Finite Group Databasebeta

← #17960320

Subgroups — #17960320

Order 4352.

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

normal subgroups only

This group has 224 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#14544417C211yesmaximal, characteristic
2Q16×#3624C211yesmaximal, characteristic, complemented
3#14543818C211yesmaximal, characteristic, complemented
4#14543740C211yesmaximal, characteristic, complemented
5#14543738C211yesmaximal, characteristic
6#14543698C211yesmaximal, characteristic, complemented
7SD32×Dic17C211yesmaximal, characteristic
8Q8×#3624—21no—
9#11763844—21no—
10#11763695C2211yescharacteristic
11#11763672C2211yescharacteristic
12#11763672C2211yescharacteristic
13Q8⋊#3624—21no—
14D4⋊#3624—21no—
15C8×#3624C2211yescharacteristic
16Q16×Dic17C2211yescharacteristic
17D8×Dic17C2211yescharacteristic
18#11762637C411yescharacteristic
19#11762613C411yescharacteristic
20#11762610C411yescharacteristic
21C16×Dic17C2211yescharacteristic
22C2×C17×SD32C411yescharacteristic
23C2×#63247—41no—
24C2×Q16×C17C2×C411yescharacteristic
25C2×D8×C17C2×C411yescharacteristic
26Q8×Dic17—21no—
27D4×Dic17—21no—
28C22⋊#3624—41no—
29C4⋊#3624—21no—
30C4⋊#3624—21no—
31C4⋊#3624—41no—
32C4×#3624—41no—
33C4×#3624D411yescharacteristic
34C4×#3624D411yescharacteristic
35C2×#63216—21no—
36C17×SD32—22no—
37C2×C16×C17C2×C411yescharacteristic
38#10587506—21no—
39#10587503—21no—
40#10587489C2×C411yescharacteristic
41#10587489C2×C411yescharacteristic
42C8×Dic17C2311yescharacteristic
43C2×#63214C2×C411yescharacteristic
44C2×Q8×C17—21no—
45C2×D4×C17—21no—
46C22×Dic17—41no—
47#63247—82no—
48C2×#3624—21no—
49C2×#3624—41no—
50C2×#3624—41no—
51C2×#3624D811yescharacteristic
52C2×#3624D811yescharacteristic
53Q16×C17—21no—
54Q16×C17M1612yescomplemented
55D8×C17—21no—
56D8×C17—21no—
57C2×C8×C17C22×C411yescharacteristic
58C22⋊Dic17—41no—
59C4⋊Dic17—21no—
60C4⋊Dic17—21no—
61C4⋊Dic17—41no—
62C4×Dic17—41no—
63C4×Dic17C2×D411yescharacteristic
64C2×#3622#4111yescharacteristic
65#63216—41no—
66C16×C17—21no—
67#63214—21no—
68#12071—171nomaximal, Sylow
69C23×C17—41no—
70Q8×C17—22no—
71Q8×C17—41no—
72D4×C17—41no—
73D4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17C4×D411yescharacteristic
76C2×Dic17—41no—
77C2×Dic17—41no—
78C2×Dic17—81no—
79C2×Dic17—81no—
80C2×Dic17C2×D811yescharacteristic
81C2×Dic17#13511yescharacteristic
82#3624—21no—
83#3624—41no—
84#3624—81no—
85#3624—81no—
86#3624SD3212yescomplemented
87C8×C17C2×M1611yescharacteristic
88C8×C17#13111yescharacteristic
89#3622—21no—
90C4×SD32—171no—
91#1589—171no—
92C8×Q16—171no—
93#1372—171no—
94#1332—171no—
95#1331—171no—
96#1312—171no—
97C22×C17—81no—
98C22×C17—81no—
99C22×C17—81no—
100C22×C17C4×D811yescharacteristic
101C4×C17—42no—
