Finite Group Databasebeta

← #44558600

Subgroups — #44558600

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#30631627C211yesmaximal, characteristic, complemented
3#30631626C211yesmaximal, characteristic, complemented
4#30631622C211yesmaximal, characteristic
5#30631620C211yesmaximal, characteristic, complemented
6#30631616C211yesmaximal, characteristic
7#30631614C211yesmaximal, characteristic
8#30631613C211yesmaximal, characteristic
9#30631609C211yesmaximal, characteristic
10#30631606C211yesmaximal, characteristic, complemented
11#30631598C211yesmaximal, characteristic
12#30631583C211yesmaximal, characteristic, complemented
13#30631576C211yesmaximal, characteristic
14#30631569C211yesmaximal, characteristic, complemented
15#395×C251C211yesmaximal, characteristic, complemented
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
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
44C22×Q8×C251C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C23×D251C2311yescharacteristic
52C22×Dic251C2311yescharacteristic
53C22×Dic251C2311yescharacteristic
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C4⋊Dic251—22no—
60C4⋊Dic251—41no—
61Dic251⋊C4—22no—
62Dic251⋊C4—41no—
63Dic251⋊C4—41no—
64Dic251⋊C4—41no—
65Dic251⋊C4—41no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—41no—
68(C2×D251)⋊C4—41no—
69(C2×D251)⋊C4—41no—
70(C2×D251)⋊C4—41no—
71(C2×D251)⋊C4—41no—
72(C2×D251)⋊C4—41no—
73C4×Dic251—41no—
74C2×Q8×C251—41no—
75C2×Q8×C251—41no—
76C2×Q8×C251—41no—
77C22×C4×C251C2311yescharacteristic
78C22×C4×C251C2311yescharacteristic
79C22×C4×C251C2311yescharacteristic
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C4⋊C4×C251—41no—
85C4⋊C4×C251—41no—
86C4⋊C4×C251—41no—
87C22×D251—21no—
88C22×D251—21no—
89C22×D251—21no—
90C22×D251—21no—
91C22×D251—21no—
92C22×D251—21no—
93C22×D251—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—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C23×C251C2411yescharacteristic
116Q8×C251—44no—
117C2×C4×C251—21no—
118C2×C4×C251—21no—
119C2×C4×C251—21no—
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#4112yes—
138C22×C251C2×#4111yescharacteristic
139C22×C251H(5,𝔽2)11yescharacteristic
140C22×C251H(5,𝔽2)11yescharacteristic
141C22×C251H(5,𝔽2)11yescharacteristic
142C22×C2512−1+411yescharacteristic
143C22×C2512−1+411yescharacteristic
144C22×C2512−1+411yescharacteristic
145C2×D251—41no—
146C2×D251—41no—
147C2×D251—41no—
148C2×D251—41no—
149C2×D251—41no—
150C2×D251—41no—
151C2×D251—41no—
152C4×C251—22no—
153C4×C251—41no—
154C4×C251—41no—
155C4×C251—41no—
156C4×C251—41no—
157C4×C251—41no—
158C4×C251—41no—
159Dic251—41no—
160Dic251—41no—
161Dic251—41no—
162Dic251—41no—
163Dic251—41no—
164Dic251—41no—
165Dic251—41no—
166C2×C251#53911yescharacteristic
167C2×C251#54211yescharacteristic
168C2×C251#54211yescharacteristic
169C2×C251#56011yescharacteristic
170C2×C251#56311yescharacteristic
171C2×C251#56311yescharacteristic
172C2×C251#56311yescharacteristic
173D251—81no—
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#397—2511no—
184#396—2511no—
185#395—2511no—
186#395D25111yescharacteristic, complemented
187#393—2511no—
188#389—2511no—
189#388—2511no—
190#387—2511no—
191C22×Q8C2×D25111yescharacteristic
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⋊C4C2×D25111yescharacteristic
200C2×C4⋊C4C2×D25111yescharacteristic
201C2×C4⋊C4C2×D25111yescharacteristic
202C2×C22⋊C4—2511no—
203C2×C22⋊C4—2511no—
204C2×C22⋊C4—2511no—
205C2×C22⋊C4—2511no—
206C2×C22⋊C4—2511no—
207C2×C22⋊C4—2511no—
208C2×C22⋊C4—2511no—
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
226C24—2511no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—2511no—
231C22×C4—2511no—
232C22×C4—2511no—
233C22×C4—2511no—
234C22×C4—2511no—
235C22×C4—2511no—
236C22×C4—2511no—
237C22×C4C22×D25111yescharacteristic
238C22×C4C22×D25111yescharacteristic
239C22×C4C22×D25111yescharacteristic
240C22×C4C22×D25111yescharacteristic
241C22×C4C22×D25111yescharacteristic
242C22×C4C22×D25111yescharacteristic
243C22×C4C22×D25111yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—10041no—
248C4⋊C4—10041no—
249C4⋊C4—10041no—
250C4⋊C4—10041no—
251C4⋊C4—10041no—
252C4⋊C4—5022no—
253C4⋊C4—5022no—
254C22⋊C4—10041no—
255C22⋊C4—10041no—
256C22⋊C4—10041no—
257C22⋊C4—10041no—
258C22⋊C4—10041no—
259C22⋊C4—10041no—
260C22⋊C4—5022no—
261C42—10041no—
262C23—5021no—
263C23—5021no—
264C23—5021no—
265C23—5021no—
266C23—5021no—
267C23—5021no—
268C23—5021no—
269C23C23×D25111yescharacteristic
270Q8—44no—
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—21no—
290C2×C4—21no—
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#1686081112yes—
313C22—10041no—
314C22—10041no—
315C22—10041no—
316C22—10041no—
317C22—10041no—
318C22—10041no—
319C22—10041no—
320C22C2×#1686081111yescharacteristic
321C22#2254876011yescharacteristic
322C22#2254876011yescharacteristic
323C22#2254876011yescharacteristic
324C22#2254876211yescharacteristic
325C22#2254876311yescharacteristic
326C22#2254876311yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—10041no—
338C4—10041no—
339C4—10041no—
340C4—10041no—
341C2—20081no—
342C2#3063228611yescharacteristic
343C2#3063231111yescharacteristic
344C2#3063231111yescharacteristic
345C2#3063242611yescharacteristic
346C2#3063243611yescharacteristic
347C2#3063243811yescharacteristic
348C2#3063243811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#395×C251C2
prime radicalC23×D251C23
hypercenter#395D251
hypocenterC251#2838
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C251⋊#401C2semidirect
2. #30631627C2semidirect
3. #30631626C2semidirect
5. #30631620C2semidirect
10. #30631606C2semidirect
12. #30631583C2semidirect
14. #30631569C2semidirect
15. #395×C251C2semidirect
174. C251#2838semidirect
186. #395D251semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316272
#306316262
#306316222
#306316202
#306316162
#306316142
#306316132
#306316092
#306316062
#306315982
#306315832
#306315762
#306315692
#395×C2512
#2838251