Finite Group Databasebeta

← #21436912

Subgroups — #21436912

Order 7552.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#16838137C211yesmaximal, characteristic
2Q8×Dic118C211yesmaximal, characteristic, complemented
3#16837762C212yesmaximal, complemented
4#16837749C211yesmaximal, characteristic
5#16837698C211yesmaximal, characteristic, complemented
6#16837683C212yesmaximal, complemented
7#16837625C212yesmaximal, complemented
8#16837603C211yesmaximal, characteristic
9#16837504C212yesmaximal, complemented
10C4×Dic236C211yesmaximal, characteristic, complemented
11C59×#446C211yesmaximal, characteristic, complemented
12C2×Dic236C2211yescharacteristic
13C2×#11745660C2212yes—
14#14273727C2212yes—
15#14273727C2211yescharacteristic
16#14273726—21no—
17#14273726—22no—
18#14273724C2211yescharacteristic
19#14273716—21no—
20#14273715C2212yes—
21#14273712—22no—
22Q8×Dic59—21no—
23C4×Dic118C2211yescharacteristic
24C4×Dic118C2211yescharacteristic
25C4×Dic118C2212yes—
26C4×D236C2212yes—
27#14273689—21no—
28#14273661C2211yescharacteristic
29#14273661C2212yes—
30#14273652C2212yes—
31#14273652C2211yescharacteristic
32#14273652C2211yescharacteristic
33#14273649C2212yes—
34#14273649C2212yes—
35#14273649C2212yes—
36C4×Q8×C59C2211yescharacteristic
37C4⋊C8×C59C2212yes—
38C4⋊C8×C59C2211yescharacteristic
39#97×C59C2212yes—
40C4×C8×C59C2211yescharacteristic
41C2×Q8×C59D411yescharacteristic
42C2×D236—21no—
43C2×D236—21no—
44C2×D236C2312yes—
45C2×C4×D59—22no—
46C2×Dic118—21no—
47C2×Dic118—22no—
48C2×Dic118C2311yescharacteristic
49C2×Dic118C2311yescharacteristic
50C2×C8×C59C2311yescharacteristic
51C2×C8×C59C2311yescharacteristic
52C2×C8×C59C2312yes—
53C4⋊C4×C59D411yescharacteristic
54C4⋊C4×C59D412yes—
55C42×C59C2311yescharacteristic
56C42×C59C2312yes—
57(C2×D59)⋊C4—22no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60C4⋊Dic59—22no—
61C4⋊Dic59—21no—
62C4⋊Dic59C2312yes—
63C4⋊Dic59C2311yescharacteristic
64C4⋊Dic59C2311yescharacteristic
65Dic59⋊C4—21no—
66Dic59⋊C4—21no—
67Dic59⋊C4—22no—
68C4×Dic59—21no—
69C4×Dic59—22no—
70C4×Dic59—21no—
71Dic236—22no—
72#11745660—24no—
73C22×D59—22no—
74Q8×C59—22no—
75C2×C4×C59C2×D412yes—
76C2×C4×C59C2×D411yescharacteristic
77C2×C4×C59C2×D412yes—
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2411yescharacteristic
80C2×Dic59—21no—
81C2×Dic59—22no—
82C2×Dic59—21no—
83C2×Dic59—21no—
84C2×Dic59—21no—
85D236—22no—
86D236—22no—
87D236—42no—
88D236—81no—
89C4×D59—82no—
90Dic118—21no—
91Dic118—21no—
92Dic118—21no—
93Dic118—82no—
94Dic118—81no—
95Dic118#4112yes—
96C8×C59—22no—
97C8×C59—21no—
98C8×C59—21no—
99C22×C59C22×D411yescharacteristic
100C2×D59—42no—
101C2×D59—42no—
102C2×D59—42no—
103C4×C59—22no—
104C4×C59—22no—
105C4×C59—21no—
106C4×C59#13712yes—
107C4×C59C2×#4111yescharacteristic
108C4×C592−1+411yescharacteristic
109Dic59—22no—
110Dic59—41no—
111Dic59—41no—
