Finite Group Databasebeta

← #28509348

Subgroups — #28509348

Order 15104.

284 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×Q32×D59C211yesmaximal, characteristic
2C2×#16837242C211yesmaximal, characteristic, complemented
3#21426184C211yesmaximal, characteristic
4#21422278C211yesmaximal, characteristic, complemented
5#21422277C211yesmaximal, characteristic, complemented
6(C2×D59)⋊C32C211yesmaximal, characteristic
7C2×C59×Q64C211yesmaximal, characteristic, complemented
8C2×Q16×D59—21no—
9C2×Dic472C2211yescharacteristic
10C2×#14273673—21no—
11Q32×D59—22no—
12Q32×D59—22no—
13C2×C16×D59C2211yescharacteristic
14#16837640—21no—
15#16837242—22no—
16C2×C59⋊C32C2211yescharacteristic
17#16837228C2211yescharacteristic
18#16837206—21no—
19C2×Q32×C59C2211yescharacteristic
20C2×Q32×C59C2211yescharacteristic
21C59×Q64—22no—
22C2×C32×C59C2211yescharacteristic
23C2×Q8×D59—41no—
24C2×Dic236—21no—
25C2×#11745672—41no—
26Q16×D59—24no—
27Q16×D59—42no—
28C2×C8×D59D411yescharacteristic
29#14273703—41no—
30Dic472—21no—
31Dic472D412yes—
32#14273673—24no—
33C2×C59⋊C16D411yescharacteristic
34C16×D59—21no—
35C16×D59—21no—
36#14273661—21no—
37#14273651—41no—
38C59⋊C32—21no—
39C2×Q16×C59—21no—
40C2×Q16×C59—21no—
41Q32×C59—21no—
42Q32×C59—21no—
43Q32×C59—21no—
44Q32×C59D412yes—
45C2×C16×C59C2311yescharacteristic
46C32×C59—21no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49Q8×D59—44no—
50Q8×D59—82no—
51C2×C4×D59—81no—
52C2×C4×D59D811yescharacteristic
53C2×Dic118—41no—
54C2×Dic118—81no—
55Q16×C59—22no—
56Q16×C59—41no—
57Q16×C59—41no—
58Q16×C59—41no—
59C2×C8×C59C2×D411yescharacteristic
60#11745672—44no—
61(C2×D59)⋊C4—81no—
62C4⋊Dic59—41no—
63Dic59⋊C4—81no—
64C2×C59⋊C8D811yescharacteristic
65Dic236—22no—
66Dic236—41no—
67C8×D59—21no—
68C8×D59—21no—
69C16×C59C2×D411yescharacteristic
70C16×C59C2×D411yescharacteristic
71C59⋊C16—21no—
72C22×D59D1611yescharacteristic
73Q8×C59—42no—
74Q8×C59—81no—
75Q8×C59—81no—
76Q8×C59—81no—
77C2×C4×C59—81no—
78C2×C4×C59—81no—
79C2×C4×C59C2×D811yescharacteristic
80C2×Dic59—81no—
81C2×Dic59—81no—
82C2×Dic59D1611yescharacteristic
83C4×D59—21no—
84C4×D59—21no—
85C4×D59—84no—
86Dic118—42no—
87Dic118—84no—
88Dic118—81no—
89C8×C59#12011yescharacteristic
90C8×C59AGL(1,ℤ/8)11yescharacteristic
91C59⋊C8—21no—
92#13665—591nomaximal, Sylow
93C22×C59C2×D1611yescharacteristic
94C2×D59—21no—
95C2×D59—21no—
96C2×D59Q6412yescomplemented
97C4×C59—82no—
98C4×C59—161no—
99C4×C59#44711yescharacteristic
100C4×C59#50911yescharacteristic
101Dic59—21no—
102Dic59—82no—
103Dic59—161no—
104C22×Q32—591no—
105C2×Q64—591no—
106C2×Q64D5911yescharacteristic, complemented
107#2216—591no—
108#1416—591no—
109#1416—591no—
110C22⋊C32—591no—
111C2×C59#218411yescharacteristic
112C2×C59C2×Q6411yescharacteristic
113C2×C59#226411yescharacteristic
114D59—22no—
115C22×Q16—1181no—
116C2×Q32—591no—
117C2×Q32—1181no—
118C2×Q32—1182no—
119C2×Q32—1182no—
120C2×Q32C2×D5911yescharacteristic
121C2×Q32C2×D5911yescharacteristic
122C22×C16—591no—
123#467—1181no—
124Q64—22no—
125Q64—1182no—
126C2×C32—591no—
127C2×C32C2×D5911yescharacteristic
128#366—591no—
129#358—1181no—
130C59#1366511yescharacteristic, complemented, Sylow
131C22×Q8—2361no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—1184no—
135C2×Q16—1181no—
136C2×Q16—2361no—
137C2×Q16—2362no—
138C22×C8—591no—
139C22⋊Q8—2361no—
140Q32—21no—
141Q32—21no—
142Q32—21no—
143Q32—592no—
144Q32—1184no—
145Q32—1181no—
146Q32C22⋊D5912yes—
147C2×C16—591no—
148C2×C16—1181no—
149C2×C16—1181no—
150C2×C16C22×D5911yescharacteristic
151#107—1181no—
152#103—2361no—
153C32—21no—
154C32—1181no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—4721no—
158C2×Q8—4722no—
159C2×Q8—2364no—
160C2×Q8—2361no—
161C22×C4—591no—
162C22×C4—4721no—
163Q16—22no—
164Q16—41no—
165Q16—41no—
166Q16—41no—
167Q16—1182no—
168Q16—2364no—
169Q16—2361no—
170C2×C8—591no—
171C2×C8—1181no—
172C2×C8—1181no—
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—4721no—
175C4⋊C4—2361no—
176C22⋊C4—4721no—
177C16—1181no—
178C16D4×D5911yescharacteristic
179C16C2×C22⋊D5911yescharacteristic
180C23—591no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184Q8—81no—
185Q8—4724no—
186Q8—4721no—
187Q8—2362no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—591no—
191C2×C4—1181no—
192C2×C4—1181no—
193C2×C4—4724no—
194C2×C4—4721no—
195C2×C4—4721no—
196C2×C4D8×D5911yescharacteristic
197C8—1181no—
198C8#1427369511yescharacteristic
199C8#1427375111yescharacteristic
200C22—592no—
201C22—1181no—
202C22—1181no—
203C22D16×D5911yescharacteristic
204C4—82no—
205C4—161no—
206C4—1181no—
207C4—9441no—
208C4—4722no—
209C4#1683753411yescharacteristic
210C4#1683783611yescharacteristic
211C2—1182no—
212C2#2142600611yescharacteristic
213C2Q64×D5911yescharacteristic
214C2#2142641511yescharacteristic
subgroupG/H
centerC22D16×D59
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC2×C59×Q64C2
socleC22×C59C2×D16
prime radicalC22×D59D16
hypercenterC2×Q64D59
hypocenterC59#13665
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×Q64

Complements

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

NH ≅ G/Ntype
2. C2×#16837242C2semidirect
4. #21422278C2semidirect
5. #21422277C2semidirect
7. C2×C59×Q64C2semidirect
96. C2×D59Q64semidirect
106. C2×Q64D59semidirect
130. C59#13665semidirect

Maximal subgroups

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

subgroupindex
C2×Q32×D592
C2×#168372422
#214261842
#214222782
#214222772
(C2×D59)⋊C322
C2×C59×Q642
#1366559