C7—22no—
53C23×C7—42no—
54C23×C7—82no—
55C23×C7C22⋊C411yescharacteristic
56C23×C7C22×C411yescharacteristic
57C23×C7C2×D411yescharacteristic
58C7×Q8—41no—
59C7×D4—41no—
60C7×D4—41no—
61C7×D4—41no—
62C7×D4—42no—
63C7×D4—41no—
64C7×D4—41no—
65C7×D4—82no—
66C7×D4—82no—
67C2×C4×C7—21no—
68C2×C4×C7—21no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7—22no—
74C2×C4×C7C22⋊C411yescharacteristic
75C2×C4×C7C22⋊C411yescharacteristic
76C2×C4×C7C22⋊C411yescharacteristic
77C2×C4×C7C2×D411yescharacteristic
78C2×Dic7—42no—
79C7⋊C8—42no—
80C2×#41C2×D711yescharacteristic
81C22×D4Dic711yescharacteristic, complemented
82C2×M16—141no—
83#127—21no—
84#124—21no—
85#121—21no—
86#121—21no—
87#120Dic712yescomplemented
88#120C2×D712yes—
89#117—141no—
90#117Dic711yescharacteristic, complemented
91#101—141no—
92#100—72no—
93#99—72no—
94#99—141no—
95C22×C7—21no—
96C22×C7—21no—
97C22×C7—21no—
98C22×C7—41no—
99C22×C7—41no—
100C22×C7—41no—
101C22×C7—42no—
102C22×C7—42no—
103C22×C7—42no—
104C22×C7—84no—
105C22×C7C2×C22⋊C411yescharacteristic
106C4×C7—21no—
107C4×C7—21no—
108C4×C7—21no—
109C4×C7—42no—
110C4×C7#9911yescharacteristic
111C4×C7#9911yescharacteristic
112Dic7—82no—
113C24C2×Dic712yes—
114#41—41no—
115#41—41no—
116C2×Q8C22⋊D711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4C2×Dic711yescharacteristic
124C2×D4C22⋊D711yescharacteristic
125C2×D4C22×D711yescharacteristic
126C22×C4C2×Dic711yescharacteristic
127C22×C4C22⋊D711yescharacteristic
128C22×C4C22⋊D711yescharacteristic
129M16—142no—
130M16—281no—
131C2×C8—281no—
132C4⋊C4—21no—
133C4⋊C4—281no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—142no—
137C22⋊C4C2×Dic712yes—
138C42—21no—
139C42—281no—
140C2×C7—21no—
141C2×C7—41no—
142C2×C7—41no—
143C2×C7—41no—
144C2×C7—82no—
145C2×C7C2×#9911yescharacteristic
146C23—22no—
147C23—42no—
148C23—82no—
149C23C22⋊Dic711yescharacteristic
150C23C22×Dic711yescharacteristic
151C23C2×C22⋊D711yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—82no—
160D4—82no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—282no—
169C2×C4C22⋊Dic711yescharacteristic
170C2×C4C22⋊Dic711yescharacteristic
171C2×C4C22⋊Dic711yescharacteristic
172C2×C4C2×C22⋊D711yescharacteristic
173C8—282no—
174C7#212211yescharacteristic, complemented, Sylow
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—562no—
191C4#640811yescharacteristic
192C4#640811yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#640811yescharacteristic
subgroupG/H
centerC2C2×#6408
commutatorC23×C7C22×C4
FrattiniC2×D4C22×D7
Fitting, prime radicalC7×#535C2
socleC2×C7C2×#99
cosocleC2×C7×D4C23
hypercenter#535D7
hypocenterC7#2122
LCSG ▷ C23×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
2. #90012C2semidirect
3. #90009C2semidirect
4. #89331C2semidirect
6. C22×C7×D4C4semidirect
11. C7×#120C4semidirect
13. C7×#117C4semidirect
81. C22×D4Dic7semidirect
87. #120Dic7semidirect
90. #117Dic7semidirect
174. C7#2122semidirect

Maximal subgroups

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

subgroupindex
C7×#5352
#900122
#900092
#893312
#21227