Finite Group Databasebeta

← #30770252

Subgroups — #30770252

Order 17152.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C16×D268C211yesmaximal, characteristic, complemented
2#22559605C211yesmaximal, characteristic, complemented
3#22559561C211yesmaximal, characteristic
4#22558283C211yesmaximal, characteristic, complemented
5#22558267C211yesmaximal, characteristic, complemented
6#22558255C211yesmaximal, characteristic, complemented
7C67×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D67—21no—
9#16863683—21no—
10#16863588—21no—
11C8×D268C2211yescharacteristic
12C4⋊(C67⋊C16)C2211yescharacteristic
13(C2×D67)⋊C16—21no—
14#16863149C2211yescharacteristic
15#16863149C2211yescharacteristic
16#16863145—21no—
17#16863140—21no—
18#500×C67C2211yescharacteristic
19#493×C67C2211yescharacteristic
20#357×C67—21no—
21#357×C67—21no—
22C4×C16×C67C2211yescharacteristic
23C2×#11761613—41no—
24C2×C8×D67—21no—
25C4×D268D411yescharacteristic
26C2×C67⋊C16—21no—
27C16×D67—42no—
28#14523974—41no—
29C4⋊(C67⋊C8)—21no—
30C4⋊(C67⋊C8)—21no—
31C4⋊(C67⋊C8)D411yescharacteristic
32#14523964—41no—
33#14523963—41no—
34(C2×D67)⋊C8—21no—
35C2×D8×C67—21no—
36C2×D8×C67—21no—
37C4⋊Q8×C67—21no—
38#127×C67—21no—
39C2×C16×C67D411yescharacteristic
40C2×C16×C67D411yescharacteristic
41#107×C67—21no—
42#107×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×D268D811yescharacteristic
45C2×C4×D67—21no—
46#11761613—82no—
47C2×C67⋊C8—21no—
48C2×C67⋊C8—41no—
49C2×C67⋊C8—41no—
50C8×D67—41no—
51C8×D67—41no—
52C4⋊Dic67D811yescharacteristic
53(C2×D67)⋊C4—21no—
54C67⋊C16—41no—
55C2×Q8×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59D8×C67—41no—
60D8×C67—41no—
61D8×C67—81no—
62C2×C8×C67C2×D411yescharacteristic
63C2×C8×C67#4111yescharacteristic
64C4⋊C4×C67—41no—
65C4⋊C4×C67—41no—
66C42×C67C2×D411yescharacteristic
67C16×C67—21no—
68C16×C67—21no—
69C23×C67—81no—
70C22×D67—21no—
71Q8×C67—161no—
72D4×C67—81no—
73D4×C67—81no—
74D4×C67—81no—
75D4×C67—81no—
76D4×C67—161no—
77C2×C4×C67—81no—
78C2×C4×C67#12111yescharacteristic
79C2×C4×C67C2×D811yescharacteristic
80C2×C4×C67#13511yescharacteristic
81C2×Dic67—21no—
82D268—21no—
83D268SD3212yes—
84C4×D67—41no—
85C4×D67—41no—
86C8×C67—21no—
87C8×C67#13511yescharacteristic
88C8×C67AGL(1,ℤ/8)11yescharacteristic
89C67⋊C8—41no—
90C67⋊C8—81no—
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—161no—
101C4×C67#46311yescharacteristic
102C4×C67#47111yescharacteristic
103C4×C67C2×SD3211yescharacteristic
104C4×C67#50811yescharacteristic
105Dic67—41no—
106#12468—671nomaximal, Sylow
107C2×C67—321no—
108C2×C67#168111yescharacteristic
109C2×C67#221711yescharacteristic
110C2×C67#222111yescharacteristic
111D67—42no—
112#2240D6711yescharacteristic, complemented
113D4×C16—671no—
114#1675—671no—
115#1667—671no—
116#1372—671no—
117#1365—671no—
118#1361—671no—
119C67#1246811yescharacteristic, complemented, Sylow
120C22×C16—1341no—
121#500C2×D6711yescharacteristic
122#493C2×D6711yescharacteristic
123#476—1341no—
124#459—1341no—
125C8×D4—671no—
126C4⋊C16—671no—
127#357—21no—
128#357—21no—
129C22⋊C16—1341no—
130C4×C16C2×D6711yescharacteristic
131#335—671no—
132#335—671no—
133#333—1341no—
134#331—1341no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2681no—
138C22×C8—1341no—
139C4⋊Q8—21no—
140#127—21no—
141C4×D4—671no—
142C2×C16—2682no—
143C2×C16—1341no—
144C2×C16C22⋊D6711yescharacteristic
145C2×C16C22⋊D6711yescharacteristic
146#107—21no—
147#107—21no—
148#106—2681no—
149C4⋊C8—671no—
150C4⋊C8—1341no—
151C4⋊C8—1341no—
152#102—2681no—
153#102—2681no—
154C22⋊C8—1341no—
155C4×C8C22×D6711yescharacteristic
156C2×Q8—41no—
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—2681no—
167C2×C8—2681no—
168C2×C8—2681no—
169C2×C8—2681no—
170C2×C8—1341no—
171C2×C8C2×C22⋊D6711yescharacteristic
172C2×C8#1176162711yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—671no—
176C22⋊C4—1341no—
177C42D4×D6711yescharacteristic
178C16—21no—
179C16—21no—
180C16—2681no—
181C23—81no—
182C23—1341no—
183Q8—161no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—672no—
190D4—1341no—
191C2×C4—81no—
192C2×C4—2681no—
193C2×C4—2681no—
194C2×C4—1341no—
195C2×C4#1452401411yescharacteristic
196C2×C4D8×D6711yescharacteristic
197C2×C4#1452406211yescharacteristic
198C8—21no—
199C8—2681no—
200C8—5361no—
201C8—5361no—
202C8#1452406311yescharacteristic
203C8#1452406811yescharacteristic
204C22—2681no—
205C22—2681no—
206C22—161no—
207C22—161no—
208C22—161no—
209C22—1342no—
210C22#1686362911yescharacteristic
211C4—21no—
212C4—2681no—
213C4—161no—
214C4#1686360811yescharacteristic
215C4#1686365611yescharacteristic
216C4SD32×D6711yescharacteristic
217C4#1686382711yescharacteristic
218C2—2682no—
219C2—321no—
220C2#2255964311yescharacteristic
221C2#2256230211yescharacteristic
222C2#2256232611yescharacteristic
subgroupG/H
centerC22#16863629
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2240C2
socleC22×C67#466
prime radical#22559561C2
hypercenter#2240D67
hypocenterC67#12468
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D268C2semidirect
2. #22559605C2semidirect
4. #22558283C2semidirect
5. #22558267C2semidirect
6. #22558255C2semidirect
7. C67×#2240C2semidirect
112. #2240D67semidirect
119. C67#12468semidirect

Maximal subgroups

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

subgroupindex
C16×D2682
#225596052
#225595612
#225582832
#225582672
#225582552
C67×#22402
#1246867