Finite Group Databasebeta

← #28502386

Subgroups — #28502386

Order 15104.

312 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 75 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#21426321C211yesmaximal, characteristic, complemented
2#21426131C211yesmaximal, characteristic
3#21423456C211yesmaximal, characteristic, complemented
4#21422154C211yesmaximal, characteristic, complemented
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422057C211yesmaximal, characteristic, complemented
7C59×#3380C211yesmaximal, characteristic, complemented
8C2×#14273668—21no—
9#16837779C2211yescharacteristic
10#16837602—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837202—22no—
13#16837202—21no—
14C4×C59⋊C16C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837145—21no—
17C59×#554—21no—
18C59×#500C2211yescharacteristic
19C59×#499—21no—
20C59×#478—22no—
21C59×#476—21no—
22C8×Q8×C59C2211yescharacteristic
23C4×D8×C59—21no—
24C4×D8×C59C2211yescharacteristic
25C2×#11745671—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273724—21no—
29#14273668—42no—
30C2×C59⋊C16—21no—
31C2×C59⋊C16—21no—
32C4⋊(C59⋊C8)—21no—
33#14273647—41no—
34C2×D8×C59D411yescharacteristic
35C4⋊Q8×C59—21no—
36#125×C59—22no—
37C22⋊Q8×C59—42no—
38C22⋊Q8×C59—41no—
39C4×Q8×C59D411yescharacteristic
40C4×D4×C59—22no—
41C4×D4×C59—21no—
42C2×C4⋊C4×C59—41no—
43#107×C59—21no—
44#107×C59D411yescharacteristic
45#107×C59D412yes—
46#106×C59—22no—
47C4⋊C8×C59—21no—
48C4⋊C8×C59D411yescharacteristic
49#102×C59—22no—
50#102×C59—21no—
51C4×C8×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—41no—
54C2×Q8×C59D811yescharacteristic
55C2×D4×C59—21no—
56C22×C4×C59—42no—
57C22×C4×C59—41no—
58C2×D236—41no—
59C2×D236—41no—
60C2×D236—41no—
61D8×C59—21no—
62D8×C59SD1612yescomplemented
63C2×C8×C59—21no—
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—21no—
67C4⋊C4×C59—21no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—41no—
72C4⋊C4×C59D811yescharacteristic
73C22⋊C4×C59—42no—
74C22⋊C4×C59—41no—
75C42×C59—21no—
76C42×C59C2×D411yescharacteristic
77#11745671—82no—
78C2×C59⋊C8—41no—
79D472—42no—
80D472—41no—
81D472—41no—
82C59⋊C16—41no—
83C59⋊C16—41no—
84C23×C59—41no—
85C22×D59—81no—
86Q8×C59—21no—
87Q8×C59—21no—
88Q8×C59—161no—
89D4×C59—22no—
90D4×C59—41no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—81no—
96C2×C4×C59—81no—
97C2×C4×C59—162no—
98C2×C4×C59#12011yescharacteristic
99C2×C4×C59C2×D811yescharacteristic
100C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
101D236—81no—
102D236—81no—
103D236—81no—
104D236—81no—
105D236—161no—
106C8×C59—21no—
107C8×C59—41no—
108C8×C59C2×SD1611yescharacteristic
109C8×C59#13711yescharacteristic
110C59⋊C8—81no—
111#12355—591nomaximal, Sylow
112C22×C59—42no—
113C22×C59—81no—
114C22×C59—81no—
115C22×C59#44711yescharacteristic
116C2×D59—161no—
117C2×D59—161no—
118C2×D59—161no—
119C4×C59—21no—
120C4×C59—41no—
121C4×C59—41no—
122C4×C59—81no—
123C4×C59—82no—
124C4×C59#44811yescharacteristic
125C4×C59#45411yescharacteristic
126C4×C59#50811yescharacteristic
127C4×C59#50911yescharacteristic
128#3380D5911yescharacteristic, complemented
