Finite Group Databasebeta

← #17962685

Subgroups — #17962685

Order 4352.

398 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#11763720C212yesmaximal, complemented
2#14543921C212yesmaximal, complemented
3#14543919C212yesmaximal, complemented
4#14539244C211yesmaximal, characteristic
5C2×#10587545—22no—
6#11763720—22no—
7#11763720C2214yes—
8C2×#10587637—22no—
9C2×#10587555C2211yescharacteristic
10#11763204—22no—
11#11763188C411yescharacteristic
12#11763188C2211yescharacteristic
13#11763181C411yescharacteristic
14#11763181C2211yescharacteristic
15C4×#63223C411yescharacteristic
16C17×#532C411yescharacteristic
17C2×#41×C17—21no—
18C22×Dic34D412yes—
19C2×#63247—42no—
20C17×C4⋊Q8C2×C411yescharacteristic
21C17×#127C2×C411yescharacteristic
22C17×#124—21no—
23C17×#121—41no—
24C4×D4×C17—41no—
25C2×C42×C17C2×C411yescharacteristic
26#10587637—44no—
27#10587637—44no—
28C2×#63231—22no—
29C2×#63231—22no—
30C2×#63229—22no—
31C2×#63229—22no—
32C2×C4⋊Dic17—41no—
33C2×#63223—21no—
34#10587555—41no—
35#10587555C2×C411yescharacteristic
36#10587555D412yes—
37#10587555C2311yescharacteristic
38#10587545—24no—
39#10587545—42no—
40#10587474—21no—
41C4×#3622C2×C411yescharacteristic
42C4×#3622C2×C411yescharacteristic
43#41×C17—42no—
44#41×C17—42no—
45C2×Q8×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C22×C4×C17—21no—
50C22×C4×C17—41no—
51C22×C4×C17C22⋊C411yescharacteristic
52C22×Dic17—22no—
53C2×Dic34—22no—
54C2×Dic34—42no—
55C2×Dic34—42no—
56C2×Dic34C22⋊C412yes—
57C2×Dic34C2×D412yes—
58#63247—44no—
59#63247—82no—
60C2×#3624—82no—
61C4⋊C4×C17—41no—
62C22⋊C4×C17—42no—
63C42×C17—21no—
64C42×C17C22⋊C411yescharacteristic
65C42×C17C22×C411yescharacteristic
66#63231—44no—
67#63231—44no—
68#63229—44no—
69#63229—44no—
70C4⋊Dic17—41no—
71C4⋊Dic17—42no—
72C4⋊Dic17—41no—
73#63223—24no—
74C2×#3622—21no—
75C2×#3622—21no—
76#13590—171nomaximal, Sylow
77C23×C17—41no—
78C23×C17#9911yescharacteristic
79Q8×C17—42no—
80D4×C17—42no—
81D4×C17—82no—
82D4×C17—82no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—42no—
87C2×C4×C17—41no—
88C2×C4×C17—81no—
89C2×C4×C17—81no—
90C2×C4×C17#9911yescharacteristic
91C2×C4×C17C2×C22⋊C411yescharacteristic
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96Dic34—42no—
97Dic34—42no—
98Dic34—82no—
99#3624—84no—
100#3622—22no—
101#3622—22no—
102#3152—171no—
103C2×#356—172no—
104#1411—172no—
105#1410—172no—
106C22×C17—41no—
107C22×C17—82no—
108C22×C17—81no—
109C22×C17#35312yes—
110C22×C17C2×#9911yescharacteristic
111C4×C17—22no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—41no—
115C4×C17—81no—
116Dic17—42no—
117Dic17—42no—
118C2×#137—342no—
119#532#59511yescharacteristic
120C2×C4⋊Q8—171no—
121#497—342no—
122#494—171no—
123#494—171no—
124#492—171no—
125#492—171no—
126C2×#101—342no—
127C4×M16—171no—
128#356—174no—
129#356—342no—
130C2×C17—21no—
131C2×C17—22no—
132C2×C17—161no—
133C2×C17C2×#35311yescharacteristic
134C2×#41—21no—
135C22×Q8—172no—
136#137—684no—
137#137—684no—
138C2×Q16—342no—
139C2×Q16—342no—
140C2×SD16—342no—
141C2×SD16—342no—
142C2×M16—341no—
143C2×M16—682no—
144C4⋊Q8—172no—
145C4⋊Q8—171no—
146C4⋊Q8—171no—
147C4⋊Q8—681no—
148C4⋊Q8C2×#59511yescharacteristic
149#127C2×#59511yescharacteristic
150#124—21no—
151#121—41no—
152C4×D4—41no—
153C2×C4⋊C4—681no—
154C2×C42C2×#59511yescharacteristic
155#101—344no—
156#101—682no—
157#97—341no—
158C4×C8—171no—
159C4×C8—171no—
160C17#1359011yescharacteristic, complemented, Sylow
161#41—42no—
162#41—42no—
163C2×Q8—21no—
164C2×Q8—172no—
165C2×Q8—172no—
166C2×Q8—342no—
167C2×Q8—682no—
168C2×Q8—682no—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—342no—
175C22×C4C22⋊#59511yescharacteristic
176Q16—684no—
177Q16—684no—
178SD16—684no—
179SD16—684no—
180M16—344no—
181M16—684no—
182M16—1362no—
183C2×C8—341no—
184C2×C8—341no—
185C2×C8—1362no—
186C4⋊C4—41no—
187C4⋊C4—681no—
188C4⋊C4—681no—
189C4⋊C4—682no—
190C22⋊C4—42no—
191C42—21no—
192C42C22⋊#59511yescharacteristic
193C42C22×#59511yescharacteristic
194C23—41no—
195C23#1058755111yescharacteristic
196Q8—42no—
197Q8—682no—
198Q8—682no—
199Q8—1362no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4#1058755111yescharacteristic
215C2×C4C2×C22⋊#59511yescharacteristic
216C8—342no—
217C8—342no—
218C8—1364no—
219C22—41no—
220C22—82no—
221C22—81no—
222C22#1176371312yes—
223C22C2×#1058755111yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—81no—
229C4—682no—
230C4—682no—
231C2—21no—
232C2—22no—
233C2—161no—
234C2C2×#1176371311yescharacteristic
subgroupG/H
centerC2C2×#11763713
commutatorC42×C17C22×C4
FrattiniC42C22×#595
Fitting, prime radicalC17×#532C4
socleC2×C17C2×#353
cosocle#10587555C23
hypercenter#532#595
hypocenterC17#13590
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#11763720C2semidirect
2. #14543921C2semidirect
3. #14543919C2semidirect
160. C17#13590semidirect

Maximal subgroups

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

subgroupindex
C2×#117637202
#145439212
#145439192
#145392442
#1359017