Finite Group Databasebeta

← #17978485

Subgroups — #17978485

Order 4352.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545150C214yesmaximal, complemented
2#14545146C212yesmaximal, complemented
3#14545142C212yesmaximal, complemented
4#14545121C212yesmaximal, complemented
5#14545011C211yesmaximal, characteristic
6C2×C22⋊#63216C211yesmaximal, characteristic
7#14544995C212yesmaximal, complemented
8C2×#11762767C211yesmaximal, characteristic
9C2×#10587538C2212yes—
10C22×#63216C2212yes—
11#11763730C2214yescomplemented
12#11763730C2214yescomplemented
13C22⋊#63216C2211yescharacteristic
14C22⋊#63216C2211yescharacteristic
15C22⋊#63216C2212yes—
16C22⋊#63216C2212yes—
17C22⋊#63216C2212yes—
18#11763698C2214yescomplemented
19#11763695C2214yescomplemented
20C23×#3622C2211yescharacteristic
21C2×#10587612C411yescharacteristic
22#11762767C411yescharacteristic
23#11762767C412yes—
24#11762767C411yescharacteristic
25#11762767C2211yescharacteristic
26#11762767C2211yescharacteristic
27#11762767C2212yes—
28C2×#10587486C412yes—
29C2×C8×Dic17C2212yes—
30C22×M16×C17C411yescharacteristic
31C23×C4×C17C2×C411yescharacteristic
32C2×M16×C17—24no—
33C2×M16×C17C2×C411yescharacteristic
34C2×M16×C17C2×C412yes—
35C2×M16×C17C2×C411yescharacteristic
36C22×C8×C17C2×C412yes—
37C2×C22⋊Dic17C812yes—
38#10587612C811yescharacteristic
39#10587612C811yescharacteristic
40#10587612C812yes—
41#10587612C2×C412yes—
42#10587612C2×C411yescharacteristic
43#10587612C2×C411yescharacteristic
44C2×C4⋊Dic17C811yescharacteristic
45C2×C4⋊Dic17C811yescharacteristic
46C2×C4×Dic17C2×C412yes—
47C22×#3622—24no—
48C22×#3622C2×C411yescharacteristic
49C22×#3622C2×C411yescharacteristic
50C22×#3622C2×C411yescharacteristic
51C22×#3622C2311yescharacteristic
52C22×#3622C2311yescharacteristic
53C22×#3622C2311yescharacteristic
54#10587538—24no—
55C2×#63216—24no—
56C2×#63216C2312yes—
57C2×#63216C2312yes—
58C2×#63216C2314yes—
59#10587486C2×C412yes—
60#10587486C2×C412yes—
61#10587486C2×C412yes—
62#10587486C2×C412yes—
63C8×Dic17C2×C412yes—
64C8×Dic17C2×C412yes—
65C8×Dic17C2312yes—
66C8×Dic17C2312yes—
67C24×C17C2×C811yescharacteristic
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17C2×C811yescharacteristic
71C22×C4×C17C2×C811yescharacteristic
72C22×C4×C17C2×C811yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C22×C411yescharacteristic
75C22×C4×C17C22×C411yescharacteristic
76C22×Dic17C2×C812yes—
77C22×Dic17C2×C812yes—
78M16×C17—22no—
79M16×C17—24no—
80M16×C17M1614yes—
81C2×C8×C17—24no—
82C2×C8×C17C22×C412yes—
83C2×C8×C17C22×C412yes—
84C22⋊Dic17C2×C812yes—
85C22⋊Dic17C2×C812yes—
86C22⋊Dic17C2×C812yes—
87C22⋊Dic17C2×C812yes—
88C4⋊Dic17C2×C812yes—
89C4⋊Dic17C2×C812yes—
90C4⋊Dic17C2×C811yescharacteristic
91C4⋊Dic17C2×C811yescharacteristic
92C4⋊Dic17C2×C811yescharacteristic
93C4⋊Dic17C2×C811yescharacteristic
94C4×Dic17C2×C812yes—
95C4×Dic17C2×C812yes—
96C4×Dic17C22×C412yes—
97C4×Dic17C22×C412yes—
98C2×#3622—22no—
99C2×#3622—22no—
100C2×#3622—22no—
101C2×#3622—22no—
102C2×#3622—22no—
103C2×#3622C22×C411yescharacteristic
104C2×#3622C22×C411yescharacteristic
105C2×#3622C22×C411yescharacteristic
106C2×#3622#4114yes—
107C2×#3622C2411yescharacteristic
108#63216—22no—
109#63216—22no—
110#63216—24no—
111#33422—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17—22no—
114C23×C17—41no—
115C23×C17C22×C811yescharacteristic
116C23×C17C22×C811yescharacteristic
117C23×C17C22×C811yescharacteristic
118C2×C4×C17—22no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17#11712yes—
126C2×C4×C17C22×C811yescharacteristic
127C2×C4×C17C22×C811yescharacteristic
128C2×C4×C17C22×C811yescharacteristic
129C2×C4×C17C2×M1612yes—
130C2×C4×C17C23×C411yescharacteristic
131C2×Dic17—24no—
132C2×Dic17—42no—
133C2×Dic17C22×C812yes—
134C2×Dic17C22×C812yes—
135C2×Dic17C22×C812yes—
136C2×Dic17C22×C812yes—
137C8×C17—22no—
138C8×C17C2×M1614yes—
139#3622—21no—
140#3622—21no—
141#3622#11712yes—
142#3622C2×#4112yes—
143C2×#405—171no—
144#2164—174no—
145#2163—172no—
146#2162—172no—
147#2151—172no—
148#2112—171no—
149C2×C22⋊C16—171no—
150#2108—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—44no—
