Finite Group Databasebeta

← #30770112

Subgroups — #30770112

Order 17152.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#22562117C211yesmaximal, characteristic, complemented
2#22559576C211yesmaximal, characteristic, complemented
3#22559566C211yesmaximal, characteristic
4#22558281C211yesmaximal, characteristic, complemented
5#22558258C211yesmaximal, characteristic, complemented
6#22558255C211yesmaximal, characteristic
7C67×#2248C211yesmaximal, characteristic
8#16863678—21no—
9#16863593—21no—
10C8×Dic134C2211yescharacteristic
11C4⋊(C67⋊C16)C2211yescharacteristic
12Dic67⋊C16—21no—
13#16863171—21no—
14#16863149C2211yescharacteristic
15#16863149C2211yescharacteristic
16#16863142—21no—
17#16863140—21no—
18C2×SD32×C67—21no—
19C2×SD32×C67—21no—
20#494×C67C2211yescharacteristic
21#493×C67C2211yescharacteristic
22C4×C16×C67C2211yescharacteristic
23C2×#11761616—41no—
24C4×Dic134D411yescharacteristic
25C2×C67⋊C16—21no—
26#14523974—41no—
27C4⋊(C67⋊C8)—21no—
28C4⋊(C67⋊C8)—21no—
29C4⋊(C67⋊C8)D411yescharacteristic
30Dic67⋊C8—21no—
31#14523966—41no—
32#14523964—41no—
33C8×Dic67—21no—
34C2×Q16×C67—21no—
35C2×Q16×C67—21no—
36C2×D8×C67—21no—
37C2×D8×C67—21no—
38C4⋊Q8×C67—21no—
39#127×C67—21no—
40SD32×C67—42no—
41SD32×C67—42no—
42C2×C16×C67D411yescharacteristic
43C2×C16×C67D411yescharacteristic
44C4×C8×C67C2311yescharacteristic
45C2×Dic134D811yescharacteristic
46#11761616—82no—
47C2×C67⋊C8—21no—
48C2×C67⋊C8—41no—
49C2×C67⋊C8—41no—
50C4⋊Dic67D811yescharacteristic
51Dic67⋊C4—21no—
52C4×Dic67—21no—
53C67⋊C16—41no—
54C2×Q8×C67—41no—
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67—41no—
58Q16×C67—41no—
59Q16×C67—41no—
60Q16×C67—81no—
61D8×C67—41no—
62D8×C67—41no—
63D8×C67—81no—
64C2×C8×C67C2×D411yescharacteristic
65C2×C8×C67#4111yescharacteristic
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C42×C67C2×D411yescharacteristic
69C16×C67—21no—
70C16×C67SD1612yescomplemented
71C23×C67—81no—
72Q8×C67—81no—
73Q8×C67—81no—
74D4×C67—81no—
75D4×C67—81no—
76D4×C67—81no—
77D4×C67—81no—
78D4×C67—161no—
79C2×C4×C67—81no—
80C2×C4×C67#12111yescharacteristic
81C2×C4×C67C2×D811yescharacteristic
82C2×C4×C67#13511yescharacteristic
83C2×Dic67—21no—
84C2×Dic67—21no—
85Dic134—21no—
86Dic134—21no—
87C8×C67—21no—
88C8×C67C2×SD1611yescharacteristic
89C8×C67AGL(1,ℤ/8)11yescharacteristic
90C67⋊C8—41no—
91C67⋊C8—81no—
92C67⋊C8—81no—
93C22×C67—161no—
94C22×C67—161no—
95C22×C67—161no—
96C22×C67#46611yescharacteristic
97C4×C67—21no—
98C4×C67—161no—
99C4×C67#46011yescharacteristic
100C4×C67#47211yescharacteristic
101C4×C67#50911yescharacteristic
102C4×C67#51011yescharacteristic
103Dic67—41no—
104Dic67—41no—
105#12444—671nomaximal, Sylow
106C2×C67—321no—
107C2×C67#166811yescharacteristic
108C2×C67#222011yescharacteristic
109C2×C67#222311yescharacteristic
110#2248D6711yescharacteristic
111#2183—671no—
112#1670—671no—
113#1668—671no—
114#1371—671no—
115#1362—671no—
116#1361—671no—
117C67#1244411yescharacteristic, complemented, Sylow
118C2×SD32—21no—
119C2×SD32—21no—
120#494C2×D6711yescharacteristic
121#493C2×D6711yescharacteristic
122#475—1341no—
123#460—1341no—
124C8×Q8—671no—
125C4⋊C16—671no—
126C4⋊C16—1341no—
127#346—1341no—
128C4×C16C2×D6711yescharacteristic
129#335—671no—
130#335—671no—
131#332—1341no—
132#331—1341no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2681no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8—671no—
141SD32—42no—
142SD32—42no—
143C2×C16—1341no—
144C2×C16C22⋊D6711yescharacteristic
145C2×C16C22⋊D6711yescharacteristic
146#106—2681no—
147C4⋊C8—671no—
148C4⋊C8—1341no—
149C4⋊C8—1341no—
150C4⋊C8—1341no—
151#103—2681no—
152#102—2681no—
153C4×C8—1341no—
154C4×C8C22×D6711yescharacteristic
155C2×Q8—41no—
156C2×Q8—671no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—5362no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—2681no—
168C2×C8—2681no—
169C2×C8—1341no—
170C2×C8C2×C22⋊D6711yescharacteristic
171C2×C8#1176162711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—671no—
175C4⋊C4—1341no—
176C42—1341no—
177C42D4×D6711yescharacteristic
178C16—21no—
179C16—2681no—
180C16#1176161612yescomplemented
181C23—81no—
182Q8—81no—
183Q8—81no—
184Q8—1341no—
185Q8—1341no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—81no—
192C2×C4—1341no—
193C2×C4—1341no—
194C2×C4#1452401411yescharacteristic
195C2×C4D8×D6711yescharacteristic
196C2×C4#1452406211yescharacteristic
197C8—21no—
198C8—2681no—
199C8—5361no—
200C8—5361no—
201C8C2×#1176161611yescharacteristic
202C8#1452406911yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1686362911yescharacteristic
207C4—21no—
208C4—2681no—
209C4—2681no—
210C4—161no—
211C4#1686359311yescharacteristic
212C4#1686366311yescharacteristic
213C4#1686383311yescharacteristic
214C4#1686383511yescharacteristic
215C2—321no—
216C2#2255956611yescharacteristic
217C2#2256231911yescharacteristic
218C2#2256233611yescharacteristic
subgroupG/H
centerC22#16863629
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2248C2
socleC22×C67#466
prime radical#493×C67C22
hypercenter#2248D67
hypocenterC67#12444
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #22562117C2semidirect
2. #22559576C2semidirect
4. #22558281C2semidirect
5. #22558258C2semidirect
70. C16×C67SD16semidirect
117. C67#12444semidirect
180. C16#11761616semidirect

Maximal subgroups

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

subgroupindex
#225621172
#225595762
#225595662
#225582812
#225582582
#225582552
C67×#22482
#1244467