Finite Group Databasebeta

← #27300968

Subgroups — #27300968

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811039C211yesmaximal, characteristic, complemented
2C2×#16810741C211yesmaximal, characteristic
3C2×#16810406C211yesmaximal, characteristic, complemented
4C2×#16810406C211yesmaximal, characteristic, complemented
5#20276328C211yesmaximal, characteristic, complemented
6#20272797C211yesmaximal, characteristic, complemented
7C53×#1374C211yesmaximal, characteristic, complemented
8C2×#13133239C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10C2×#13133223C2211yescharacteristic
11#16811039—22no—
12#16811039—22no—
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15C2×#13133156—21no—
16C2×C16×D53—21no—
17#16810741—22no—
18#16810741—22no—
19C2×#13133153C411yescharacteristic
20C2×#13133145C411yescharacteristic
21C2×#13133145C411yescharacteristic
22#16810406—22no—
23#16810406—22no—
24#16810402—21no—
25#16810402—21no—
26C22×C16×C53C2211yescharacteristic
27C22×C16×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C2×#11719945C2×C411yescharacteristic
30C2×#11719945C2×C411yescharacteristic
31#13133239—22no—
32#13133239—22no—
33C2×Dic212C811yescharacteristic
34C2×#11719913C811yescharacteristic
35C2×#11719913C811yescharacteristic
36C2×D424C811yescharacteristic
37#13133223—21no—
38#13133223—21no—
39#13133223—21no—
40#13133223—21no—
41#13133223—21no—
42#13133223—21no—
43#13133223D412yes—
44#13133223D412yes—
45C2×#11719917C2×C411yescharacteristic
46C2×#11719917C2×C411yescharacteristic
47C2×#11719912—21no—
48C2×#11719912—21no—
49C2×C8×D53—21no—
50C2×C8×D53—21no—
51#13133157—21no—
52#13133157—21no—
53#13133157—21no—
54#13133157D412yes—
55#13133156—24no—
56C2×#11719908—21no—
57C2×#11719908—21no—
58C16×D53—24no—
59#13133153—22no—
60#13133145—22no—
61#13133145—22no—
62C22×C8×C53C2×C411yescharacteristic
63C22×C8×C53C2311yescharacteristic
64C2×C42×C53C2×C411yescharacteristic
65C2×C16×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70C2×C16×C53D411yescharacteristic
71C2×C16×C53D411yescharacteristic
72C4×C8×C53—21no—
73C4×C8×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C22×C411yescharacteristic
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78#11719945—21no—
79#11719945—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945C22⋊C412yes—
85#11719945M1612yes—
86C2×D212C2×C811yescharacteristic
87C2×D212C2×C811yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×Dic106C2×C811yescharacteristic
91C2×Dic106C2×C811yescharacteristic
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53C2×C811yescharacteristic
95C2×C8×C53C2×C811yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C42×C53—21no—
103C42×C53—21no—
104#11719917—21no—
105#11719917—21no—
106#11719917—21no—
107#11719917C22⋊C412yes—
108C2×#88322—21no—
109C2×#88322—21no—
110Dic212—22no—
111D424—22no—
112#11719913—22no—
113#11719913—22no—
114#11719912—22no—
115#11719912—22no—
116#11719912—41no—
117#11719912—41no—
118C8×D53—22no—
119C8×D53—22no—
120C8×D53—41no—
121C8×D53—41no—
122C16×C53—21no—
123C16×C53—21no—
124C16×C53—41no—
125#11719908—22no—
126#11719908—41no—
127C23×C53#13111yescharacteristic
128C22×D53—21no—
129C22×D53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C22⋊D53—22no—
140C22⋊D53—22no—
141C22⋊D53—41no—
142C22⋊D53—41no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212C22⋊C812yes—
149C4×D53—22no—
150C4×D53—22no—
151C4×D53—41no—
152C4×D53—41no—
153Dic106—21no—
154Dic106—21no—
155Dic106—21no—
156Dic106C22⋊C812yes—
157C8×C53—21no—
158C8×C53—21no—
159C8×C53#12011yescharacteristic
160C8×C53#12111yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12311yescharacteristic
163#88322—22no—
164#88322—41no—
165#12980—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#40811yescharacteristic
169C22×C53C8×D411yescharacteristic
170C22×C53#43511yescharacteristic
171C2×D53—22no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—42no—
178C4×C53#38611yescharacteristic
179C4×C53C2×C22⋊C811yescharacteristic
180C4×C53C8×D411yescharacteristic
181C4×C53#43511yescharacteristic
182Dic53—22no—
183Dic53—41no—
184C2×#504—531no—
185C2×#443—531no—
186C2×#350—531no—
187C2×#350—531no—
188#2113—531no—
189#1376—531no—
190#1374D5311yescharacteristic, complemented
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#186911yescharacteristic
194C2×C53#217011yescharacteristic
195C2×C53#217111yescharacteristic
