Finite Group Databasebeta

← #44558597

Subgroups — #44558597

Order 32128.

366 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C251⋊#401C211yesmaximal, characteristic, complemented
2#30631628C211yesmaximal, characteristic, complemented
3#30631627C211yesmaximal, characteristic, complemented
4#30631623C211yesmaximal, characteristic
5#30631620C211yesmaximal, characteristic, complemented
6#30631615C211yesmaximal, characteristic
7#30631612C211yesmaximal, characteristic
8#30631610C211yesmaximal, characteristic
9#30631608C211yesmaximal, characteristic, complemented
10#30631607C211yesmaximal, characteristic, complemented
11#30631599C211yesmaximal, characteristic
12#30631584C211yesmaximal, characteristic, complemented
13#30631575C211yesmaximal, characteristic
14#30631574C211yesmaximal, characteristic, complemented
15#397×C251C211yesmaximal, characteristic
16C22×Dic502C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C251⋊#95C2211yescharacteristic
32C251⋊#95C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C2×C4⋊C4×C251C2211yescharacteristic
45C2×C22⋊C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C2×Dic502—41no—
52C2×Dic502—41no—
53C2×Dic502—41no—
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C4⋊Dic251—41no—
63C4⋊Dic251—41no—
64C4⋊Dic251—41no—
65Dic251⋊C4—22no—
66Dic251⋊C4—22no—
67Dic251⋊C4—41no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70Dic251⋊C4—41no—
71C22⋊Dic251—22no—
72C22⋊Dic251—41no—
73C22⋊Dic251—41no—
74C22⋊Dic251—41no—
75C4×Dic251—41no—
76C24×C251C2311yescharacteristic
77C22×C4×C251C2311yescharacteristic
78C22×C4×C251C2311yescharacteristic
79C22×C4×C251C2311yescharacteristic
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C4⋊C4×C251—41no—
84C22⋊C4×C251—41no—
85C22⋊C4×C251—41no—
86C22⋊C4×C251—41no—
87Dic502—44no—
88C2×Dic251—21no—
89C2×Dic251—21no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—21no—
96C2×Dic251—21no—
97C2×Dic251—21no—
98C2×Dic251—21no—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251#4112yes—
112C23×C251—21no—
113C23×C251—21no—
114C23×C251—21no—
115C23×C251—21no—
116C23×C251—21no—
117C23×C251—21no—
118C23×C251—21no—
119C23×C251C2411yescharacteristic
120C2×C4×C251—21no—
121C2×C4×C251—21no—
122C2×C4×C251—21no—
123C2×C4×C251—21no—
124C2×C4×C251—21no—
125C2×C4×C251—21no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C22×C251—41no—
139C22×C251—41no—
140C22×C251—41no—
141C22×C251—41no—
142C22×C251—41no—
143C22×C251—41no—
144C22×C251—41no—
145C22×C251C2×#4111yescharacteristic
146C22×C251H(5,𝔽2)11yescharacteristic
147C22×C251H(5,𝔽2)11yescharacteristic
148C22×C251H(5,𝔽2)11yescharacteristic
149C22×C2512−1+411yescharacteristic
150C22×C2512−1+411yescharacteristic
151C22×C2512−1+411yescharacteristic
152C4×C251—41no—
153C4×C251—41no—
154C4×C251—41no—
155C4×C251—41no—
156C4×C251—41no—
157C4×C251—41no—
158Dic251—22no—
159Dic251—41no—
160Dic251—41no—
161Dic251—41no—
162Dic251—41no—
163Dic251—41no—
164Dic251—41no—
165Dic251—41no—
166C2×C251—81no—
167C2×C251#53911yescharacteristic
168C2×C251#54211yescharacteristic
169C2×C251#54211yescharacteristic
170C2×C251#56011yescharacteristic
171C2×C251#56311yescharacteristic
172C2×C251#56311yescharacteristic
173C2×C251#56311yescharacteristic
174C251#283811yescharacteristic, complemented, Sylow
175#2838—2511nomaximal, Sylow
176#401—2511no—
177#400—2511no—
178#400—2511no—
179#399—2511no—
180#398—2511no—
