Finite Group Databasebeta

← #14544812

Subgroups — #14544812

Order 2176.

338 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×D17⋊M16C211yesmaximal, characteristic
2C2×#10587545C212yesmaximal, complemented
3#11763781C211yesmaximal, characteristic
4#11763689C212yesmaximal, complemented
5C2×#10587582C211yesmaximal, characteristic
6C22×C4×D17C2211yescharacteristic
7C22×Dic34C2211yescharacteristic
8C2×#63247C2212yes—
9C2×#63247C2212yes—
10D17⋊M16—24no—
11C2×#63243—21no—
12C2×#63242—21no—
13C2×C4⋊C4×C17C411yescharacteristic
14C2×#63227C411yescharacteristic
15C2×#63227C411yescharacteristic
16C2×#63225C411yescharacteristic
17C2×#63225C2211yescharacteristic
18#10587582—21no—
19#10587582—21no—
20#10587582—21no—
21#10587582—21no—
22#10587545—24no—
23#10587543—21no—
24#10587543—21no—
25C23×D17C2×C411yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C22×Dic17C2×C411yescharacteristic
30C22×Dic17C2×C411yescharacteristic
31C22×Dic17C2311yescharacteristic
32C2×C4×D17—22no—
33C2×C4×D17—22no—
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×Dic34—22no—
39C2×Dic34—21no—
40C2×Dic34—21no—
41C2×Dic34—22no—
42#63247—22no—
43#63247—22no—
44#63247—22no—
45#63247D414yes—
46C2×#3624—22no—
47C2×#3624—22no—
48#63243—24no—
49#63242—24no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17—21no—
52#63227—21no—
53#63227—21no—
54#63227—21no—
55#63227—21no—
56#63225—21no—
57#63225—21no—
58#63225—21no—
59#63225—21no—
60C23×C17C22×C411yescharacteristic
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—41no—
64C22×D17C22⋊C411yescharacteristic
65C22×D17C22⋊C411yescharacteristic
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—41no—
71C2×C4×C17C22⋊C411yescharacteristic
72C2×C4×C17C22⋊C411yescharacteristic
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—22no—
76C2×Dic17—41no—
77C2×Dic17C2×D412yes—
78C2×Dic17C2×D411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80C2×Dic17#4112yes—
81C4×D17—21no—
82C4×D17—21no—
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—21no—
86C4×D17—21no—
87Dic34—41no—
88Dic34—42no—
89Dic34—41no—
90#3624—24no—
91#3624—42no—
92#1901—171nomaximal, Sylow
93C22×C17—21no—
94C22×C17—21no—
95C22×C17C2×C22⋊C411yescharacteristic
96C22×C17C4×D412yes—
97C2×D17—21no—
98C2×D17—21no—
99C2×D17—21no—
100C2×D17—21no—
101C2×D17—21no—
102C2×D17—44no—
103C2×D17#10112yescomplemented
104C4×C17—21no—
105C4×C17—21no—
106C4×C17—41no—
107C4×C17—41no—
108Dic17—41no—
109Dic17—41no—
110Dic17#12011yescharacteristic
111Dic17#12112yes—
112Dic17#12311yescharacteristic
113C22×M16—171no—
114C2×C22⋊Q8—171no—
115C2×#101—172no—
116#407—171no—
117#343—172no—
118C2×C17—22no—
119C2×C17#38611yescharacteristic
120C2×C17C2×#10111yescharacteristic
121C2×C17#41311yescharacteristic
122D17—22no—
123D17—41no—
124C22×Q8—171no—
125C23×C4—171no—
126C2×M16—172no—
127C2×M16—172no—
128C2×M16—344no—
129C2×M16—341no—
130C22×C8—341no—
131C22⋊Q8—341no—
132C22⋊Q8—341no—
133C22⋊Q8—341no—
134C22⋊Q8—341no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4#59511yescharacteristic
138C2×C22⋊C4—171no—
139C2×C22⋊C4—171no—
140#101—344no—
141C22⋊C8—341no—
142C22⋊C8—341no—
143C17#190111yescharacteristic, complemented, Sylow
144C24—171no—
145C2×Q8—342no—
146C2×Q8—341no—
147C2×Q8—341no—
148C2×Q8—342no—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—171no—
154C22×C4—171no—
155C22×C4—171no—
156C22×C4—342no—
157C22×C4—342no—
158C22×C4C2×#59511yescharacteristic
159C22×C4C2×#59511yescharacteristic
160C22×C4C2×#59511yescharacteristic
161M16—174no—
162M16—342no—
163M16—344no—
164M16—342no—
165M16—342no—
166C2×C8—342no—
167C2×C8—344no—
168C2×C8—342no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—341no—
172C4⋊C4—341no—
173C4⋊C4—341no—
174C4⋊C4—341no—
175C22⋊C4—341no—
176C22⋊C4—341no—
177C22⋊C4—341no—
178C22⋊C4—341no—
179C23—171no—
180C23—171no—
181C23—342no—
182C23—342no—
183C23—681no—
184C23C22×#59511yescharacteristic
185Q8—681no—
186Q8—682no—
187Q8—681no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—171no—
196C2×C4—171no—
197C2×C4—341no—
198C2×C4—341no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—341no—
202C2×C4—341no—
203C2×C4—342no—
204C2×C4—341no—
205C2×C4—341no—
206C2×C4—681no—
207C2×C4C22⋊#59511yescharacteristic
208C2×C4C22⋊#59511yescharacteristic
209C8—344no—
210C8—682no—
211C22—21no—
212C22—21no—
213C22—172no—
214C22—341no—
215C22—341no—
216C22—341no—
217C22—341no—
218C22—341no—
219C22—684no—
220C22D4×#59512yes—
221C22C2×C22⋊#59511yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—171no—
227C4—172no—
228C4—171no—
229C4—681no—
230C4—681no—
231C2—22no—
232C2—342no—
233C2—681no—
234C2#1176375711yescharacteristic
235C2D17⋊#10111yescharacteristic
236C2#1176379511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17C2×C4
cosocleC22×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1901
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#10587545C2semidirect
4. #11763689C2semidirect
103. C2×D17#101semidirect
143. C17#1901semidirect

Maximal subgroups

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

subgroupindex
C2×D17⋊M162
C2×#105875452
#117637812
#117636892
C2×#105875822
#190117