112Dic59—42no—
113Dic59—41no—
114#3375—591nomaximal, Sylow
115C2×C59#54811yescharacteristic
116C2×C59C2×#13711yescharacteristic
117C2×C59#57611yescharacteristic
118D59—82no—
119#559—591no—
120Q82—591no—
121#489—592no—
122#486—591no—
123#477—591no—
124#474—592no—
125#464—592no—
126#460—591no—
127#446D5911yescharacteristic, complemented
128#440—592no—
129C4×Q16—591no—
130C59#337511yescharacteristic, complemented, Sylow
131C2×Q16—591no—
132C2×SD16—592no—
133C4⋊Q8—592no—
134C4⋊Q8—591no—
135C4⋊Q8—1181no—
136C4⋊Q8—1182no—
137#127—591no—
138#124—592no—
139#124—1181no—
140#123—1182no—
141C4×Q8—591no—
142C4×Q8—592no—
143C4×Q8—591no—
144C4×Q8—1181no—
145C4×Q8C2×D5911yescharacteristic
146C4×D4—592no—
147C4×D4—1181no—
148#107—591no—
149#107—592no—
150C4⋊C8C2×D5911yescharacteristic
151C4⋊C8C2×D5912yes—
152#103—591no—
153#103—591no—
154#103—592no—
155#102—592no—
156#102—592no—
157#102—592no—
158#97C2×D5912yes—
159C4×C8C2×D5911yescharacteristic
160C2×Q8—591no—
161C2×Q8—591no—
162C2×Q8—1182no—
163C2×Q8—1181no—
164C2×Q8D23611yescharacteristic
165C2×D4—592no—
166C2×D4—1181no—
167C2×D4—1181no—
168C22×C4—1182no—
169Q16—1182no—
170SD16—1184no—
171C2×C8C22×D5911yescharacteristic
172C2×C8C22×D5911yescharacteristic
173C2×C8C22×D5912yes—
174C4⋊C4—592no—
175C4⋊C4—591no—
176C4⋊C4—591no—
177C4⋊C4—1181no—
178C4⋊C4—1182no—
179C4⋊C4—1181no—
180C4⋊C4—1182no—
181C4⋊C4—1181no—
182C4⋊C4D23611yescharacteristic
183C4⋊C4D23612yes—
184C22⋊C4—1182no—
185C22⋊C4—1182no—
186C22⋊C4—1182no—
187C42—1181no—
188C42—1182no—
189C42—1181no—
190C42C22×D5911yescharacteristic
191C42C22×D5912yes—
192C23—1182no—
193Q8—22no—
194Q8—592no—
195Q8—1181no—
196Q8—1181no—
197Q8—1181no—
198Q8—4722no—
199Q8—4721no—
200D4—1182no—
201D4—1182no—
202D4—4721no—
203D4—2362no—
204C2×C4—1181no—
205C2×C4—1181no—
206C2×C4—1181no—
207C2×C4—1182no—
208C2×C4—1181no—
209C2×C4—4722no—
210C2×C4C2×D23612yes—
211C2×C4C2×D23612yes—
212C2×C4C2×D23611yescharacteristic
213C2×C4C2×D23611yescharacteristic
214C2×C4C23×D5911yescharacteristic
215C8—22no—
216C8—21no—
217C8—21no—
218C22—2362no—
219C22—2362no—
220C22—2362no—
221C22C22×D23611yescharacteristic
222C4—22no—
223C4—22no—
224C4—21no—
225C4—1182no—
226C4—2361no—
227C4—2361no—
228C4—2362no—
229C4—2361no—
230C4#1427376012yes—
231C4#41×D5911yescharacteristic
232C4#1427377511yescharacteristic
233C2—4722no—
234C2#1683806411yescharacteristic
235C2C2×#1427376011yescharacteristic
236C2#1683820711yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#446C2
socleC22×C59C22×D4
hypercenter#446D59
hypocenterC59#3375
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. Q8×Dic118C2semidirect
3. #16837762C2semidirect
5. #16837698C2semidirect
6. #16837683C2semidirect
7. #16837625C2semidirect
9. #16837504C2semidirect
10. C4×Dic236C2semidirect
11. C59×#446C2semidirect
127. #446D59semidirect
130. C59#3375semidirect