181#397—2511no—
182#397—2511no—
183#397D25111yescharacteristic
184#396—2511no—
185#395—2511no—
186#395—2511no—
187#393—2511no—
188#389—2511no—
189#388—2511no—
190#387—2511no—
191C22×Q8—2511no—
192C2×C4⋊C4—2511no—
193C2×C4⋊C4—2511no—
194C2×C4⋊C4—2511no—
195C2×C4⋊C4—2511no—
196C2×C4⋊C4—2511no—
197C2×C4⋊C4—2511no—
198C2×C4⋊C4—2511no—
199C2×C4⋊C4—2511no—
200C2×C4⋊C4—2511no—
201C2×C4⋊C4C2×D25111yescharacteristic
202C2×C22⋊C4—2511no—
203C2×C22⋊C4—2511no—
204C2×C22⋊C4—2511no—
205C2×C22⋊C4—2511no—
206C2×C22⋊C4C2×D25111yescharacteristic
207C2×C22⋊C4C2×D25111yescharacteristic
208C2×C22⋊C4C2×D25111yescharacteristic
209C2×C42—2511no—
210#95—2511no—
211#95—2511no—
212#95—2511no—
213#95—2511no—
214#95—2511no—
215#95—2511no—
216#95—2511no—
217#95—2511no—
218#95—2511no—
219#95—2511no—
220#95—2511no—
221#95—2511no—
222#95—2511no—
223#95C2×D25111yescharacteristic
224#95C2×D25111yescharacteristic
225#95C2×D25111yescharacteristic
226C24C22×D25111yescharacteristic
227C2×Q8—10041no—
228C2×Q8—10041no—
229C2×Q8—10041no—
230C22×C4—2511no—
231C22×C4—2511no—
232C22×C4—2511no—
233C22×C4—2511no—
234C22×C4—2511no—
235C22×C4—2511no—
236C22×C4—2511no—
237C22×C4—2511no—
238C22×C4C22×D25111yescharacteristic
239C22×C4C22×D25111yescharacteristic
240C22×C4C22×D25111yescharacteristic
241C22×C4C22×D25111yescharacteristic
242C22×C4C22×D25111yescharacteristic
243C22×C4C22×D25111yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—10041no—
246C4⋊C4—10041no—
247C4⋊C4—10041no—
248C4⋊C4—10041no—
249C4⋊C4—10041no—
250C4⋊C4—10041no—
251C4⋊C4—10041no—
252C4⋊C4—5022no—
253C4⋊C4—5022no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—10041no—
258C22⋊C4—10041no—
259C22⋊C4—10041no—
260C22⋊C4—5022no—
261C42—10041no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C23×D25111yescharacteristic
270Q8—10044no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—5021no—
290C2×C4—5021no—
291C2×C4—5021no—
292C2×C4—5021no—
293C2×C4—5021no—
294C2×C4—5021no—
295C2×C4—5021no—
296C2×C4—5021no—
297C2×C4—5021no—
298C2×C4—5021no—
299C2×C4—5021no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—2512no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#41×D25111yescharacteristic
321C22#2254876011yescharacteristic
322C22#2254876011yescharacteristic
323C22#2254876111yescharacteristic
324C22#2254876211yescharacteristic
325C22#2254876211yescharacteristic
326C22#2254876211yescharacteristic
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—10041no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—10041no—
338C4—10041no—
339C4—10041no—
340C4—5022no—
341C2—81no—
342C2#3063228011yescharacteristic
343C2#3063230111yescharacteristic
344C2#3063230311yescharacteristic
345C2#3063242511yescharacteristic
346C2#3063243911yescharacteristic
347C2#3063244011yescharacteristic
348C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#397×C251C2
prime radicalC24×C251C23
hypercenter#397D251
hypocenterC251#2838
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C251⋊#401C2semidirect
2. #30631628C2semidirect
3. #30631627C2semidirect
5. #30631620C2semidirect
9. #30631608C2semidirect
10. #30631607C2semidirect
12. #30631584C2semidirect
14. #30631574C2semidirect
174. C251#2838semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316282
#306316272
#306316232
#306316202
#306316152
#306316122
#306316102
#306316082
#306316072
#306315992
#306315842
#306315752
#306315742
#397×C2512
#2838251