Finite Group Databasebeta

← #14544296

Subgroups — #14544296

Order 2176.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763812C211yesmaximal, characteristic, complemented
2C2×#10587544C211yesmaximal, characteristic, complemented
3#11763668C212yesmaximal, complemented
4#11763667C212yesmaximal, complemented
5#11763354C211yesmaximal, characteristic
6C2×#63254—21no—
7C2×#63247—21no—
8C22×#3624—21no—
9C17×#121C411yescharacteristic
10#10587626C2211yescharacteristic
11#10587625—21no—
12D4×Dic17C411yescharacteristic
13#10587616C411yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15#10587579C411yescharacteristic
16#10587574C2211yescharacteristic
17#10587571—21no—
18#10587567—21no—
19#10587544—21no—
20#10587544C2212yes—
21#10587544C2212yes—
22C4⋊#3624—21no—
23C2×D4×C17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29C22×Dic17—21no—
30C22×Dic17D411yescharacteristic
31C22×Dic17D411yescharacteristic
32#63254—42no—
33#63254—42no—
34C2×D68C2×C411yescharacteristic
35C2×C4×D17—21no—
36C2×Dic34—21no—
37#63247—42no—
38C2×#3624—22no—
39C2×#3624—22no—
40C2×#3624—42no—
41C4⋊C4×C17C2×C411yescharacteristic
42C22⋊C4×C17—21no—
43C22⋊Dic17—21no—
44#63227—21no—
45#63227—21no—
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48C4×Dic17C2×C411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17C2311yescharacteristic
52C23×C17—21no—
53C23×C17C22⋊C411yescharacteristic
54C22×D17—21no—
55D4×C17—41no—
56D4×C17—42no—
57C2×C4×C17—21no—
58C2×C4×C17—41no—
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22×C411yescharacteristic
61C22⋊D17—42no—
62C22⋊D17—41no—
63C22⋊D17—41no—
64C22⋊D17—41no—
65C22⋊D17—41no—
66C2×Dic17—21no—
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17—42no—
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C22⋊C411yescharacteristic
72C2×Dic17D812yes—
73C2×Dic17SD1612yes—
74C2×Dic17C2×D411yescharacteristic
75C2×Dic17C2×D411yescharacteristic
76D68—41no—
77C4×D17—42no—
78Dic34—41no—
79Dic34—41no—
80#3624—42no—
81#3624—42no—
82#1547—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—41no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17#10212yes—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—41no—
91C2×D17—41no—
92C2×D17—41no—
93C4×C17—21no—
94C4×C17—21no—
95C4×C17—41no—
96Dic17—21no—
97Dic17—41no—
98Dic17#10212yes—
99Dic17C2×D811yescharacteristic
100Dic17C2×SD1611yescharacteristic
101Dic17AGL(1,ℤ/8)11yescharacteristic
102Dic17#13711yescharacteristic
103#532—171no—
104#423—171no—
105C2×C4⋊C8—171no—
106#332—172no—
107#331—172no—
108C2×C17—22no—
109C2×C17—81no—
110C2×C17#41311yescharacteristic
111C2×C17C2×#10211yescharacteristic
112C2×C17#41711yescharacteristic
113D17—81no—
114C2×#41—341no—
115C2×M16—341no—
116C22×C8—341no—
117C4⋊Q8—171no—
118#127—171no—
119#124—341no—
120#121—171no—
121#121—341no—
122#121#59511yescharacteristic
123C4×D4—171no—
124C4×D4—171no—
125C4×D4—341no—
126C2×C42—171no—
127C4⋊C8—172no—
128C4⋊C8—172no—
129C4⋊C8—341no—
130C4⋊C8—341no—
131C17#154711yescharacteristic, complemented, Sylow
132#41—682no—
133#41—682no—
134C2×Q8—341no—
135C2×D4—21no—
136C2×D4—171no—
137C2×D4—341no—
138C2×D4—341no—
139C2×D4—341no—
140C2×D4C2×#59511yescharacteristic
141C22×C4—171no—
142C22×C4—171no—
143C22×C4—341no—
144C22×C4—341no—
145C22×C4C2×#59511yescharacteristic
146M16—682no—
147C2×C8—342no—
148C2×C8—342no—
149C2×C8—682no—
150C4⋊C4—171no—
151C4⋊C4—341no—
152C4⋊C4C2×#59511yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—341no—
155C22⋊C4—341no—
156C22⋊C4—341no—
157C42—171no—
158C42—171no—
159C42—171no—
160C42—171no—
161C23—21no—
162C23—341no—
163C23C22⋊#59511yescharacteristic
164Q8—681no—
165Q8—681no—
166D4—41no—
167D4—42no—
168D4—681no—
169D4—681no—
170D4—681no—
171D4—681no—
172D4—681no—
173D4—682no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—171no—
177C2×C4—171no—
178C2×C4—171no—
179C2×C4—172no—
180C2×C4—172no—
181C2×C4—171no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—341no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4C22⋊#59511yescharacteristic
188C2×C4C22×#59511yescharacteristic
189C8—682no—
190C8—682no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—681no—
197C22—681no—
198C22—681no—
199C22#1058754712yes—
200C22C2×C22⋊#59511yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—171no—
205C4—172no—
206C4—171no—
207C4—171no—
208C4—171no—
209C4—341no—
210C4—681no—
211C2—22no—
212C2—81no—
213C2—1361no—
214C2#1176379611yescharacteristic
215C2C2×#1058754711yescharacteristic
216C2#1176380011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#11763354C2
cosocleC4×Dic17C23
hypercenter#121#595
hypocenterC17#1547
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763812C2semidirect
2. C2×#10587544C2semidirect
3. #11763668C2semidirect
4. #11763667C2semidirect
131. C17#1547semidirect

Maximal subgroups

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

subgroupindex
#117638122
C2×#105875442
#117636682
#117636672
#117633542
#154717