Finite Group Databasebeta

← #30770254

Subgroups — #30770254

Order 17152.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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#22562432C211yesmaximal, characteristic, complemented
2#22562028C211yesmaximal, characteristic, complemented
3#22559609C211yesmaximal, characteristic, complemented
4#22559561C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic, complemented
6#22558266C211yesmaximal, characteristic, complemented
7C67×#1361C211yesmaximal, characteristic, complemented
8C2×C16×D67—21no—
9#16863801C2211yescharacteristic
10#16863687—21no—
11#16863588—21no—
12C8×D268C2211yescharacteristic
13#16863197—21no—
14#16863197—21no—
15(C2×D67)⋊C16—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863144—21no—
19#16863140—21no—
20#493×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×#11761613—41no—
24C2×C8×D67—21no—
25#14524043—21no—
26C4×D268D411yescharacteristic
27C2×C67⋊C16D411yescharacteristic
28C2×C67⋊C16D411yescharacteristic
29C16×D67—42no—
30#14523977—21no—
31#14523977—21no—
32#14523975—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35#14523965—41no—
36#14523964—41no—
37(C2×D67)⋊C8—21no—
38C2×D8×C67—21no—
39C2×D8×C67—21no—
40#127×C67—21no—
41C2×C16×C67—21no—
42C4⋊C8×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×Dic134—41no—
45C2×D268D811yescharacteristic
46C2×C4×D67—21no—
47#11761613—82no—
48C2×C67⋊C8—21no—
49C2×C67⋊C8—41no—
50C8×D67—41no—
51C8×D67—41no—
52C4⋊Dic67—41no—
53C4⋊Dic67—41no—
54C4⋊Dic67D811yescharacteristic
55(C2×D67)⋊C4—21no—
56C67⋊C16—21no—
57C67⋊C16—21no—
58C2×D4×C67—41no—
59C2×D4×C67—41no—
60C2×D4×C67—41no—
61D8×C67—41no—
62D8×C67—41no—
63D8×C67—81no—
64C2×C8×C67—41no—
65C2×C8×C67C2×D411yescharacteristic
66C2×C8×C67#4111yescharacteristic
67C42×C67C2×D411yescharacteristic
68C16×C67—41no—
69C23×C67—81no—
70C22×D67—21no—
71D4×C67—81no—
72D4×C67—81no—
73D4×C67—81no—
74D4×C67—81no—
75D4×C67—161no—
76C2×C4×C67#12111yescharacteristic
77C2×C4×C67C2×D811yescharacteristic
78C2×C4×C67#13511yescharacteristic
79C2×Dic67—21no—
80C2×Dic67—81no—
81D268—21no—
82D268SD3212yes—
83C4×D67—41no—
84C4×D67—41no—
85Dic134—161no—
86C8×C67—21no—
87C8×C67—81no—
88C8×C67#13511yescharacteristic
89C8×C67AGL(1,ℤ/8)11yescharacteristic
90C67⋊C8—41no—
91C67⋊C8—81no—
92C22×C67—161no—
93C22×C67—161no—
94C22×C67—161no—
95C22×C67#46611yescharacteristic
96C2×D67—22no—
97C2×D67—41no—
98C2×D67—41no—
99C4×C67—21no—
100C4×C67#46311yescharacteristic
101C4×C67#47111yescharacteristic
102C4×C67C2×SD3211yescharacteristic
103C4×C67#50811yescharacteristic
104Dic67—41no—
105Dic67—161no—
106#12468—671nomaximal, Sylow
107C2×C67—321no—
108C2×C67#168111yescharacteristic
109C2×C67#221711yescharacteristic
110C2×C67#222111yescharacteristic
111D67—42no—
112#2240—671no—
113D4×C16—671no—
114#1675—671no—
115#1667—671no—
116#1372—671no—
117#1365—671no—
118#1361D6711yescharacteristic, complemented
119C67#1246811yescharacteristic, complemented, Sylow
120C22×C16—1341no—
121#500—671no—
122#493C2×D6711yescharacteristic
123#476—1341no—
124#459—1341no—
125C8×D4—671no—
126C4⋊C16C2×D6711yescharacteristic
127#357—1341no—
128#357—1341no—
129C22⋊C16—1341no—
130C4×C16—671no—
131#335—671no—
132#335C2×D6711yescharacteristic
133#333—1341no—
134#331—1341no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2681no—
138C22×C8—1341no—
139C4⋊Q8—1341no—
140#127—21no—
141C4×D4—671no—
142C2×C16—21no—
143C2×C16—2682no—
144C2×C16—671no—
145C2×C16—671no—
146#107—1341no—
147#107—1341no—
148#106—2681no—
149C4⋊C8—21no—
150C4⋊C8—671no—
151C4⋊C8—1341no—
152#102—2681no—
153#102—2681no—
154C22⋊C8—1341no—
155C4×C8C22×D6711yescharacteristic
156C2×Q8—2681no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—671no—
161C22×C4—1341no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—5362no—
166C2×C8—41no—
167C2×C8—2681no—
168C2×C8—2681no—
169C2×C8—2681no—
170C2×C8—1341no—
171C2×C8D4×D6711yescharacteristic
172C2×C8#1176162811yescharacteristic
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C4⋊C4—671no—
176C22⋊C4—1341no—
177C42D4×D6711yescharacteristic
178C16—41no—
179C16—1341no—
180C16—1341no—
181C23—81no—
182C23—1341no—
183Q8—10721no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—672no—
190D4—1341no—
191C2×C4—2681no—
192C2×C4—2681no—
193C2×C4—5361no—
194C2×C4—1341no—
195C2×C4#1452401611yescharacteristic
196C2×C4D8×D6711yescharacteristic
197C2×C4#1452406411yescharacteristic
198C8—21no—
199C8—81no—
200C8—2681no—
201C8—5361no—
202C8#1452406411yescharacteristic
203C8#1452406711yescharacteristic
204C22—2681no—
205C22—2681no—
206C22—161no—
207C22—161no—
208C22—161no—
209C22—1342no—
210C22#1686363111yescharacteristic
211C4—21no—
212C4—2681no—
213C4—10721no—
214C4#1686361011yescharacteristic
215C4#1686365811yescharacteristic
216C4SD32×D6711yescharacteristic
217C4#1686382511yescharacteristic
218C2—2682no—
219C2—321no—
220C2#2255964511yescharacteristic
221C2#2256230411yescharacteristic
222C2#2256232811yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1361C2
socleC22×C67#466
prime radical#22559561C2
hypercenter#1361D67
hypocenterC67#12468
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
1. #22562432C2semidirect
2. #22562028C2semidirect
3. #22559609C2semidirect
5. #22558282C2semidirect
6. #22558266C2semidirect
7. C67×#1361C2semidirect
118. #1361D67semidirect
119. C67#12468semidirect

Maximal subgroups

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

subgroupindex
#225624322
#225620282
#225596092
#225595612
#225582822
#225582662
C67×#13612
#1246867