Finite Group Databasebeta

← #16948593

Subgroups — #16948593

Order 4352.

232 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 71 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#14531041C211yesmaximal, characteristic
2Q8×#63214C211yesmaximal, characteristic, complemented
3#14528144C211yesmaximal, characteristic
4#14526857C211yesmaximal, characteristic, complemented
5#14526833C211yesmaximal, characteristic, complemented
6#14526829C211yesmaximal, characteristic
7C17×#1670C211yesmaximal, characteristic, complemented
8C2×#10587499—21no—
9C2×#10587499—21no—
10#11763186C2211yescharacteristic
11#11763009—21no—
12C4⋊#63214—21no—
13C4⋊#63214C2211yescharacteristic
14C4×#63214—21no—
15C4×#63214C2211yescharacteristic
16#11762562C2211yescharacteristic
17#11762552—21no—
18C17×#494C2211yescharacteristic
19C17×#475—21no—
20C8×Q8×C17C2211yescharacteristic
21C17×#335C2211yescharacteristic
22C17×#332—21no—
23C2×Q16×C17—21no—
24C2×Q16×C17—21no—
25C17×C4⋊Q8—21no—
26C4×Q8×C17D411yescharacteristic
27C2×#63230—41no—
28C2×D136—21no—
29C2×D136—21no—
30#10587560—21no—
31C17×#106—41no—
32C17×C4⋊C8—21no—
33C17×C4⋊C8—21no—
34C17×C4⋊C8D411yescharacteristic
35C17×#103—41no—
36C4×C8×C17—21no—
37C4×C8×C17C2311yescharacteristic
38#10587499—42no—
39#10587499—42no—
40C2×#63214—21no—
41C2×#63214D411yescharacteristic
42C2×#63214D411yescharacteristic
43#10587480—41no—
44C4⋊#3622—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17D811yescharacteristic
47C2×D68—41no—
48C2×D68—41no—
49C2×D68—41no—
50Q16×C17—41no—
51Q16×C17—41no—
52Q16×C17—81no—
53C2×C8×C17—21no—
54C2×C8×C17—41no—
55C2×C8×C17C2×D411yescharacteristic
56C2×C8×C17#4111yescharacteristic
57C4⋊C4×C17—21no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17D811yescharacteristic
61C42×C17—21no—
62C42×C17C2×D411yescharacteristic
63#63230—82no—
64C2×#3622—41no—
65D136—41no—
66D136—41no—
67D136—81no—
68#63214—21no—
69#63214—41no—
70#63214SD1612yescomplemented
71#12441—171nomaximal, Sylow
72C22×D17—81no—
73Q8×C17—21no—
74Q8×C17—81no—
75Q8×C17—81no—
76Q8×C17SD3212yescomplemented
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—81no—
80C2×C4×C17#12111yescharacteristic
81C2×C4×C17C2×D811yescharacteristic
82C2×C4×C17#13511yescharacteristic
83D68—81no—
84D68—81no—
85D68—81no—
86D68—81no—
87D68—161no—
88C8×C17—21no—
89C8×C17—41no—
90C8×C17—81no—
91C8×C17C2×SD1611yescharacteristic
92C8×C17AGL(1,ℤ/8)11yescharacteristic
93#3622—81no—
94#2248—171no—
95Q8×C16—171no—
96#1670D1711yescharacteristic, complemented
97#1668—171no—
98#1372—171no—
99#1362—171no—
100#1361—171no—
101C22×C17#46611yescharacteristic
102C2×D17—161no—
103C2×D17—161no—
104C2×D17—161no—
105C4×C17—22no—
106C4×C17—21no—
107C4×C17—41no—
108C4×C17—161no—
109C4×C17#46011yescharacteristic
110C4×C17#47211yescharacteristic
111C4×C17C2×SD3211yescharacteristic
112C4×C17#50811yescharacteristic
113C2×SD32—341no—
114C2×SD32—341no—
115#494C2×D1711yescharacteristic
116#493—171no—
117#475—21no—
118#460—341no—
119C8×Q8C2×D1711yescharacteristic
120C4⋊C16—171no—
121C4⋊C16—341no—
122C4×C16—171no—
123C4×C16—341no—
124#335—171no—
125#335C2×D1711yescharacteristic
126#332—21no—
127#331—341no—
128C2×C17#166811yescharacteristic
129C2×C17#221711yescharacteristic
130C2×C17#222311yescharacteristic
131D17—321no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×SD16—681no—
135C2×D8—341no—
136C2×D8—341no—
137C4⋊Q8—21no—
138#127—341no—
139C4×Q8C22⋊D1711yescharacteristic
140SD32—682no—
141SD32—682no—
142C2×C16—171no—
143C2×C16—171no—
144C2×C16—341no—
145#106—41no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—341no—
149C4⋊C8C22⋊D1711yescharacteristic
150#103—41no—
151#102—681no—
152C4×C8—21no—
153C4×C8C22×D1711yescharacteristic
154C17#1244111yescharacteristic, complemented, Sylow
155C2×Q8—41no—
156C2×Q8#6322811yescharacteristic
157C2×D4—681no—
158C2×D4—681no—
159C2×D4—681no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—1362no—
164D8—681no—
165D8—681no—
166D8—1361no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—681no—
170C2×C8D4×D1711yescharacteristic
171C2×C8#6325411yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4#6322811yescharacteristic
176C42—21no—
177C42C2×C22⋊D1711yescharacteristic
178C16—172no—
179C16—341no—
180C16—681no—
181C23—1361no—
182Q8—21no—
183Q8—81no—
184Q8—81no—
185Q8#1058749912yescomplemented
186D4—2721no—
187D4—1361no—
188D4—1361no—
189D4—1361no—
190D4—1361no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—81no—
194C2×C4C2×#6322811yescharacteristic
195C2×C4#1058762511yescharacteristic
196C2×C4#1058763611yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—1361no—
201C8SD16×D1711yescharacteristic
202C8#1058760011yescharacteristic
203C22—2721no—
204C22—2721no—
205C22—2721no—
206C22#1176304311yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—161no—
211C4#1176300711yescharacteristic
212C4#1176307711yescharacteristic
213C4C2×#1058749911yescharacteristic
214C4#1176323911yescharacteristic
215C2—5441no—
216C2#1452814211yescharacteristic
217C2#1453088211yescharacteristic
218C2#1453091611yescharacteristic
subgroupG/H
centerC22#11763043
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1670C2
socleC22×C17#466
prime radical#11763186C22
hypercenter#1670D17
hypocenterC17#12441
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. Q8×#63214C2semidirect
4. #14526857C2semidirect
5. #14526833C2semidirect
7. C17×#1670C2semidirect
70. #63214SD16semidirect
76. Q8×C17SD32semidirect
96. #1670D17semidirect
154. C17#12441semidirect
185. Q8#10587499semidirect

Maximal subgroups

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

subgroupindex
#145310412
Q8×#632142
#145281442
#145268572
#145268332
#145268292
C17×#16702
#1244117