129#2240—591no—
130#2206—591no—
131#1668—591no—
132#1365—591no—
133Q8⋊C16—591no—
134#1331—591no—
135C2×C59—82no—
136C2×C59#162711yescharacteristic
137C2×C59#218611yescharacteristic
138C2×C59#219211yescharacteristic
139D59—321no—
140#554—21no—
141C2×D16—1181no—
142#500C2×D5911yescharacteristic
143#499—21no—
144#493—591no—
145#478—22no—
146#476—21no—
147#460—1181no—
148C8×Q8C2×D5911yescharacteristic
149C4×D8—21no—
150C4×D8C2×D5911yescharacteristic
151C4⋊C16—591no—
152#357—1181no—
153#357—1182no—
154C4×C16—591no—
155#335—591no—
156#331—1181no—
157C59#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2361no—
159C2×D8—1181no—
160C2×D8—1181no—
161C2×D8C22⋊D5911yescharacteristic
162C4⋊Q8—21no—
163#127—1181no—
164#125—22no—
165C22⋊Q8—42no—
166C22⋊Q8—41no—
167C4×Q8C22⋊D5911yescharacteristic
168C4×D4—22no—
169C4×D4—21no—
170C2×C4⋊C4—41no—
171D16—2362no—
172C2×C16—1181no—
173C2×C16—1181no—
174#107—21no—
175#107C22⋊D5911yescharacteristic
176#107C22⋊D5912yes—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—1181no—
180C4⋊C8C22⋊D5911yescharacteristic
181#102—22no—
182#102—21no—
183#102—2361no—
184C4×C8—21no—
185C4×C8C22×D5911yescharacteristic
186C2×Q8—41no—
187C2×Q8#1174566911yescharacteristic
188C2×D4—21no—
189C2×D4—2361no—
190C2×D4—2361no—
191C2×D4—2361no—
192C22×C4—42no—
193C22×C4—41no—
194SD16—4722no—
195D8—21no—
196D8—2362no—
197D8—2361no—
198D8—2361no—
199D8#1174567112yescomplemented
200C2×C8—21no—
201C2×C8—2361no—
202C2×C8C2×C22⋊D5911yescharacteristic
203C2×C8C2×C22⋊D5911yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4#1174566911yescharacteristic
211C22⋊C4—42no—
212C22⋊C4—41no—
213C42—21no—
214C42C2×C22⋊D5911yescharacteristic
215C16—2361no—
216C16—2361no—
217C23—41no—
218C23—4721no—
219Q8—21no—
220Q8—21no—
221Q8—161no—
222D4—22no—
223D4—41no—
224D4—9441no—
225D4—4721no—
226D4—4721no—
227D4—4721no—
228D4—4721no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—162no—
236C2×C4#1427369611yescharacteristic
237C2×C4C2×#1174566911yescharacteristic
238C2×C4#1427375311yescharacteristic
239C8—21no—
240C8—41no—
241C8—4721no—
242C8C2×#1174567111yescharacteristic
243C8#1427375811yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—9441no—
248C22—9441no—
249C22—9441no—
250C22#1683753611yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4#1683754111yescharacteristic
257C4#1683757511yescharacteristic
258C4#1683783211yescharacteristic
259C4#1683783711yescharacteristic
260C2—82no—
261C2—18881no—
262C2#2142319911yescharacteristic
263C2#2142601911yescharacteristic
264C2#2142605311yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3380C2
socleC22×C59#447
prime radical#21426131C2
hypercenter#3380D59
hypocenterC59#12355
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #21426321C2semidirect
3. #21423456C2semidirect
4. #21422154C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
6. #21422057C2semidirect
7. C59×#3380C2semidirect
62. D8×C59SD16semidirect
128. #3380D59semidirect
157. C59#12355semidirect
199. D8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214263212
#214261312
#214234562
#214221542
Q8⋊(C59⋊C16)2
#214220572
C59×#33802
#1235559