Finite Group Databasebeta

← #14544183

Subgroups — #14544183

Order 2176.

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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#11763780C211yesmaximal, characteristic
2C4×C22⋊#595C211yesmaximal, characteristic
3#11763662C214yesmaximal, complemented
4C2×#10587583C211yesmaximal, characteristic
5C22×C4×D17C2211yescharacteristic
6C22×Dic34C411yescharacteristic, complemented
7C2×C22⋊#595—21no—
8C2×C4×#595—21no—
9C2×#63243—21no—
10C2×C4⋊C4×C17C411yescharacteristic, complemented
11C2×#63227—21no—
12C2×C4⋊Dic17—21no—
13#10587583—21no—
14#10587583—21no—
15#10587583C412yescomplemented
16#10587583C2212yes—
17C22⋊#3624—21no—
18(C2×#595)⋊C4C2212yes—
19#10587543C2212yes—
20C23×D17D411yescharacteristic
21C22×#595—22no—
22C22×C4×C17—21no—
23C22×C4×C17C2×C411yescharacteristic
24C22×Dic17—21no—
25C22×Dic17D411yescharacteristic
26C2×C4×D17—42no—
27C2×C4×D17C2×C411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×C4×D17D411yescharacteristic
30C2×C4×D17C2311yescharacteristic
31C2×Dic34—21no—
32C2×Dic34—42no—
33C2×Dic34C2×C412yes—
34C22⋊#595—24no—
35C2×#3624—22no—
36C4×#595—42no—
37#63243—42no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17C2×C412yes—
40#63227—22no—
41#63227—22no—
42C4⋊Dic17—22no—
43C4⋊Dic17—22no—
44C23×C17C22⋊C411yescharacteristic
45C22×D17—22no—
46C22×D17—22no—
47C22×D17—41no—
48C22×D17C22⋊C411yescharacteristic
49C22×D17C2×D411yescharacteristic
50C2×#595—24no—
51C2×#595—42no—
52C2×#595—42no—
53C2×C4×C17—22no—
54C2×C4×C17—22no—
55C2×C4×C17—42no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22×C411yescharacteristic
58C2×Dic17—22no—
59C2×Dic17—41no—
60C2×Dic17—42no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63C4×D17—22no—
64C4×D17—22no—
65C4×D17—41no—
66C4×D17—41no—
67Dic34—42no—
68Dic34—42no—
69#3624—42no—
70#1508—171nomaximal, Sylow
71C22×C17—21no—
72C22×C17—21no—
73C22×C17#10412yes—
74C22×C17C2×C22⋊C411yescharacteristic
75C2×D17—21no—
76C2×D17—21no—
77C2×D17—21no—
78C2×D17—21no—
79C2×D17—21no—
80C2×D17—44no—
81C2×D17#13712yes—
82#595—44no—
83C4×C17—21no—
84C4×C17—42no—
85C4×C17#9912yes—
86Dic17—21no—
87Dic17—21no—
88Dic17—42no—
89C2×C22⋊Q8—171no—
90#407—171no—
91C4×C22⋊C4—171no—
92#328—174no—
93C2×C17—22no—
94C2×C17C2×#9911yescharacteristic
95C2×C17#41911yescharacteristic
96C2×C17C2×#10411yescharacteristic
97D17—22no—
98D17—41no—
99C22×Q8—171no—
100C23×C4—171no—
101C2×M16—341no—
102C22⋊Q8—172no—
103C22⋊Q8—172no—
104C22⋊Q8—341no—
105C22⋊Q8—341no—
106C2×C4⋊C4—341no—
107C2×C4⋊C4#59511yescharacteristic, complemented
108C2×C22⋊C4—341no—
109C2×C22⋊C4—341no—
110C2×C42—341no—
111C22⋊C8—172no—
112C22⋊C8—341no—
113#95—172no—
114C17#150811yescharacteristic, complemented, Sylow
115C24—171no—
116C2×Q8—172no—
117C2×Q8—341no—
118C2×Q8—682no—
119C22×C4—21no—
120C22×C4—171no—
121C22×C4—171no—
122C22×C4—171no—
123C22×C4—171no—
124C22×C4—171no—
125C22×C4—342no—
126C22×C4—341no—
127C22×C4—682no—
128C22×C4C2×#59511yescharacteristic
129M16—682no—
130C2×C8—342no—
131C4⋊C4—21no—
132C4⋊C4—342no—
133C4⋊C4—342no—
134C4⋊C4C2×#59512yes—
135C22⋊C4—342no—
136C22⋊C4—342no—
137C22⋊C4—344no—
138C42—682no—
139C23—171no—
140C23—171no—
141C23—342no—
142C23—342no—
143C23—681no—
144C23C22⋊#59511yescharacteristic
145Q8—682no—
146Q8—682no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—171no—
151C2×C4—171no—
152C2×C4—342no—
153C2×C4—342no—
154C2×C4—344no—
155C2×C4—342no—
156C2×C4—681no—
157C2×C4—681no—
158C2×C4—681no—
159C2×C4—682no—
160C2×C4—682no—
161C2×C4—682no—
162C2×C4C22⋊#59511yescharacteristic
163C2×C4C22×#59511yescharacteristic
164C8—682no—
165C22—21no—
166C22—21no—
167C22—172no—
168C22—341no—
169C22—341no—
170C22—341no—
171C22—341no—
172C22—341no—
173C22—684no—
174C22#1058754812yes—
175C22C2×C22⋊#59511yescharacteristic
176C4—21no—
177C4—42no—
178C4—341no—
179C4—341no—
180C4—684no—
181C4—682no—
182C4#1058753912yes—
183C2—22no—
184C2—342no—
185C2—681no—
186C2C2×#1058753911yescharacteristic
187C2#1176380211yescharacteristic
188C2C2×#1058754811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1508
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #11763662C2semidirect
6. C22×Dic34C4semidirect
10. C2×C4⋊C4×C17C4semidirect
15. #10587583C4semidirect
107. C2×C4⋊C4#595semidirect
114. C17#1508semidirect

Maximal subgroups

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

subgroupindex
#117637802
C4×C22⋊#5952
#117636622
C2×#105875832
#150817