Finite Group Databasebeta

← #21435898

Subgroups — #21435898

Order 7552.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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#16838217C212yesmaximal, complemented
2#16838216C211yesmaximal, characteristic, complemented
3#16837780C211yesmaximal, characteristic, complemented
4#16837677C211yesmaximal, characteristic, complemented
5#16837675C212yesmaximal, complemented
6#16837674C211yesmaximal, characteristic, complemented
7#16837585C212yesmaximal, complemented
8#16837583C212yesmaximal, complemented
9#16837519C211yesmaximal, characteristic, complemented
10#16837517C211yesmaximal, characteristic, complemented
11C59×#578C211yesmaximal, characteristic, complemented
12#14273774C2212yes—
13#14273758—22no—
14#14273758C2212yes—
15#14273757—22no—
16#14273757C2212yes—
17#14273757C2212yes—
18#14273756—22no—
19#14273747—22no—
20#14273747C2211yescharacteristic
21#14273746—21no—
22C2×Dic236C2211yescharacteristic
23C2×#11745672—22no—
24C2×#11745672C2212yes—
25Q16×D59—21no—
26#14273734C2212yes—
27#14273733C2211yescharacteristic
28#14273725C2212yes—
29#14273663C2211yescharacteristic
30#14273662C2211yescharacteristic
31#14273654C2211yescharacteristic
32#14273654C2212yes—
33#14273653C2211yescharacteristic
34#14273646C2212yes—
35#14273645C2212yes—
36C8×Dic59C2211yescharacteristic
372−1+4×C59C2212yes—
38#137×C59—22no—
39#137×C59C2212yes—
40#135×C59—21no—
41#135×C59C2211yescharacteristic
42C2×Q16×C59—21no—
43C2×Q16×C59C2211yescharacteristic
44#131×C59C2211yescharacteristic
45#41×C59—22no—
46#41×C59—22no—
47#41×C59—42no—
48#41×C59C2311yescharacteristic
49#41×C59C2312yes—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59C2312yes—
53Q8×D59—22no—
54#11745687—22no—
55#11745687—42no—
56#11745685—22no—
57#11745685C2311yescharacteristic
58C2×Dic118—22no—
59C2×Dic118C2312yes—
60Q16×C59—22no—
61Q16×C59—21no—
62Q16×C59—21no—
63Q16×C59D411yescharacteristic
64SD16×C59—22no—
65SD16×C59D412yes—
66D8×C59D411yescharacteristic
67M16×C59—21no—
68M16×C59C2311yescharacteristic
69C2×C8×C59—21no—
70C2×C8×C59C2311yescharacteristic
71#11745672—22no—
72#11745672—22no—
73#11745672—22no—
74#11745672—22no—
75#11745672—22no—
76#11745672D411yescharacteristic
77#11745671—22no—
78#11745671D411yescharacteristic
79#11745670—22no—
80#11745670—22no—
81#11745670D411yescharacteristic
82#11745669D411yescharacteristic
83Dic59⋊C4—42no—
84C4×Dic59C2311yescharacteristic
85#11745664—22no—
86#11745664C2312yes—
87#11745664C2311yescharacteristic
88C2×C59⋊C8—22no—
89C2×C59⋊C8C2311yescharacteristic
90Dic236—21no—
91Dic236—21no—
92#11745659—21no—
93C8×D59—21no—
94Q8×C59—22no—
95Q8×C59—22no—
96Q8×C59—42no—
97Q8×C59C2×D411yescharacteristic
98Q8×C59C2×D412yes—
99D4×C59—22no—
100D4×C59—21no—
101D4×C59—42no—
102D4×C59C2×D411yescharacteristic
103D4×C59C2×D412yes—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59C2411yescharacteristic
110C22⋊D59—21no—
111C22⋊D59—42no—
112C2×Dic59—21no—
113C2×Dic59—22no—
114C2×Dic59—42no—
115D236C2×D411yescharacteristic
116C4×D59—21no—
117C4×D59—42no—
118Dic118—22no—
119Dic118—22no—
120Dic118—42no—
121Dic118C2×D411yescharacteristic
122C8×C59—21no—
123C8×C59C2×D411yescharacteristic
124C8×C59C2×D411yescharacteristic
125C59⋊C8—21no—
126C59⋊C8—22no—
127C59⋊C8C2×D411yescharacteristic
128C59⋊C8C2×D411yescharacteristic
129C22×C59—22no—
130C22×C59—21no—
131C22×C59H(5,𝔽2)11yescharacteristic
