Finite Group Databasebeta

← #16948366

Subgroups — #16948366

Order 4352.

276 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 69 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#14541673C211yesmaximal, characteristic
2#14531059C211yesmaximal, characteristic
3#14530926C211yesmaximal, characteristic
4#14528179C211yesmaximal, characteristic
5#14526840C211yesmaximal, characteristic
6#14526764C211yesmaximal, characteristic
7C17×#1332C211yesmaximal, characteristic
8#11763538—21no—
9#11763214C2211yescharacteristic
10#11763214C2211yescharacteristic
11#11763210—21no—
12#11763118—22no—
13#11763094—21no—
14#11763020—21no—
15C8×Dic34C2211yescharacteristic
16Q16×Dic17—21no—
17#11762636—21no—
18#11762635—22no—
19C4⋊#63214C2211yescharacteristic
20#11762613—21no—
21#11762556—21no—
22C4×Q16×C17C2211yescharacteristic
23C4×C16×C17C2211yescharacteristic
24C17×#335C2211yescharacteristic
25C2×Q16×C17D411yescharacteristic
26C4×Q8×C17—21no—
27Q8×Dic17—22no—
28C2×Dic68—41no—
29#10587576—22no—
30#10587576—42no—
31#10587555—21no—
32#10587555—21no—
33#10587555—41no—
34C4×Dic34—41no—
35C4×Dic34D411yescharacteristic
36C2×C16×C17—21no—
37C17×#107D411yescharacteristic
38C17×C4⋊C8—21no—
39C17×#103—21no—
40C4×C8×C17C2311yescharacteristic
41#10587506—22no—
42#10587489—21no—
43#10587489—21no—
44#10587489—21no—
45#10587489D412yes—
46#10587487—41no—
47#10587485—21no—
48C8×Dic17—21no—
49C2×#63214—21no—
50#10587479—22no—
51C4⋊#3622D411yescharacteristic
52C2×Q8×C17—21no—
53C2×Dic34—41no—
54C2×Dic34—41no—
55C2×Dic34D811yescharacteristic
56Q16×C17—21no—
57Q16×C17Q1612yescomplemented
58C2×C8×C17—41no—
59C2×C8×C17C2×D411yescharacteristic
60C2×C8×C17C2×D411yescharacteristic
61C4⋊C4×C17—21no—
62C4⋊C4×C17—41no—
63C42×C17—41no—
64C42×C17C2×D411yescharacteristic
65C4⋊Dic17—22no—
66C4⋊Dic17—42no—
67C4⋊Dic17—42no—
68C4⋊Dic17—41no—
69C4⋊Dic17—41no—
70C4⋊Dic17—41no—
71C4⋊Dic17D811yescharacteristic
72#63225—21no—
73#63225—42no—
74C4×Dic17—21no—
75C4×Dic17—42no—
76C2×#3622—21no—
77Dic68—82no—
78C16×C17—22no—
79#63214—41no—
80#12405—171nomaximal, Sylow
81Q8×C17—22no—
82Q8×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17#12011yescharacteristic
86C2×C4×C17C2×D811yescharacteristic
87C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—42no—
91C2×Dic17—81no—
92Dic34—21no—
93Dic34—21no—
94Dic34—81no—
95Dic34—81no—
96Dic34—81no—
97Dic34—81no—
98C8×C17—21no—
99C8×C17—81no—
100C8×C17C2×Q1611yescharacteristic
101C8×C17#13711yescharacteristic
102#3622—41no—
103#3383—171no—
104#2252—171no—
105#2225—171no—
106#1674—171no—
107#1364—171no—
108Q8⋊C16—171no—
109#1332D1711yescharacteristic
110C22×C17#44711yescharacteristic
111C4×C17—21no—
112C4×C17—42no—
113C4×C17—81no—
114C4×C17#45111yescharacteristic
115C4×C17#45611yescharacteristic
116C4×C17#50811yescharacteristic
117C4×C17#50911yescharacteristic
118Dic17—41no—
119Dic17—41no—
120Dic17—82no—
121Dic17—161no—
122#557—341no—
123#500—171no—
124#500—171no—
125#499—341no—
126#479—342no—
127#475—341no—
128#462—341no—
129C8×Q8—171no—
130C4×Q16—341no—
131C4×Q16C2×D1711yescharacteristic
132#367—341no—
133#366—342no—
134C4⋊C16—171no—
135#358—341no—
136C4×C16C2×D1711yescharacteristic
137#335C2×D1711yescharacteristic
138#332—341no—
139C2×C17#164911yescharacteristic
140C2×C17#218611yescharacteristic
141C2×C17#219511yescharacteristic
142C2×Q16—681no—
143C2×Q16D6811yescharacteristic
144C4⋊Q8—341no—
145C4⋊Q8—341no—
146C4⋊Q8—681no—
147#125—342no—
148#125—682no—
149C4×Q8—21no—
150C4×Q8—171no—
151C4×Q8—342no—
152C4×Q8—681no—
153C2×C16—21no—
154C2×C16—341no—
155#107—172no—
156#107—341no—
157#107—341no—
158#107—341no—
159#107D6811yescharacteristic
160#106—342no—
161C4⋊C8—21no—
162C4⋊C8—171no—
163C4⋊C8—341no—
164#103—21no—
165#103—342no—
166#103—681no—
167C4×C8—341no—
168C4×C8C22×D1711yescharacteristic
169C17#1240511yescharacteristic, complemented, Sylow
170C2×Q8—21no—
171C2×Q8—171no—
172C2×Q8—681no—
173C2×Q8—681no—
174Q16—21no—
175Q16—1362no—
176Q16Dic6812yescomplemented
177C2×C8—41no—
178C2×C8—341no—
179C2×C8C2×D6811yescharacteristic
180C2×C8D4×D1711yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—171no—
184C4⋊C4—341no—
185C4⋊C4—342no—
186C4⋊C4—682no—
187C4⋊C4—682no—
188C4⋊C4—682no—
189C4⋊C4—681no—
190C4⋊C4—681no—
191C4⋊C4—681no—
192C42—41no—
193C42—341no—
194C42—682no—
195C42D4×D1711yescharacteristic
196C16—22no—
197C16—681no—
198Q8—22no—
199Q8—41no—
200Q8—341no—
201Q8—341no—
202Q8—1361no—
203Q8—1361no—
204Q8—1361no—
205Q8—1361no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—682no—
211C2×C4—1361no—
212C2×C4#1058756811yescharacteristic
213C2×C4D8×D1711yescharacteristic
214C2×C4#1058760011yescharacteristic
215C8—21no—
216C8—81no—
217C8—681no—
218C8C2×Dic6811yescharacteristic
219C8#1058759511yescharacteristic
220C22#1176294411yescharacteristic
221C4—21no—
222C4—42no—
223C4—81no—
224C4—2721no—
225C4—681no—
226C4—681no—
227C4—1362no—
228C4#1176296711yescharacteristic
229C4#1176299311yescharacteristic
230C4#1176324211yescharacteristic
231C4#1176324611yescharacteristic
232C2#1452803611yescharacteristic
233C2#1453071011yescharacteristic
234C2#1453076311yescharacteristic
subgroupG/H
centerC22#11762944
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1332C2
socle, prime radicalC22×C17#447
hypercenter#1332D17
hypocenterC17#12405
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
57. Q16×C17Q16semidirect
169. C17#12405semidirect
176. Q16Dic68semidirect

Maximal subgroups

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

subgroupindex
#145416732
#145310592
#145309262
#145281792
#145268402
#145267642
C17×#13322
#1240517