159C22×C17#43112yes—
160C22×C17#50412yes—
161C22×C17C23×C811yescharacteristic
162C4×C17—21no—
163C4×C17—21no—
164C4×C17#43112yes—
165C4×C17C2×#11711yescharacteristic
166C4×C17C22×M1611yescharacteristic
167Dic17—22no—
168Dic17—22no—
169Dic17—22no—
170Dic17—22no—
171C22×M16#59511yescharacteristic
172C23×C8—171no—
173C2×#117—171no—
174C2×M32—172no—
175C22×C16—172no—
176#405—171no—
177#405—171no—
178#405—172no—
179#405—171no—
180#405—171no—
181#405—172no—
182C2×#97—172no—
183C2×C4×C8—172no—
184C4⋊C16—174no—
185C4⋊C16—174no—
186C22⋊C16—171no—
187C22⋊C16—171no—
188C22⋊C16—172no—
189C22⋊C16—172no—
190C22⋊C16—172no—
191#346—174no—
192C4×C16—174no—
193C2×C17—22no—
194C2×C17—22no—
195C2×C17—41no—
196C2×C17C2×#43111yescharacteristic
197C2×C17C2×#50411yescharacteristic
198C2×C17#340711yescharacteristic
199C23×C4C2×#59511yescharacteristic
200C2×M16—24no—
201C2×M16C2×#59511yescharacteristic
202C2×M16C2×#59512yes—
203C2×M16C2×#59511yescharacteristic
204C22×C8—171no—
205C22×C8—171no—
206C22×C8—171no—
207C22×C8—171no—
208C22×C8—171no—
209C22×C8—171no—
210C22×C8—344no—
211C22×C8C2×#59512yes—
212#117—171no—
213#117—171no—
214#117—172no—
215#117—172no—
216#117—171no—
217#117—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C22⋊C4—172no—
221C2×C42—172no—
222M32—344no—
223C2×C16—172no—
224C2×C16—174no—
225C2×C16—172no—
226C2×C16—344no—
227#97—172no—
228#97—172no—
229#97—172no—
230#97—172no—
231C4×C8—172no—
232C4×C8—172no—
233C4×C8—172no—
234C4×C8—172no—
235C17#3342211yescharacteristic, complemented, Sylow
236C24#6324211yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4#6324211yescharacteristic
242C22×C4#6324211yescharacteristic
243C22×C4#6324211yescharacteristic
244C22×C4C22×#59511yescharacteristic
245C22×C4C22×#59511yescharacteristic
246C22×C4C22×#59511yescharacteristic
247M16—22no—
248M16—24no—
249M16#6324714yes—
250C2×C8—24no—
251C2×C8—174no—
252C2×C8—171no—
253C2×C8—171no—
254C2×C8—171no—
255C2×C8—171no—
256C2×C8—342no—
257C2×C8—342no—
258C2×C8—342no—
259C2×C8—342no—
260C2×C8—342no—
261C2×C8C22×#59512yes—
262C2×C8C22×#59512yes—
263C4⋊C4—172no—
264C4⋊C4—172no—
265C4⋊C4—171no—
266C4⋊C4—171no—
267C4⋊C4—171no—
268C4⋊C4—171no—
269C22⋊C4—172no—
270C22⋊C4—172no—
271C22⋊C4—172no—
272C22⋊C4—172no—
273C42—172no—
274C42—172no—
275C42—172no—
276C42—172no—
277C16—342no—
278C16—342no—
279C16—344no—
280C23—22no—
281C23—22no—
282C23—41no—
283C23C2×#6324211yescharacteristic
284C23C2×#6324211yescharacteristic
285C23C2×#6324211yescharacteristic
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—172no—
294C2×C4—172no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—344no—
298C2×C4—682no—
299C2×C4C2×#6324211yescharacteristic
300C2×C4C2×#6324211yescharacteristic
301C2×C4C2×#6324211yescharacteristic
302C2×C4#1058767712yes—
303C2×C4C2×#6324712yes—
304C2×C4C23×#59511yescharacteristic
305C8—22no—
306C8—172no—
307C8—172no—
308C8—341no—
309C8—341no—
310C8C2×#6324714yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22#1176382612yes—
320C22#1176386812yes—
321C22C22×#6324211yescharacteristic
322C4—21no—
323C4—21no—
324C4—342no—
325C4—342no—
326C4—342no—
327C4—342no—
328C4#1176382612yes—
329C4C2×#1058767711yescharacteristic
330C4C22×#6324711yescharacteristic
331C2—22no—
332C2—22no—
333C2—41no—
334C2C2×#1176382611yescharacteristic
335C2C2×#1176386811yescharacteristic
336C2#1454575311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#595
commutator, socleC22×C17C23×C8
FittingC22×M16×C17C4
prime radicalC24×C17C2×C8
cosocleC2×#3622C24
hypercenterC22×M16#595
hypocenterC17#33422
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #14545150C2semidirect
2. #14545146C2semidirect
3. #14545142C2semidirect
4. #14545121C2semidirect
7. #14544995C2semidirect
11. #11763730C22semidirect
12. #11763730C22semidirect
18. #11763698C22semidirect
19. #11763695C22semidirect
235. C17#33422semidirect

Maximal subgroups

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

subgroupindex
#145451502
#145451462
#145451422
#145451212
#145450112
C2×C22⋊#632162
#145449952
C2×#117627672
#3342217