196D53—42no—
197D53—81no—
198C2×#135—531no—
199C2×#131—531no—
200C2×#131—531no—
201#504—1062no—
202#504—1062no—
203C2×M32—531no—
204C2×M32—531no—
205C2×M32—1061no—
206C22×C16—1061no—
207C22×C16C2×D5311yescharacteristic
208C22×C16C2×D5311yescharacteristic
209#443—1062no—
210#443—1062no—
211C2×#108—531no—
212C2×#104—531no—
213C2×#104—531no—
214C2×C4×C8C2×D5311yescharacteristic
215#350—1062no—
216#350—1062no—
217C22⋊C16—1061no—
218C22⋊C16—1061no—
219C53#1298011yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41—531no—
222#135—1062no—
223#135—1062no—
224C2×Q16—531no—
225C2×SD16—531no—
226C2×SD16—531no—
227C2×D8—531no—
228#131—532no—
229#131—532no—
230#131—1061no—
231#131—1061no—
232#131—1061no—
233#131—1061no—
234#131—1061no—
235#131—1061no—
236C2×M16—531no—
237C2×M16—531no—
238C2×M16—1061no—
239C2×M16—1061no—
240C22×C8—1061no—
241C22×C8—1061no—
242C22×C8C4×D5311yescharacteristic
243C22×C8C22×D5311yescharacteristic
244C2×C42C4×D5311yescharacteristic
245M32—532no—
246M32—1064no—
247M32—1061no—
248M32—1061no—
249M32—1061no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16—21no—
255C2×C16—1064no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16D21211yescharacteristic
259C2×C16D21211yescharacteristic
260#108—1062no—
261#104—1062no—
262#104—1062no—
263C4×C8—21no—
264C4×C8—21no—
265#41—532no—
266#41—532no—
267#41—1061no—
268#41—1061no—
269#41—1061no—
270#41—1061no—
271#41—1061no—
272#41—1061no—
273C2×Q8—531no—
274C2×Q8—531no—
275C2×D4—531no—
276C2×D4—531no—
277C2×D4—1061no—
278C2×D4—1061no—
279C22×C4—21no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4C2×C4×D5311yescharacteristic
283Q16—1062no—
284SD16—1062no—
285SD16—1062no—
286D8—1062no—
287M16—532no—
288M16—1061no—
289M16—1062no—
290M16—1062no—
291M16—1061no—
292M16—1061no—
293M16—2121no—
294M16—2121no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—1061no—
298C2×C8—1062no—
299C2×C8—1062no—
300C2×C8—1061no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8C8×D5311yescharacteristic
304C2×C8C8×D5311yescharacteristic
305C2×C8C2×D21211yescharacteristic
306C2×C8#1171994511yescharacteristic
307C2×C8D4×D5311yescharacteristic
308C2×C8D4×D5311yescharacteristic
309C2×C8D4×D5311yescharacteristic
310C2×C8#1171994911yescharacteristic
311C42—21no—
312C42—21no—
313C16—21no—
314C16—21no—
315C16—41no—
316C16—1062no—
317C16—2121no—
318C23—1061no—
319C23—1061no—
320C23#1313322311yescharacteristic
321Q8—532no—
322Q8—1061no—
323Q8—1061no—
324Q8—1061no—
325D4—532no—
326D4—1062no—
327D4—1062no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—2121no—
332D4—2121no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—1061no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—1061no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4C22⋊C4×D5311yescharacteristic
343C2×C4#1313317911yescharacteristic
344C2×C4C4×D21211yescharacteristic
345C2×C4C2×C8×D5311yescharacteristic
346C2×C4M16×D5311yescharacteristic
347C2×C4#1313322511yescharacteristic
348C8—21no—
349C8—21no—
350C8—1062no—
351C8—2121no—
352C8#1313318711yescharacteristic
353C8#1313319011yescharacteristic
354C8#1313319211yescharacteristic
355C8#1313320211yescharacteristic
356C22—21no—
357C22—21no—
358C22—1062no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22#1681058811yescharacteristic
365C22C8×D21211yescharacteristic
366C22#1681069311yescharacteristic
367C4—42no—
368C4—1062no—
369C4—2121no—
370C4#1681050111yescharacteristic
371C4C22⋊C8×D5311yescharacteristic
372C4#1681068811yescharacteristic
373C4#1681069611yescharacteristic
374C2—21no—
375C2—21no—
376C2—4241no—
377C2—2122no—
378C2#2027500611yescharacteristic
379C2#2027655311yescharacteristic
380C2#2027655811yescharacteristic
subgroupG/H
centerC2×C8C2×D212
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1374C2
socleC22×C53C8×D4
prime radicalC2×#13133239C4
cosocleC22×C8×C53C23
hypercenter#1374D53
hypocenterC53#12980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C16 ◁ #1374

Complements

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

NH ≅ G/Ntype
1. C2×#16811039C2semidirect
3. C2×#16810406C2semidirect
4. C2×#16810406C2semidirect
5. #20276328C2semidirect
6. #20272797C2semidirect
7. C53×#1374C2semidirect
190. #1374D53semidirect
219. C53#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#168110392
C2×#168107412
C2×#168104062
#202763282
#202727972
C53×#13742
#1298053