Finite Group Databasebeta

← #30770097

Subgroups — #30770097

Order 17152.

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

normal subgroups only

This group has 232 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
1#22562464C211yesmaximal, characteristic
2#22562111C211yesmaximal, characteristic, complemented
3#22559580C211yesmaximal, characteristic, complemented
4#22559566C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic, complemented
6#22558259C211yesmaximal, characteristic, complemented
7C67×#1361C211yesmaximal, characteristic
8C2×#14523987—21no—
9C2×#14523987—21no—
10#16863776C2211yescharacteristic
11#16863682—21no—
12#16863593—21no—
13C8×Dic134C2211yescharacteristic
14Dic67⋊C16—21no—
15C4×C67⋊C16C2211yescharacteristic
16C16×Dic67—21no—
17#16863149C2211yescharacteristic
18#16863141—21no—
19#16863140—21no—
20#493×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×#11761616—41no—
26#14524043—21no—
27C4×Dic134D411yescharacteristic
28#14523987—42no—
29#14523987—42no—
30C2×C67⋊C16D411yescharacteristic
31C2×C67⋊C16D411yescharacteristic
32#14523975—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35Dic67⋊C8—21no—
36#14523968—41no—
37#14523964—41no—
38C8×Dic67—21no—
39C2×D8×C67—21no—
40C2×D8×C67—21no—
41#127×C67—21no—
42C2×C16×C67—21no—
43C4⋊C8×C67—21no—
44C4×C8×C67C2311yescharacteristic
45C2×Dic134—41no—
46C2×Dic134D811yescharacteristic
47Dic268—41no—
48Dic268—41no—
49Dic268—81no—
50#11761616—82no—
51C2×C67⋊C8—21no—
52C2×C67⋊C8—41no—
53C4⋊Dic67—41no—
54C4⋊Dic67—41no—
55C4⋊Dic67D811yescharacteristic
56Dic67⋊C4—21no—
57C4×Dic67—21no—
58C67⋊C16—21no—
59C67⋊C16SD1612yescomplemented
60C2×D4×C67—41no—
61C2×D4×C67—41no—
62C2×D4×C67—41no—
63D8×C67—41no—
64D8×C67—41no—
65D8×C67—81no—
66C2×C8×C67—41no—
67C2×C8×C67C2×D411yescharacteristic
68C2×C8×C67#4111yescharacteristic
69C42×C67C2×D411yescharacteristic
70C16×C67—41no—
71C23×C67—81no—
72D4×C67—81no—
73D4×C67—81no—
74D4×C67—81no—
75D4×C67—81no—
76D4×C67—161no—
77C2×C4×C67#12111yescharacteristic
78C2×C4×C67C2×D811yescharacteristic
79C2×C4×C67#13511yescharacteristic
80C2×Dic67—21no—
81C2×Dic67—21no—
82C2×Dic67—81no—
83Dic134—21no—
84Dic134—81no—
85Dic134—81no—
86Dic134SD3212yescomplemented
87C8×C67—21no—
88C8×C67—81no—
89C8×C67C2×SD1611yescharacteristic
90C8×C67AGL(1,ℤ/8)11yescharacteristic
91C67⋊C8—41no—
92C67⋊C8—81no—
93C22×C67—161no—
94C22×C67—161no—
95C22×C67—161no—
96C22×C67#46611yescharacteristic
97C4×C67—21no—
98C4×C67#46011yescharacteristic
99C4×C67#47211yescharacteristic
100C4×C67C2×SD3211yescharacteristic
101C4×C67#50811yescharacteristic
102Dic67—22no—
103Dic67—41no—
104Dic67—161no—
105#12441—671nomaximal, Sylow
106C2×C67—321no—
107C2×C67#166811yescharacteristic
108C2×C67#221711yescharacteristic
109C2×C67#222311yescharacteristic
110#2248—671no—
111Q8×C16—671no—
112#1670—671no—
113#1668—671no—
114#1372—671no—
115#1362—671no—
116#1361D6711yescharacteristic
117C67#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1341no—
119C2×SD32—1341no—
120#494—671no—
121#493C2×D6711yescharacteristic
122#475—1341no—
123#460—1341no—
124C8×Q8—671no—
125C4⋊C16—1341no—
126C4⋊C16C2×D6711yescharacteristic
127C4×C16—671no—
128C4×C16—1341no—
129#335—671no—
130#335C2×D6711yescharacteristic
131#332—1341no—
132#331—1341no—
133C2×Q16—1341no—
134C2×Q16—1341no—
135C2×SD16—2681no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1341no—
139#127—21no—
140C4×Q8—671no—
141SD32—2682no—
142SD32—2682no—
143C2×C16—21no—
144C2×C16—671no—
145C2×C16—671no—
146#106—2681no—
147C4⋊C8—21no—
148C4⋊C8—671no—
149C4⋊C8—1341no—
150C4⋊C8—1341no—
151#103—2681no—
152#102—2681no—
153C4×C8—1341no—
154C4×C8C22×D6711yescharacteristic
155C2×Q8—2681no—
156C2×Q8—671no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—2681no—
161Q16—2681no—
162Q16—5361no—
163SD16—5362no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—41no—
168C2×C8—2681no—
169C2×C8—1341no—
170C2×C8D4×D6711yescharacteristic
171C2×C8#1176162811yescharacteristic
172C4⋊C4—2681no—
173C4⋊C4—2681no—
174C4⋊C4—671no—
175C4⋊C4—1341no—
176C42—1341no—
177C42D4×D6711yescharacteristic
178C16—41no—
179C16—672no—
180C16—1341no—
181C23—81no—
182Q8—5361no—
183Q8—5361no—
184Q8—672no—
185Q8—1341no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—5361no—
192C2×C4—1341no—
193C2×C4—1341no—
194C2×C4#1452401611yescharacteristic
195C2×C4D8×D6711yescharacteristic
196C2×C4#1452406411yescharacteristic
197C8—21no—
198C8—81no—
199C8—2681no—
200C8—5361no—
201C8SD16×D6711yescharacteristic
202C8#1452406711yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1686363111yescharacteristic
207C4—21no—
208C4—2681no—
209C4—10721no—
210C4—1342no—
211C4#1686359511yescharacteristic
212C4#1686366511yescharacteristic
213C4SD32×D6711yescharacteristic
214C4#1686382511yescharacteristic
215C2—321no—
216C2#2255956811yescharacteristic
217C2#2256230411yescharacteristic
218C2#2256233811yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1361C2
socleC22×C67#466
prime radical#493×C67C22
hypercenter#1361D67
hypocenterC67#12441
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
2. #22562111C2semidirect
3. #22559580C2semidirect
5. #22558282C2semidirect
6. #22558259C2semidirect
59. C67⋊C16SD16semidirect
86. Dic134SD32semidirect
117. C67#12441semidirect

Maximal subgroups

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

subgroupindex
#225624642
#225621112
#225595802
#225595662
#225582822
#225582592
C67×#13612
#1244167