Finite Group Databasebeta

← #24909151

Subgroups — #24909151

Order 10624.

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

normal subgroups only

This group has 230 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#19128276C211yesmaximal, characteristic
2#101×D83C212yesmaximal, complemented
3#19127500C211yesmaximal, characteristic, complemented
4#19127497C212yesmaximal, complemented
5#356×C83C211yesmaximal, characteristic, complemented
6C2×#13050591C411yescharacteristic
7C2×Q8×D83C2211yescharacteristic
8M16×D83—22no—
9#14581790C411yescharacteristic
10#14581781C2211yescharacteristic
11#14581780C411yescharacteristic
12#14581777—22no—
13#14581770—22no—
14#14581749C411yescharacteristic
15#14581711C2212yes—
16#14581710—22no—
17C4⋊Q8×C83C2211yescharacteristic
18#101×C83C2212yes—
19#13050591—42no—
20C2×Dic166C2×C411yescharacteristic
21C2×Dic166D412yes—
22Q8×D83—42no—
23C2×D332—21no—
24C2×D332C2×C411yescharacteristic
25C2×C4×D83—21no—
26C2×C4×D83C2×C411yescharacteristic
27C2×C4×D83D412yes—
28#13050574—22no—
29#13050573—42no—
30C8×D83—42no—
31Dic83⋊C4—21no—
32Dic83⋊C4—22no—
33(C2×D83)⋊C4—21no—
34(C2×D83)⋊C4—22no—
35(C2×D83)⋊C4—22no—
36C4×Dic83C2×C411yescharacteristic
37C2×Q8×C83C2×C411yescharacteristic
38C2×Q8×C83C2311yescharacteristic
39M16×C83—22no—
40C4⋊C4×C83—22no—
41C42×C83C2×C411yescharacteristic
42C22×D83—21no—
43C22×D83C22⋊C411yescharacteristic
44Q8×C83—41no—
45Q8×C83—41no—
46C2×C4×C83—21no—
47C2×C4×C83—21no—
48C2×C4×C83C22×C411yescharacteristic
49C2×C4×C83C2×D412yes—
50C2×Dic83—21no—
51C2×Dic83C22⋊C412yes—
52C2×Dic83C22⋊C411yescharacteristic
53D332—41no—
54D332—42no—
55C4×D83—22no—
56C4×D83—22no—
57C4×D83—22no—
58C4×D83—42no—
59Dic166—41no—
60Dic166—42no—
61C8×C83—42no—
62C83⋊C8—42no—
63C22×C83C2×C22⋊C411yescharacteristic
64C2×D83—41no—
65C2×D83—42no—
66C2×D83—41no—
67C2×D83#9912yes—
68C4×C83—22no—
69C4×C83—21no—
70C4×C83—41no—
71C4×C83—41no—
72Dic83—21no—
73Dic83—22no—
74Dic83—41no—
75C2×C83—21no—
76C2×C83C2×#9911yescharacteristic
77D83—22no—
78D83—81no—
79#2133—831nomaximal, Sylow
80C83#213311yescharacteristic, complemented, Sylow
81#536—831no—
82C2×#101—832no—
83#356—831no—
84#356D8311yescharacteristic, complemented
85#355—832no—
86C2×#41—831no—
87C22×Q8—831no—
88C2×M16—1662no—
89C4⋊Q8—831no—
90C4⋊Q8C2×D8311yescharacteristic
91#124—831no—
92#124—831no—
93#123—1662no—
94C22⋊Q8—1662no—
95#117—831no—
96#101—832no—
97#101—1662no—
98#101C2×D8312yes—
99#41—3322no—
100C2×Q8—3322no—
101C2×Q8—831no—
102C2×Q8—832no—
103C2×Q8C4×D8311yescharacteristic
104C2×Q8C22×D8311yescharacteristic
105C2×D4—831no—
106C2×D4—1661no—
107C22×C4—831no—
108C22×C4—832no—
109C22×C4—1661no—
110M16—22no—
111M16—3322no—
112M16—1662no—
113C2×C8—3322no—
114C4⋊C4—22no—
115C4⋊C4—1661no—
116C4⋊C4—1662no—
117C22⋊C4—1662no—
118C22⋊C4—1661no—
119C22⋊C4—1662no—
120C42—831no—
121C42C4×D8311yescharacteristic
122C23—831no—
123C23—1661no—
124Q8—41no—
125Q8—41no—
126Q8—3322no—
127Q8—3321no—
128D4—3321no—
129D4—3322no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—3322no—
133C2×C4—832no—
134C2×C4—831no—
135C2×C4—1662no—
136C2×C4—1662no—
137C2×C4—1662no—
138C2×C4—1661no—
139C2×C4C2×C4×D8311yescharacteristic
140C2×C4D4×D8312yes—
141C8—42no—
142C8—3322no—
143C22—3322no—
144C22—3321no—
145C22—3321no—
146C22—832no—
147C22C22⋊C4×D8311yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—3321no—
153C4—1661no—
154C4—1662no—
155C2—21no—
156C2—6641no—
157C2—1662no—
158C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC2×C4×C83C22×C4
FrattiniC2×Q8C22×D83
Fitting#356×C83C2
socleC2×C83C2×#99
prime radicalC2×#13050591C4
cosocleC2×Q8×C83C23
hypercenter#356D83
hypocenterC83#2133
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #101×D83C2semidirect
3. #19127500C2semidirect
4. #19127497C2semidirect
5. #356×C83C2semidirect
80. C83#2133semidirect
84. #356D83semidirect

Maximal subgroups

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

subgroupindex
#191282762
#101×D832
#191275002
#191274972
#356×C832
#213383