Finite Group Databasebeta

← #17981188

Subgroups — #17981188

Order 4352.

368 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 55 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#14545793C212yesmaximal
2#14543973C211yesmaximal, characteristic
3C2×C22⋊#3633—22no—
4C2×(C2×#595)⋊C4—22no—
5C2×(C2×#595)⋊C4C411yescharacteristic
6C2×(C2×#595)⋊C4C2211yescharacteristic
7C2×C4⋊C4×D17C411yescharacteristic
8C23×#595—21no—
9C23×#595D412yes—
10C22×#3633—42no—
11C22×C4×D17—21no—
12C22×C4×D17C2×C411yescharacteristic
13C22⋊#3633—24no—
14C22⋊#3633—42no—
15C2×C4⋊C4×C17C811yescharacteristic, complemented
16C2×C4⋊Dic17C811yescharacteristic, complemented
17C2×#63225—21no—
18C4⋊C4×D17—41no—
19C4⋊C4×D17—41no—
20(C2×#595)⋊C4—41no—
21(C2×#595)⋊C4—41no—
22(C2×#595)⋊C4—44no—
23C23×D17C22⋊C411yescharacteristic
24C22×#595—22no—
25C22×#595—22no—
26C22×#595—22no—
27C22×#595—42no—
28C22×#595—42no—
29C22×#595—42no—
30C22×#595—41no—
31C22×#595—41no—
32C22×#595—42no—
33C22×#595SD1612yes—
34C22×#595Q1612yes—
35C2×#3633—44no—
36C2×#3633—82no—
37C2×#3633—82no—
38C22×C4×C17—21no—
39C22×C4×C17C2×C811yescharacteristic
40C22×Dic17—21no—
41C22×Dic17M1611yescharacteristic
42C2×C4×D17—21no—
43C2×C4×D17—21no—
44C2×C4×D17—42no—
45C2×C4×D17—42no—
46C2×C4×D17—42no—
47C2×C4×D17—41no—
48C2×C4×D17—41no—
49C4⋊C4×C17—41no—
50C4⋊Dic17—41no—
51#63225—42no—
52#7132—171nomaximal, Sylow
53C23×C17C22⋊C811yescharacteristic
54C22×D17—22no—
55C22×D17—21no—
56C22×D17—21no—
57C22×D17#9911yescharacteristic
58C22×D17#10011yescharacteristic
59C22×D17#10312yes—
60C22×D17#10412yes—
61C2×#595—22no—
62C2×#595—22no—
63C2×#595—42no—
64C2×#595—42no—
65C2×#595—42no—
66C2×#595—42no—
67C2×#595—42no—
68C2×#595—44no—
69C2×#595—82no—
70C2×#595—81no—
71C2×#595—82no—
72C2×#595—81no—
73#3633—84no—
74C2×C4×C17—21no—
75C2×C4×C17—41no—
76C2×C4×C17—42no—
77C2×C4×C17—41no—
78C2×Dic17—21no—
79C2×Dic17—41no—
80C2×Dic17—42no—
81C2×Dic17—41no—
82C4×D17—42no—
83C4×D17—42no—
84C4×D17—82no—
85C4×D17—81no—
86C4×D17—81no—
87#1445—171no—
88#1280—172no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17#32311yescharacteristic
92C22×C17Q8⋊C812yes—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—22no—
96C2×D17—22no—
97C2×D17—22no—
98C2×D17—22no—
99C2×D17#32812yes—
100C2×D17#32911yescharacteristic
101C2×D17#33311yescharacteristic
102C2×D17#35212yes—
103C2×D17#35311yescharacteristic
104C2×D17#35511yescharacteristic
105#595—42no—
106#595—42no—
107#595—82no—
108C4×C17—41no—
109C4×C17—81no—
110Dic17—41no—
111Dic17—81no—
112C22×C4⋊C4—171no—
113C2×C22⋊C8—342no—
114C2×#95—171no—
115C2×#95—171no—
116C2×#95—342no—
117C2×C17—22no—
118C2×C17#127211yescharacteristic
119C2×C17#131711yescharacteristic
120C2×C17#131811yescharacteristic
121D17—21no—
122D17—21no—
123D17#135212yes—
124D17#135312yes—
125C23×C4—172no—
126C23×C4—171no—
127C23×C4—341no—
128C23×C4—341no—
129C22×C8—682no—
130C2×C4⋊C4—171no—
131C2×C4⋊C4—341no—
132C2×C4⋊C4—681no—
133C2×C4⋊C4—681no—
134C2×C4⋊C4#363311yescharacteristic, complemented
135C22⋊C8—344no—
136C22⋊C8—682no—
137#95—681no—
138#95—681no—
139#95—684no—
140C17#713211yescharacteristic, complemented, Sylow
141C24—171no—
142C22×C4—21no—
143C22×C4—172no—
144C22×C4—172no—
145C22×C4—171no—
146C22×C4—341no—
147C22×C4—341no—
148C22×C4—342no—
149C22×C4—342no—
150C22×C4—342no—
151C22×C4—341no—
152C22×C4—682no—
153C22×C4—681no—
154C22×C4—681no—
155C22×C4—682no—
156C22×C4—682no—
157C22×C4—682no—
158C22×C4—682no—
159C22×C4—681no—
160C22×C4—682no—
161C22×C4—682no—
162C22×C4—681no—
163C22×C4C2×#363311yescharacteristic
164C2×C8—684no—
165C2×C8—1362no—
166C2×C8—1362no—
167C4⋊C4—41no—
168C4⋊C4—681no—
169C4⋊C4—682no—
170C23—171no—
171C23—171no—
172C23—172no—
173C23—172no—
174C23—341no—
175C23—342no—
176C23—341no—
177C23C22⋊#363311yescharacteristic
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—341no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4—682no—
188C2×C4—682no—
189C2×C4—681no—
190C2×C4—682no—
191C2×C4—682no—
192C2×C4—682no—
193C2×C4—684no—
194C2×C4—682no—
195C2×C4—681no—
196C2×C4—1361no—
197C2×C4—1361no—
198C2×C4—1362no—
199C2×C4—1362no—
200C2×C4—1361no—
201C2×C4—1362no—
202C2×C4—1361no—
203C8—1364no—
204C22—21no—
205C22—21no—
206C22—172no—
207C22—172no—
208C22—171no—
209C22—171no—
210C22—171no—
211C22—171no—
212C22—342no—
213C22—342no—
214C22—342no—
215C22—342no—
216C22—341no—
217C22—341no—
218C22#1176389411yescharacteristic
219C22Q8⋊#363312yes—
220C4—41no—
221C4—81no—
222C4—682no—
223C4—682no—
224C4—681no—
225C4—1362no—
226C4—1361no—
227C2—22no—
228C2—172no—
229C2—172no—
230C2—341no—
231C2—341no—
232C2#1454578511yescharacteristic
233C2#1454581011yescharacteristic
234C2#1454581111yescharacteristic
subgroupG/H
centerC22#11763894
commutatorC22×C4×C17C2×C8
FrattiniC22×C4C2×#3633
FittingC2×C4⋊C4×C17C8
socleC22×C17#323
prime radicalC23×D17C22⋊C4
cosocleC2×(C2×#595)⋊C4C22
hypercenterC2×C4⋊C4#3633
hypocenterC17#7132
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
15. C2×C4⋊C4×C17C8semidirect
16. C2×C4⋊Dic17C8semidirect
134. C2×C4⋊C4#3633semidirect
140. C17#7132semidirect

Maximal subgroups

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

subgroupindex
#145457932
#145439732
#713217