Finite Group Databasebeta

← #17982306

Subgroups — #17982306

Order 4352.

224 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 73 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
1Q8×#63241C211yesmaximal, characteristic, complemented
2#14546000C211yesmaximal, characteristic
3#14545843C211yesmaximal, characteristic, complemented
4Q8⋊#63241C211yesmaximal, characteristic, complemented
5D4⋊#63241C211yesmaximal, characteristic
6#14545800C211yesmaximal, characteristic, complemented
7SD16×#3624C211yesmaximal, characteristic
8C2×#10587670—21no—
9C4⋊#63241—21no—
10C4⋊#63241C2211yescharacteristic
11C4⋊#63241C2211yescharacteristic
12C22⋊#63241—21no—
13C4×#63241—21no—
14C4×#63241C2211yescharacteristic
15C4×#63241C2211yescharacteristic
16Q8×#3624C2211yescharacteristic
17D4×#3624C2211yescharacteristic
18#11763670C411yescharacteristic
19Q8⋊#3624C411yescharacteristic
20D4⋊#3624C411yescharacteristic
21C8×#3624C2211yescharacteristic
22SD16×Dic17C411yescharacteristic
23C22×#3624—21no—
24#10587670—42no—
25C2×#63241—21no—
26C2×#63241—21no—
27C2×#63241—21no—
28C2×#63241D411yescharacteristic
29C2×#63241D411yescharacteristic
30C2×SD16×C17C811yescharacteristic
31Q8×Dic17C2×C411yescharacteristic
32D4×Dic17C2×C411yescharacteristic
33C22⋊#3624—21no—
34C4⋊#3624—21no—
35C4⋊#3624C2×C411yescharacteristic
36C4⋊#3624C2×C411yescharacteristic
37C4×#3624—21no—
38C4×#3624C2×C411yescharacteristic
39C4×#3624C2311yescharacteristic
40#10587506C811yescharacteristic
41#10587503C811yescharacteristic
42#10587488C811yescharacteristic
43C8×Dic17C2×C411yescharacteristic
44C2×Q8×C17C2×C811yescharacteristic
45C2×D4×C17C2×C811yescharacteristic
46C22×Dic17—21no—
47C2×#3624—21no—
48C2×#3624—21no—
49C2×#3624—21no—
50C2×#3624—41no—
51C2×#3624—41no—
52C2×#3624C2×D411yescharacteristic
53C2×#3624#4111yescharacteristic
54#63241—22no—
55#63241—21no—
56#63241—41no—
57#63241—41no—
58#63241SD1612yescomplemented
59SD16×C17—22no—
60C2×C8×C17C2×C811yescharacteristic
61C22⋊Dic17—21no—
62C4⋊Dic17—21no—
63C4⋊Dic17C2×C811yescharacteristic
64C4⋊Dic17C2×C811yescharacteristic
65C4×Dic17—21no—
66C4×Dic17C22×C411yescharacteristic
67C2×#3622C2×C811yescharacteristic
68#12050—171nomaximal, Sylow
69C23×C17—21no—
70Q8×C17—21no—
71Q8×C17M3212yescomplemented
72D4×C17—21no—
73D4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17C22×C811yescharacteristic
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17C4×D411yescharacteristic
81C2×Dic17#13111yescharacteristic
82#3624—21no—
83#3624—41no—
84#3624—41no—
85#3624—41no—
86#3624C2×SD1611yescharacteristic
87#3624#13511yescharacteristic
88C8×C17—21no—
89#3622—21no—
90Q8×C16—171no—
91#2169—171no—
92C8×SD16—171no—
93#1367—171no—
94Q8⋊C16—171no—
95D4⋊C16—171no—
96#1311—171no—
97C22×C17—41no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17C8×D411yescharacteristic
101C4×C17—22no—
102C4×C17C2×M3211yescharacteristic
103C4×C17#50411yescharacteristic
104Dic17—21no—
105Dic17—41no—
106Dic17—41no—
107Dic17C4×SD1611yescharacteristic
108Dic17#44311yescharacteristic
109C2×M32—341no—
110C8×Q8—171no—
111C4×SD16—171no—
112C8×D4—171no—
113C4⋊C16—171no—
114C4⋊C16—171no—
115C4⋊C16—341no—
116C22⋊C16—341no—
117C4×C16—171no—
118C4×C16—171no—
119C4×C16—341no—
120#334—171no—
121Q8⋊C8—171no—
122D4⋊C8—171no—
123C82—171no—
124C2×C17—81no—
125C2×C17C8×SD1611yescharacteristic
126C2×C17#216811yescharacteristic
127C2×C17#217111yescharacteristic
128C2×SD16#363311yescharacteristic
129C22×C8—341no—
130C4×Q8—171no—
131C4×D4—171no—
132M32—682no—
133C2×C16—171no—
134C2×C16—171no—
135C2×C16—341no—
136C2×C16—341no—
137C2×C16—341no—
138#106—171no—
139C4⋊C8—171no—
140C4⋊C8—171no—
141C4⋊C8—341no—
142#103—171no—
143#102—171no—
144C22⋊C8—341no—
145C4×C8—171no—
146C4×C8—171no—
147C4×C8—171no—
148C4×C8—341no—
149C17#1205011yescharacteristic, complemented, Sylow
150C2×Q8C2×#363311yescharacteristic
151C2×D4C2×#363311yescharacteristic
152C22×C4—341no—
153SD16—22no—
154C2×C8—171no—
155C2×C8—171no—
156C2×C8—171no—
157C2×C8—341no—
158C2×C8—341no—
159C2×C8—341no—
160C2×C8—681no—
161C2×C8—681no—
162C2×C8C2×#363311yescharacteristic
163C4⋊C4—171no—
164C4⋊C4—171no—
165C4⋊C4—341no—
166C22⋊C4—341no—
167C42—171no—
168C42—341no—
169C16—172no—
170C16—341no—
171C16—342no—
172C16—681no—
173C16—681no—
174C23—21no—
175Q8—21no—
176Q8#1058767012yescomplemented
177D4—21no—
178D4—21no—
179C2×C4—21no—
180C2×C4—171no—
181C2×C4—171no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4C22×#363311yescharacteristic
187C8—21no—
188C8—171no—
189C8—171no—
190C8—341no—
191C8—341no—
192C8—681no—
193C8—681no—
194C8—681no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22D4×#363311yescharacteristic
199C4—22no—
200C4—171no—
201C4—171no—
202C4—341no—
203C4—681no—
204C4—681no—
205C4C2×#1058767011yescharacteristic
206C4#1176393011yescharacteristic
207C2—81no—
208C2SD16×#363311yescharacteristic
209C2#1454599611yescharacteristic
210C2#1454600411yescharacteristic
subgroupG/H
centerC22D4×#3633
commutatorC2×C4×C17C22×C8
FrattiniC2×C4C22×#3633
FittingC2×SD16×C17C8
socleC22×C17C8×D4
prime radicalC2×D4×C17C2×C8
cosocleC4×#3624C23
hypercenterC2×SD16#3633
hypocenterC17#12050
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#63241C2semidirect
3. #14545843C2semidirect
4. Q8⋊#63241C2semidirect
6. #14545800C2semidirect
58. #63241SD16semidirect
71. Q8×C17M32semidirect
149. C17#12050semidirect
176. Q8#10587670semidirect

Maximal subgroups

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

subgroupindex
Q8×#632412
#145460002
#145458432
Q8⋊#632412
D4⋊#632412
#145458002
SD16×#36242
#1205017