Finite Group Databasebeta

← #14527749

Subgroups — #14527749

Order 2176.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#11763406C211yesmaximal, characteristic
2#11762967C211yesmaximal, characteristic, complemented
3#11762950C211yesmaximal, characteristic
4#11762729C211yesmaximal, characteristic, complemented
5#11762547C211yesmaximal, characteristic, complemented
6#11762544C211yesmaximal, characteristic
7C17×#324C211yesmaximal, characteristic, complemented
8C22×Dic34C2211yescharacteristic
9C2×C4⋊C4×C17C2211yescharacteristic
10#10587616C2211yescharacteristic
11#10587615—21no—
12C4×C22⋊D17—21no—
13#10587610C2211yescharacteristic
14C2×#63225—21no—
15C2×Dic68—21no—
16C2×#63219—21no—
17#10587584—21no—
18#10587582—21no—
19#10587580—21no—
20#10587575—21no—
21C17×C22⋊C8C2211yescharacteristic
22C17×C22⋊C8C2211yescharacteristic
23#10587502—21no—
24#10587502C2211yescharacteristic
25#10587487—21no—
26#10587487—21no—
27C22×C4×C17D411yescharacteristic
28C22×C4×C17D411yescharacteristic
29C22×C4×C17C2311yescharacteristic
30C2×C22⋊D17—21no—
31C22×Dic17—21no—
32C22×Dic17—21no—
33C2×D68D411yescharacteristic
34C2×C4×D17—21no—
35C2×Dic34—21no—
36C2×Dic34—21no—
37C2×Dic34—42no—
38C2×Dic34D411yescharacteristic
39C2×C8×C17—21no—
40C2×C8×C17—21no—
41C4⋊C4×C17—21no—
42C4⋊C4×C17—21no—
43C22⋊Dic17—21no—
44#63227—21no—
45#63227—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17D411yescharacteristic
49C4⋊Dic17D411yescharacteristic
50#63225—21no—
51#63225—21no—
52#63225—21no—
53#63225—21no—
54#63225—42no—
55C4×Dic17—21no—
56Dic68—42no—
57#63219—42no—
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17C2×D411yescharacteristic
67C2×C4×C17C2×D411yescharacteristic
68C22⋊D17—41no—
69C22⋊D17—41no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—21no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—41no—
76C2×Dic17—41no—
77C2×Dic17—41no—
78C2×Dic17—41no—
79D68—41no—
80C4×D17—81no—
81Dic34—22no—
82Dic34—22no—
83Dic34—41no—
84Dic34—41no—
85Dic34—41no—
86C8×C17—41no—
87C8×C17—41no—
88#1603—171nomaximal, Sylow
89C22×C17—21no—
90C22×C17—21no—
91C22×C17#12011yescharacteristic
92C22×C17#13511yescharacteristic
93C22×C17#13711yescharacteristic
94C2×D17—41no—
95C2×D17—41no—
96C2×D17—41no—
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—41no—
100C4×C17—41no—
101Dic17—42no—
102Dic17—41no—
103Dic17—41no—
104Dic17—42no—
105Dic17—41no—
106#539—171no—
107#451—171no—
108#448—171no—
109#395—171no—
110#328—171no—
111#327—171no—
112#324D1711yescharacteristic, complemented
113C2×C17—21no—
114C2×C17—21no—
115C2×C17#45211yescharacteristic
116C2×C17#45611yescharacteristic
117C2×C17B(4,𝔽2)11yescharacteristic
118D17—81no—
119C22×Q8—171no—
120C2×Q16—341no—
121C2×SD16—341no—
122#126—341no—
123#125—341no—
124#123—341no—
125#123—341no—
126C22⋊Q8—171no—
127C22⋊Q8—341no—
128#121—171no—
129C4×D4—341no—
130C2×C4⋊C4—341no—
131C2×C4⋊C4C2×D1711yescharacteristic
132#103—341no—
133#103—341no—
134C22⋊C8C2×D1711yescharacteristic
135C22⋊C8C2×D1711yescharacteristic
136#95—171no—
137#95—341no—
138C17#160311yescharacteristic, complemented, Sylow
139C2×Q8—171no—
140C2×Q8—341no—
141C2×Q8—341no—
142C2×Q8—682no—
143C2×D4—171no—
144C2×D4—341no—
145C22×C4—341no—
146C22×C4—341no—
147C22×C4—341no—
148C22×C4D6811yescharacteristic
149C22×C4D6811yescharacteristic
150C22×C4C22×D1711yescharacteristic
151Q16—682no—
152SD16—682no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—171no—
158C4⋊C4—171no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C4⋊C4—341no—
162C4⋊C4—341no—
163C4⋊C4—682no—
164C22⋊C4—341no—
165C22⋊C4—341no—
166C22⋊C4—341no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C42—341no—
170C23—341no—
171C23C2×D6811yescharacteristic
172Q8—342no—
173Q8—342no—
174Q8—681no—
175Q8—681no—
176Q8—681no—
177D4—681no—
178D4—681no—
179D4—681no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—341no—
191C2×C4—681no—
192C2×C4—681no—
193C2×C4—681no—
194C2×C4—681no—
195C2×C4—1361no—
196C2×C4D4×D1711yescharacteristic
197C2×C4D4×D1711yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—681no—
203C22—681no—
204C22—681no—
205C22#1058756811yescharacteristic
206C22#1058759011yescharacteristic
207C22#1058759511yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—682no—
213C4—681no—
214C4—681no—
215C4—682no—
216C4—681no—
217C2—21no—
218C2—21no—
219C2—1361no—
220C2#1176297311yescharacteristic
221C2#1176299411yescharacteristic
222C2#1176299711yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#324C2
socleC22×C17#120
prime radical#10587616C22
hypercenter#324D17
hypocenterC17#1603
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #11762967C2semidirect
4. #11762729C2semidirect
5. #11762547C2semidirect
7. C17×#324C2semidirect
112. #324D17semidirect
138. C17#1603semidirect

Maximal subgroups

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

subgroupindex
#117634062
#117629672
#117629502
#117627292
#117625472
#117625442
C17×#3242
#160317