Finite Group Databasebeta

← #30770041

Subgroups — #30770041

Order 17152.

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

normal subgroups only

This group has 236 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#22562461C211yesmaximal, characteristic, complemented
2#22562028C211yesmaximal, characteristic
3#22559575C211yesmaximal, characteristic
4#22559571C211yesmaximal, characteristic
5#22558280C211yesmaximal, characteristic, complemented
6#22558259C211yesmaximal, characteristic, complemented
7C67×#1362C211yesmaximal, characteristic, complemented
8C2×#14523989—21no—
9C2×#14523989—21no—
10C2×C16×D67—21no—
11#16863776C2211yescharacteristic
12#16863677—21no—
13#16863673—21no—
14C8×D268C2211yescharacteristic
15(C2×D67)⋊C16—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863142—21no—
19#16863141—21no—
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×C8×D67—21no—
26#14524043—21no—
27C4×D268D411yescharacteristic
28#14523989—42no—
29#14523989—42no—
30C2×C67⋊C16D411yescharacteristic
31C2×C67⋊C16D411yescharacteristic
32C16×D67—42no—
33#14523977—41no—
34#14523976—41no—
35C4⋊(C67⋊C8)—21no—
36C4⋊(C67⋊C8)D411yescharacteristic
37#14523965—41no—
38#14523963—41no—
39(C2×D67)⋊C8—21no—
40C2×Q16×C67—21no—
41C2×Q16×C67—21no—
42C4⋊Q8×C67—21no—
43C2×C16×C67—21no—
44C4⋊C8×C67—21no—
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×D268D811yescharacteristic
48C2×C4×D67—21no—
49Dic268—41no—
50Dic268—41no—
51Dic268—81no—
52C2×C67⋊C8—21no—
53C2×C67⋊C8—41no—
54C8×D67—41no—
55C8×D67—41no—
56C4⋊Dic67—41no—
57C4⋊Dic67—41no—
58C4⋊Dic67D811yescharacteristic
59(C2×D67)⋊C4—21no—
60C67⋊C16—21no—
61C67⋊C16D812yes—
62C2×Q8×C67—41no—
63Q16×C67—41no—
64Q16×C67—41no—
65Q16×C67—81no—
66C2×C8×C67—41no—
67C2×C8×C67C2×D411yescharacteristic
68C2×C8×C67#4111yescharacteristic
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C42×C67C2×D411yescharacteristic
72C16×C67—41no—
73C22×D67—21no—
74Q8×C67—81no—
75Q8×C67—81no—
76C2×C4×C67—81no—
77C2×C4×C67#12111yescharacteristic
78C2×C4×C67C2×D811yescharacteristic
79C2×C4×C67#13511yescharacteristic
80C2×Dic67—21no—
81C2×Dic67—81no—
82D268—21no—
83D268Q3212yescomplemented
84C4×D67—41no—
85C4×D67—41no—
86Dic134—81no—
87Dic134—81no—
88C8×C67—21no—
89C8×C67—81no—
90C8×C67C2×D811yescharacteristic
91C8×C67AGL(1,ℤ/8)11yescharacteristic
92C67⋊C8—41no—
93C67⋊C8—81no—
94C22×C67#46611yescharacteristic
95C2×D67—22no—
96C2×D67—41no—
97C2×D67—41no—
98C4×C67—21no—
99C4×C67—161no—
100C4×C67#45911yescharacteristic
101C4×C67#47211yescharacteristic
102C4×C67C2×Q3211yescharacteristic
103C4×C67#50811yescharacteristic
104Dic67—41no—
105Dic67—161no—
106#12431—671nomaximal, Sylow
107C2×C67#166711yescharacteristic
108C2×C67#221611yescharacteristic
109C2×C67#222211yescharacteristic
110D67—42no—
111#2247—671no—
112D4×C16—671no—
113#1669—671no—
114#1669—671no—
115#1371—671no—
116#1362—671no—
117#1362D6711yescharacteristic, complemented
118C67#1243111yescharacteristic, complemented, Sylow
119C2×Q32—1341no—
120C2×Q32—1341no—
121C22×C16—1341no—
122#494—671no—
123#494C2×D6711yescharacteristic
124#474—1341no—
125#474—1341no—
126C8×D4—671no—
127C4⋊C16C2×D6711yescharacteristic
128C22⋊C16—1341no—
129C4×C16—671no—
130#335—671no—
131#335C2×D6711yescharacteristic
132#332—1341no—
133#332—1341no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—1341no—
137C2×Q16—1341no—
138C22×C8—1341no—
139C4⋊Q8—21no—
140C4⋊Q8—1341no—
141C4×D4—671no—
142Q32—2682no—
143Q32—2682no—
144C2×C16—21no—
145C2×C16—2682no—
146C2×C16—671no—
147C2×C16—671no—
148#107—2681no—
149#107—2681no—
150C4⋊C8—21no—
151C4⋊C8—671no—
152C4⋊C8—1341no—
153#102—2681no—
154#102—2681no—
155C22⋊C8—1341no—
156C4×C8C22×D6711yescharacteristic
157C2×Q8—41no—
158C2×Q8—2681no—
159C2×D4—671no—
160C22×C4—1341no—
161Q16—41no—
162Q16—41no—
163Q16—81no—
164Q16—2681no—
165Q16—2681no—
166Q16—5361no—
167C2×C8—41no—
168C2×C8—2681no—
169C2×C8—2681no—
170C2×C8—2681no—
171C2×C8—1341no—
172C2×C8D4×D6711yescharacteristic
173C2×C8#1176162811yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2681no—
177C4⋊C4—2681no—
178C4⋊C4—671no—
179C22⋊C4—1341no—
180C42D4×D6711yescharacteristic
181C16—41no—
182C16—672no—
183C16—1341no—
184C23—1341no—
185Q8—81no—
186Q8—81no—
187Q8—5361no—
188Q8—5361no—
189D4—672no—
190D4—1341no—
191C2×C4—81no—
192C2×C4—2681no—
193C2×C4—2681no—
194C2×C4—5361no—
195C2×C4—1341no—
196C2×C4#1452401611yescharacteristic
197C2×C4D8×D6711yescharacteristic
198C2×C4#1452406411yescharacteristic
199C8—21no—
200C8—81no—
201C8—2681no—
202C8—5361no—
203C8D8×D6711yescharacteristic
204C8#1452407011yescharacteristic
205C22—2681no—
206C22—2681no—
207C22—1342no—
208C22#1686363111yescharacteristic
209C4—21no—
210C4—2681no—
211C4—161no—
212C4—10721no—
213C4#1686359011yescharacteristic
214C4#1686366511yescharacteristic
215C4Q32×D6711yescharacteristic
216C4#1686382711yescharacteristic
217C2—2682no—
218C2#2255956311yescharacteristic
219C2#2256229911yescharacteristic
220C2#2256233311yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socleC22×C67#466
prime radicalC2×D268D8
hypercenter#1362D67
hypocenterC67#12431
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #22562461C2semidirect
5. #22558280C2semidirect
6. #22558259C2semidirect
7. C67×#1362C2semidirect
83. D268Q32semidirect
117. #1362D67semidirect
118. C67#12431semidirect

Maximal subgroups

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

subgroupindex
#225624612
#225620282
#225595752
#225595712
#225582802
#225582592
C67×#13622
#1243167