Finite Group Databasebeta

← #14544124

Subgroups — #14544124

Order 2176.

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

normal subgroups only

This group has 316 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#11763831C211yesmaximal, characteristic, complemented
2C4×#63243C211yesmaximal, characteristic, complemented
3#11763657C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5C4×D4×D17C211yesmaximal, characteristic
6C2×D4×D17C411yescharacteristic
7C22×C4×D17—21no—
8C2×#63243—21no—
9C4×D4×C17C411yescharacteristic
10D4×Dic17C2211yescharacteristic
11C4×C22⋊D17—21no—
12#10587579C2211yescharacteristic
13C4⋊C4×D17C411yescharacteristic
14#10587567—21no—
15C22⋊C4×D17—21no—
16C4×D68C411yescharacteristic
17C42×D17C2211yescharacteristic
18#10587543—21no—
19#10587542—21no—
20#10587542—21no—
21C4⋊#3624C2212yescomplemented
22C4×#3624C2212yescomplemented
23C23×D17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C22×Dic17—21no—
28D4×D17—22no—
29D4×D17—22no—
30C2×D68C2×C411yescharacteristic
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×C4×D17—44no—
35C2×C4×D17C2×C411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×#3624—22no—
39C2×#3624—22no—
40#63243—24no—
41C4⋊C4×C17C2×C411yescharacteristic
42C22⋊C4×C17—21no—
43C42×C17C2×C411yescharacteristic
44C22⋊Dic17—21no—
45#63227—21no—
46#63227—21no—
47C4⋊Dic17C2×C411yescharacteristic
48#63225—21no—
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17C2311yescharacteristic
52C23×C17—21no—
53C22×D17—24no—
54C22×D17—21no—
55C22×D17—42no—
56C22×D17—42no—
57C22×D17C22⋊C411yescharacteristic
58D4×C17—21no—
59D4×C17M1612yescomplemented
60C2×C4×C17—21no—
61C2×C4×C17—42no—
62C2×C4×C17C22⋊C411yescharacteristic
63C2×C4×C17C22⋊C411yescharacteristic
64C2×C4×C17C22×C411yescharacteristic
65C22⋊D17—24no—
66C2×Dic17—21no—
67C2×Dic17—42no—
68C2×Dic17C22⋊C411yescharacteristic
69C2×Dic17C2×D411yescharacteristic
70C2×Dic17C2×D411yescharacteristic
71D68—21no—
72D68M1612yescomplemented
73C4×D17—22no—
74C4×D17—21no—
75C4×D17—21no—
76C4×D17—22no—
77C4×D17—42no—
78C4×D17—42no—
79C4×D17—42no—
80#3624—24no—
81#3624—42no—
82#1490—171nomaximal, Sylow
83C22×C17—22no—
84C22×C17—41no—
85C22×C17—41no—
86C22×C17C2×C22⋊C411yescharacteristic
87C2×D17—21no—
88C2×D17—21no—
89C2×D17—22no—
90C2×D17—48no—
91C2×D17—41no—
92C2×D17—41no—
93C2×D17#10412yes—
94C4×C17—21no—
95C4×C17—21no—
96C4×C17—41no—
97C4×C17C2×M1611yescharacteristic
98C4×C17C2×M1611yescharacteristic
99Dic17—21no—
100Dic17—21no—
101Dic17—41no—
102Dic17AGL(1,ℤ/8)12yes—
103C2×C4×D4—171no—
104#432—171no—
105C4×M16—171no—
106D4⋊C8—172no—
107D4⋊C8—172no—
108C2×C17—42no—
109C2×C17#40711yescharacteristic
110C2×C17#41811yescharacteristic
111C2×C17C2×#10411yescharacteristic
112D17—22no—
113D17—42no—
114C22×D4—171no—
115C23×C4—341no—
116C2×M16—341no—
117C4×D4—171no—
118C4×D4—171no—
119C4×D4—171no—
120C4×D4—341no—
121C4×D4—341no—
122C4×D4#59511yescharacteristic
123C2×C4⋊C4—171no—
124C2×C22⋊C4—341no—
125C2×C42—171no—
126C4⋊C8—172no—
127C22⋊C8—341no—
128#97—341no—
129#97—341no—
130C4×C8—172no—
131C17#149011yescharacteristic, complemented, Sylow
132C24—341no—
133C2×D4—171no—
134C2×D4—341no—
135C2×D4—342no—
136C2×D4—342no—
137C2×D4C2×#59511yescharacteristic
138C22×C4—21no—
139C22×C4—171no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—341no—
143C22×C4—341no—
144C22×C4—341no—
145C22×C4—341no—
146C22×C4—684no—
147M16—344no—
148C2×C8—342no—
149C2×C8—342no—
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—
160C42C2×#59511yescharacteristic
161C23—21no—
162C23—171no—
163C23—344no—
164C23—341no—
165C23—682no—
166C23—682no—
167D4—21no—
168D4—172no—
169D4—344no—
170D4—341no—
171D4#6324312yescomplemented
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—171no—
175C2×C4—171no—
176C2×C4—171no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—341no—
180C2×C4—341no—
181C2×C4—341no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4—682no—
185C2×C4—682no—
186C2×C4C22⋊#59511yescharacteristic
187C2×C4C22⋊#59511yescharacteristic
188C2×C4C22×#59511yescharacteristic
189C8—344no—
190C8—682no—
191C22—22no—
192C22—41no—
193C22—41no—
194C22—172no—
195C22—341no—
196C22—341no—
197C22—342no—
198C22—688no—
199C22—681no—
200C22—681no—
201C22C2×C22⋊#59511yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—172no—
206C4—341no—
207C4—341no—
208C4—681no—
209C4C2×#6324311yescharacteristic
210C4D17⋊M1611yescharacteristic
211C2—42no—
212C2—342no—
213C2—682no—
214C2#1176378011yescharacteristic
215C2#1176380111yescharacteristic
216C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1490
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763831C2semidirect
2. C4×#63243C2semidirect
3. #11763657C2semidirect
4. D4⋊#3624C2semidirect
21. C4⋊#3624C22semidirect
22. C4×#3624C22semidirect
59. D4×C17M16semidirect
72. D68M16semidirect
131. C17#1490semidirect
171. D4#63243semidirect

Maximal subgroups

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

subgroupindex
#117638312
C4×#632432
#117636572
D4⋊#36242
C4×D4×D172
#149017