Finite Group Databasebeta

← #14544234

Subgroups — #14544234

Order 2176.

234 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C4×#63243C211yesmaximal, characteristic, complemented
2#11763768C211yesmaximal, characteristic, complemented
3#11763666C212yesmaximal, complemented
4#11763664C212yesmaximal, complemented
5#11763341C211yesmaximal, characteristic
6C2×#63243—21no—
7C2×#63242—21no—
8C4×D4×C17C411yescharacteristic
9#10587624C411yescharacteristic
10#10587622C2211yescharacteristic
11C4×C22⋊D17—21no—
12#10587612—21no—
13#10587583C411yescharacteristic
14#10587576C2211yescharacteristic
15#10587569—21no—
16#10587565—21no—
17C42×D17C2211yescharacteristic
18C4×Dic34C411yescharacteristic
19#10587542—21no—
20#10587542C2212yes—
21#10587542C2212yes—
22C4×#3624—21no—
23C2×D4×C17C2×C411yescharacteristic
24C22×C4×C17—21no—
25C2×C22⋊D17—21no—
26C2×C4×D17C2×C411yescharacteristic
27C2×C4×D17D411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—22no—
31C2×#3624—22no—
32#63243—24no—
33#63242—24no—
34C4⋊C4×C17C2×C411yescharacteristic
35C22⋊C4×C17—21no—
36C42×C17C2×C411yescharacteristic
37C22⋊Dic17—21no—
38C22⋊Dic17—21no—
39C22⋊Dic17—21no—
40#63227—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17C2×C411yescharacteristic
44#63225—21no—
45#63225—21no—
46C4×Dic17—21no—
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17C2311yescharacteristic
50C23×C17—21no—
51C22×D17C22⋊C411yescharacteristic
52D4×C17—41no—
53C2×C4×C17—21no—
54C2×C4×C17—42no—
55C2×C4×C17C22⋊C411yescharacteristic
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22×C411yescharacteristic
58C22⋊D17—42no—
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C4×D17—22no—
65C4×D17—22no—
66C4×D17—41no—
67Dic34—41no—
68#3624—24no—
69#3624—24no—
70#1524—171nomaximal, Sylow
71C22×C17—41no—
72C22×C17—41no—
73C22×C17—41no—
74C22×C17C2×C22⋊C411yescharacteristic
75C2×D17—21no—
76C2×D17—21no—
77C2×D17#10412yes—
78C4×C17—21no—
79C4×C17—21no—
80C4×C17—41no—
81C4×C17#10412yes—
82Dic17—21no—
83Dic17—21no—
84Dic17—21no—
85Dic17—41no—
86Dic17—41no—
87#529—171no—
88C4×M16—171no—
89C2×#97—171no—
90#330—172no—
91#329—172no—
92C2×C17—81no—
93C2×C17#41311yescharacteristic
94C2×C17C2×#10411yescharacteristic
95C2×C17C2×#10411yescharacteristic
96D17—22no—
97C2×M16—341no—
98C22×C8—341no—
99#126—341no—
100#125—171no—
101#124—171no—
102#123—341no—
103C22⋊Q8—171no—
104#121—171no—
105C4×Q8—171no—
106C4×D4—341no—
107C4×D4#59511yescharacteristic
108#117—341no—
109C2×C42—171no—
110#97—172no—
111#97—172no—
112#97—341no—
113C4×C8—341no—
114C17#152411yescharacteristic, complemented, Sylow
115C2×Q8—171no—
116C2×D4—341no—
117C2×D4C2×#59511yescharacteristic
118C22×C4—21no—
119C22×C4—171no—
120C22×C4—171no—
121C22×C4—171no—
122M16—344no—
123C2×C8—342no—
124C2×C8—342no—
125C2×C8—344no—
126C4⋊C4—171no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4C2×#59511yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—341no—
134C22⋊C4—341no—
135C22⋊C4—341no—
136C22⋊C4—341no—
137C42—171no—
138C42—171no—
139C42—171no—
140C42—341no—
141C42C2×#59511yescharacteristic
142C23—21no—
143C23—171no—
144Q8—681no—
145D4—41no—
146D4—682no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—171no—
150C2×C4—171no—
151C2×C4—171no—
152C2×C4—342no—
153C2×C4—342no—
154C2×C4—341no—
155C2×C4—341no—
156C2×C4—681no—
157C2×C4C22⋊#59511yescharacteristic
158C2×C4C22⋊#59511yescharacteristic
159C2×C4C22×#59511yescharacteristic
160C8—344no—
161C8—344no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22—172no—
166C22—341no—
167C22—341no—
168C22C2×C22⋊#59511yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—341no—
173C4—341no—
174C4—341no—
175C4—681no—
176C4—681no—
177C4#1058754812yes—
178C2—81no—
179C2—342no—
180C2#1176379511yescharacteristic
181C2C2×#1058754811yescharacteristic
182C2D17⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#10587624C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1524
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×#63243C2semidirect
2. #11763768C2semidirect
3. #11763666C2semidirect
4. #11763664C2semidirect
114. C17#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#632432
#117637682
#117636662
#117636642
#117633412
#152417