102C4×C17#43511yescharacteristic
103C4×C17#44411yescharacteristic
104Dic17—21no—
105Dic17—81no—
106Dic17—81no—
107Dic17C2×SD3211yescharacteristic
108Dic17#50811yescharacteristic
109C2×SD32#59511yescharacteristic
110C8×Q8—341no—
111C4×Q16—171no—
112C4×D8—171no—
113#436—341no—
114#367—171no—
115#358—171no—
116#357—171no—
117C4×C16—171no—
118C4×C16—171no—
119#335—171no—
120#335—171no—
121Q8⋊C8—341no—
122D4⋊C8—341no—
123C82—171no—
124C2×C17—161no—
125C2×C17#158111yescharacteristic
126C2×C17C4×SD3211yescharacteristic
127C2×C17#217911yescharacteristic
128C2×Q16C2×#59511yescharacteristic
129C2×D8C2×#59511yescharacteristic
130C2×M16—681no—
131C4×Q8—341no—
132C4×D4—341no—
133SD32—22no—
134C2×C16—171no—
135C2×C16—341no—
136C2×C16C2×#59511yescharacteristic
137#107—171no—
138#107—171no—
139C4⋊C8—341no—
140C4⋊C8—341no—
141C4⋊C8—681no—
142#103—341no—
143#102—341no—
144C22⋊C8—681no—
145C4×C8—171no—
146C4×C8—171no—
147C4×C8—171no—
148C4×C8—681no—
149C17#1207111yescharacteristic, complemented, Sylow
150C2×Q8—21no—
151C2×D4—21no—
152C22×C4—681no—
153Q16—21no—
154Q16#6324712yescomplemented
155D8—21no—
156D8—21no—
157M16—1362no—
158C2×C8—171no—
159C2×C8—171no—
160C2×C8—171no—
161C2×C8—341no—
162C2×C8—681no—
163C2×C8—681no—
164C2×C8C22×#59511yescharacteristic
165C4⋊C4—341no—
166C4⋊C4—341no—
167C4⋊C4—681no—
168C22⋊C4—681no—
169C42—171no—
170C42—681no—
171C16—21no—
172C16—341no—
173C16—681no—
174C23—41no—
175Q8—22no—
176Q8—41no—
177D4—41no—
178D4—41no—
179C2×C4—41no—
180C2×C4—171no—
181C2×C4—171no—
182C2×C4—681no—
183C2×C4—681no—
184C2×C4—1361no—
185C2×C4—1361no—
186C2×C4D4×#59511yescharacteristic
187C8—172no—
188C8—341no—
189C8—341no—
190C8—681no—
191C8—1361no—
192C8—1361no—
193C8#1058767811yescharacteristic
194C8C2×#6324711yescharacteristic
195C22—81no—
196C22—81no—
197C22—81no—
198C22D8×#59511yescharacteristic
199C4—42no—
200C4—171no—
201C4—171no—
202C4—341no—
203C4—1361no—
204C4—1361no—
205C4#1176384011yescharacteristic
206C4#1176385611yescharacteristic
207C2—161no—
208C2#1454438511yescharacteristic
209C2SD32×#59511yescharacteristic
210C2#1454518611yescharacteristic
subgroupG/H
centerC22D8×#595
commutatorC2×C8×C17C22×C4
FrattiniC2×C8C22×#595
FittingC2×C17×SD32C4
socleC22×C17C4×D8
prime radicalC2×D8×C17C2×C4
cosocleC8×Dic17C23
hypercenterC2×SD32#595
hypocenterC17#12071
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. Q16×#3624C2semidirect
3. #14543818C2semidirect
4. #14543740C2semidirect
6. #14543698C2semidirect
54. Q16×C17M16semidirect
86. #3624SD32semidirect
149. C17#12071semidirect
154. Q16#63247semidirect

Maximal subgroups

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

subgroupindex
#145444172
Q16×#36242
#145438182
#145437402
#145437382
#145436982
SD32×Dic172
#1207117