Finite Group Databasebeta

← #14545056

Subgroups — #14545056

Order 2176.

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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#11763790C212yesmaximal, complemented
2#11763716C212yesmaximal, complemented
3#11763713C212yesmaximal, complemented
4#11763358C211yesmaximal, characteristic
5C2×#63256C2211yescharacteristic
6C2×#63254C411yescharacteristic, complemented
7#10587677—22no—
8C17×#124C411yescharacteristic, complemented
9#10587630C411yescharacteristic, complemented
10#10587626C2211yescharacteristic
11#10587570—22no—
12#10587567—22no—
13#10587561C2211yescharacteristic
14C42×D17C411yescharacteristic, complemented
15#10587539—22no—
16#10587539C2214yes—
17C2×Q8×C17C2×C411yescharacteristic
18C2×D4×C17C2×C411yescharacteristic
19C2×C22⋊D17—21no—
20C2×C22⋊D17—22no—
21C22×Dic17—21no—
22#63256—41no—
23#63256—41no—
24#63254—41no—
25#63254—41no—
26C2×D68D412yes—
27C2×D68C2311yescharacteristic
28C2×C4×D17—21no—
29C2×C4×D17C2×C411yescharacteristic
30C2×C4×D17D412yes—
31C2×Dic34C2×C411yescharacteristic
32C22⋊#595—24no—
33C4⋊#595—42no—
34C4×#595—42no—
35C22⋊C4×C17—22no—
36C42×C17C2×C411yescharacteristic
37#63227—22no—
38#63225—22no—
39C4×Dic17—21no—
40C4×Dic17C2×C411yescharacteristic
41C23×C17—21no—
42C22×D17C22⋊C411yescharacteristic
43C22×D17C2×D412yes—
44C2×#595—44no—
45Q8×C17—41no—
46D4×C17—41no—
47C2×C4×C17—21no—
48C2×C4×C17C22⋊C412yes—
49C2×C4×C17C22×C411yescharacteristic
50C22⋊D17—44no—
51C22⋊D17—42no—
52C2×Dic17—21no—
53C2×Dic17—21no—
54C2×Dic17—41no—
55C2×Dic17—41no—
56C2×Dic17C22⋊C411yescharacteristic
57D68—42no—
58D68—41no—
59C4×D17—22no—
60C4×D17—22no—
61C4×D17—21no—
62C4×D17—21no—
63C4×D17—42no—
64Dic34—41no—
65#2126—171nomaximal, Sylow
66C22×C17—42no—
67C22×C17—41no—
68C22×C17C2×C22⋊C411yescharacteristic
69C2×D17—21no—
70C2×D17—22no—
71C2×D17—41no—
72C2×D17—42no—
73#595—84no—
74C4×C17—21no—
75C4×C17—22no—
76C4×C17—41no—
77Dic17—22no—
78Dic17—41no—
79Dic17#9911yescharacteristic
80Dic17#9911yescharacteristic
81#532—171no—
82#410—172no—
83#354—172no—
84#353—172no—
85C2×C17—21no—
86C2×C17—81no—
87C2×C17C2×#9911yescharacteristic
88D17—42no—
89D17—41no—
90C2×#41—171no—
91C2×#41—171no—
92C4⋊Q8—171no—
93#127—171no—
94#124—171no—
95#124#59511yescharacteristic, complemented
96#121—342no—
97C4×D4—342no—
98#117—342no—
99C2×C42—171no—
100#99—174no—
101#99—342no—
102C17#212611yescharacteristic, complemented, Sylow
103#41—681no—
104#41—681no—
105#41—681no—
106#41—681no—
107C2×Q8—171no—
108C2×Q8C2×#59511yescharacteristic
109C2×D4—172no—
110C2×D4—171no—
111C2×D4—342no—
112C2×D4—341no—
113C2×D4C2×#59511yescharacteristic
114C22×C4—172no—
115C22×C4—171no—
116C22×C4—341no—
117C22×C4—341no—
118C4⋊C4—342no—
119C4⋊C4—682no—
120C22⋊C4—22no—
121C22⋊C4—342no—
122C22⋊C4—344no—
123C42—171no—
124C42—341no—
125C42—682no—
126C42C2×#59511yescharacteristic
127C23—21no—
128C23—172no—
129C23—171no—
130Q8—41no—
131Q8—681no—
132D4—41no—
133D4—682no—
134D4—681no—
135D4—684no—
136D4—682no—
137C2×C4—21no—
138C2×C4—171no—
139C2×C4—342no—
140C2×C4—342no—
141C2×C4—341no—
142C2×C4—341no—
143C2×C4—341no—
144C2×C4—341no—
145C2×C4—682no—
146C2×C4—681no—
147C2×C4—681no—
148C2×C4—684no—
149C2×C4C22⋊#59512yes—
150C2×C4C22×#59511yescharacteristic
151C22—42no—
152C22—41no—
153C22—342no—
154C22—341no—
155C22—681no—
156C22—682no—
157C22C2×C22⋊#59511yescharacteristic
158C4—22no—
159C4—21no—
160C4—41no—
161C4—171no—
162C4—171no—
163C4—342no—
164C4—681no—
165C4—1364no—
166C2—21no—
167C2—81no—
168C2—682no—
169C2—681no—
170C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC2×C17C2×#99
prime radical#11763358C2
cosocleC2×D68C23
hypercenter#124#595
hypocenterC17#2126
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11763790C2semidirect
2. #11763716C2semidirect
3. #11763713C2semidirect
6. C2×#63254C4semidirect
8. C17×#124C4semidirect
9. #10587630C4semidirect
14. C42×D17C4semidirect
95. #124#595semidirect
102. C17#2126semidirect

Maximal subgroups

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

subgroupindex
#117637902
#117637162
#117637132
#117633582
#212617