Finite Group Databasebeta

← #23766149

Subgroups — #23766149

Order 10112.

460 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 237 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#19098470C214yesmaximal
2#19098462C212yesmaximal
3#19098449C214yesmaximal
4#19098429C211yesmaximal, characteristic
5C4⋊C4×Dic79C212yesmaximal
6#19098409C211yesmaximal, characteristic
7C2×C4⋊Q8×C79C211yesmaximal, characteristic
8C2×C4⋊Dic79C2212yes—
9C2×C4⋊Dic79C2214yes—
10C2×C4×Dic79C2211yescharacteristic
11C2×C4×Dic79C2214yes—
12C2×C4×Dic79C2211yescharacteristic
13C79⋊#95C2212yes—
14C79⋊#95C2212yes—
15C79⋊#95C2214yes—
16C79⋊#95C2212yes—
17C79⋊#95C2212yes—
18C79⋊#95C2214yes—
19C22×Q8×C79C2212yes—
20C4⋊Q8×C79C418yescomplemented
21C2×C4⋊C4×C79C2212yes—
22C2×C4⋊C4×C79C2212yes—
23C2×C42×C79C2211yescharacteristic
24C22×Dic79C2312yes—
25C22×Dic79C2314yes—
26C22×Dic79C2312yes—
27C4⋊Dic79—24no—
28C4⋊Dic79—44no—
29C4×Dic79—22no—
30C4×Dic79—44no—
31C4×Dic79Q814yescomplemented
32C2×Q8×C79—44no—
33C2×Q8×C79C2×C418yes—
34C22×C4×C79C2311yescharacteristic
35C22×C4×C79C2311yescharacteristic
36C22×C4×C79C2314yes—
37C22×C4×C79C2311yescharacteristic
38C4⋊C4×C79C2×C418yes—
39C4⋊C4×C79C2×C418yes—
40C42×C79C2×C414yes—
41C23×C79C2411yescharacteristic
42Q8×C79—48no—
43C2×C4×C79—24no—
44C2×C4×C79—24no—
45C2×C4×C79—22no—
46C2×C4×C79—22no—
47C2×C4×C79C22×C412yes—
48C2×C4×C79C22×C412yes—
49C2×C4×C79C22×C418yes—
50C2×C4×C79C22×C412yes—
51C2×C4×C79C2×Q812yes—
52C2×C4×C79#4112yes—
53C2×Dic79—22no—
54C2×Dic79—22no—
55C2×Dic79—22no—
56C2×Dic79—24no—
57C2×Dic79—24no—
58C2×Dic79—22no—
59C2×Dic79—22no—
60C2×Dic79—24no—
61C2×Dic79C2×Q814yes—
62C22×C79C23×C411yescharacteristic
63C22×C79C22×Q811yescharacteristic
64C22×C79C2×#4111yescharacteristic
65C22×C79H(5,𝔽2)12yes—
66C22×C792−1+412yes—
67C4×C79—28no—
68C4×C79—22no—
69C4×C79—22no—
70C4×C79C4×Q814yes—
71Dic79—24no—
72Dic79—42no—
73Dic79—44no—
74C2×C79C2×C4×Q811yescharacteristic
75C2×C79#51811yescharacteristic
76C2×C79#51911yescharacteristic
77C2×C79#54212yes—
78C2×C79#54412yes—
79#2331—791nomaximal, Sylow
80C79#233111yescharacteristic, complemented, Sylow
81C2×C4⋊Q8D7911yescharacteristic
82#391—794no—
83#389—792no—
84#387—794no—
85#382—791no—
86C4×C4⋊C4—792no—
87#376—791no—
88C22×Q8C2×D7912yes—
89C4⋊Q8Dic7918yescomplemented
90C2×C4⋊C4—794no—
91C2×C4⋊C4—792no—
92C2×C4⋊C4C2×D7912yes—
93C2×C4⋊C4C2×D7912yes—
94C2×C42—794no—
95C2×C42—791no—
96C2×C42—791no—
97C2×C42C2×D7911yescharacteristic
98#95—792no—
99#95—792no—
100#95—794no—
101#95—792no—
102#95—792no—
103#95—794no—
104C2×Q8—44no—
105C2×Q8C2×Dic7918yes—
106C22×C4—792no—
107C22×C4—792no—
108C22×C4—794no—
109C22×C4C22×D7911yescharacteristic
110C22×C4C22×D7911yescharacteristic
111C22×C4C22×D7914yes—
112C22×C4C22×D7911yescharacteristic
113C4⋊C4—3164no—
114C4⋊C4—1584no—
115C4⋊C4C2×Dic7918yes—
116C4⋊C4C2×Dic7918yes—
117C42—3164no—
118C42—794no—
119C42—1582no—
120C42C2×Dic7914yes—
121C23C23×D7911yescharacteristic
122Q8—48no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—794no—
128C2×C4—1582no—
129C2×C4—1584no—
130C2×C4—1584no—
131C2×C4—1582no—
132C2×C4—1582no—
133C2×C4—1584no—
134C2×C4—1582no—
135C2×C4—1582no—
136C2×C4C22×Dic7912yes—
137C2×C4C22×Dic7912yes—
138C2×C4C22×Dic7918yes—
139C2×C4C22×Dic7912yes—
140C2×C4Q8×D7912yes—
141C2×C4#1192962512yes—
142C22C23×Dic7911yescharacteristic
143C22C2×Q8×D7911yescharacteristic
144C22C2×#1192962511yescharacteristic
145C22#1457920612yes—
146C22#1457920912yes—
147C4—22no—
148C4—28no—
149C4—22no—
150C4—3162no—
151C4—3164no—
152C4—1584no—
153C4Q8×Dic7914yes—
154C2C2×Q8×Dic7911yescharacteristic
155C2#1909903811yescharacteristic
156C2#1909904111yescharacteristic
157C2#1909918712yes—
158C2#1909919012yes—
subgroupG/H
center, FrattiniC23C23×D79
commutatorC22×C79C23×C4
FittingC2×C4⋊Q8×C79C2
socle, prime radical, cosocleC23×C79C24
hypercenterC2×C4⋊Q8D79
hypocenterC79#2331
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
20. C4⋊Q8×C79C4semidirect
31. C4×Dic79Q8semidirect
80. C79#2331semidirect
89. C4⋊Q8Dic79semidirect