132C2×D59—21no—
133C4×C59—22no—
134C4×C59—21no—
135C4×C59—42no—
136C4×C59C22×D411yescharacteristic
137C4×C59C22×D411yescharacteristic
138Dic59—22no—
139Dic59—21no—
140Dic59—42no—
141#3293—591nomaximal, Sylow
142C2×C59—21no—
143C2×C59—42no—
144C2×C59—41no—
145C2×C59D4211yescharacteristic
146D59—41no—
147#578—592no—
148#578—591no—
149#578D5911yescharacteristic, complemented
150#494—591no—
151#473—592no—
152#473—591no—
153#473—591no—
154#456—592no—
155#456—592no—
156#443—591no—
157#443—591no—
158C59#329311yescharacteristic, complemented, Sylow
1592−1+4—592no—
1602−1+4C2×D5912yes—
161#137—22no—
162#137—592no—
163#137—592no—
164#137—592no—
165#137—1182no—
166#137—1182no—
167#137—1182no—
168#137C2×D5912yes—
169#135—21no—
170#135—591no—
171#135—1182no—
172#135—1181no—
173#135C2×D5911yescharacteristic
174C2×Q16—21no—
175C2×Q16—592no—
176C2×Q16—591no—
177C2×Q16—1182no—
178C2×Q16—1181no—
179C2×Q16C2×D5911yescharacteristic
180#131—592no—
181#131—591no—
182#131C2×D5911yescharacteristic
183C4⋊Q8—592no—
184#108—591no—
185#108—591no—
186#104—591no—
187#104—592no—
188#104—591no—
189#101—592no—
190#101—592no—
191C4×C8—591no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—591no—
196#41—1182no—
197#41—1182no—
198#41—2362no—
199#41C22×D5911yescharacteristic
200#41C22×D5912yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—592no—
204C2×Q8—1182no—
205C2×Q8—1182no—
206C2×Q8C22×D5912yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—591no—
211Q16—1182no—
212Q16—1182no—
213Q16—1182no—
214Q16—1182no—
215Q16—1182no—
216Q16—1181no—
217Q16—1181no—
218Q16C22⋊D5911yescharacteristic
219SD16—22no—
220SD16—591no—
221SD16—591no—
222SD16—1182no—
223SD16—1182no—
224SD16—1182no—
225SD16C22⋊D5912yes—
226D8—591no—
227D8C22⋊D5911yescharacteristic
228M16—21no—
229M16—592no—
230M16—591no—
231M16—1182no—
232M16—1181no—
233M16C22×D5911yescharacteristic
234C2×C8—21no—
235C2×C8—591no—
236C2×C8—1182no—
237C2×C8—1181no—
238C2×C8C22×D5911yescharacteristic
239C4⋊C4—2362no—
240C42—591no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—591no—
245Q8—1182no—
246Q8—1182no—
247Q8—2362no—
248Q8D4×D5911yescharacteristic
249Q8C2×C22⋊D5912yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—591no—
254D4—1181no—
255D4—2362no—
256D4D4×D5911yescharacteristic
257D4C2×C22⋊D5912yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—1181no—
264C2×C4—1181no—
265C2×C4—1182no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4C23×D5911yescharacteristic
269C8—21no—
270C8—591no—
271C8—591no—
272C8—1182no—
273C8—1181no—
274C8C2×C22⋊D5911yescharacteristic
275C8C2×C22⋊D5911yescharacteristic
276C22—21no—
277C22—22no—
278C22—1181no—
279C22#1427377211yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—1182no—
284C4—1181no—
285C4—2362no—
286C4C2×D4×D5911yescharacteristic
287C4C22×C22⋊D5911yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—2361no—
292C2D4×C22⋊D5911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#578C2
socleC2×C59D42
hypercenter#578D59
hypocenterC59#3293
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #16838217C2semidirect
2. #16838216C2semidirect
3. #16837780C2semidirect
4. #16837677C2semidirect
5. #16837675C2semidirect
6. #16837674C2semidirect
7. #16837585C2semidirect
8. #16837583C2semidirect
9. #16837519C2semidirect
10. #16837517C2semidirect
11. C59×#578C2semidirect
149. #578D59semidirect
158. C59#3293semidirect