Finite Group Databasebeta

← #28504969

Subgroups — #28504969

Order 15104.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#21432736C211yesmaximal, characteristic
2#21424201C212yesmaximal
3#21422226C212yesmaximal
4#21422217C211yesmaximal, characteristic
5C59×#1392C211yesmaximal, characteristic
6#41×Dic59C2211yescharacteristic
7#16837460—22no—
8#16837452—22no—
9#16837430C411yescharacteristic, complemented
10#16837428C411yescharacteristic, complemented
11#16837376C2212yes—
12#16837369C2211yescharacteristic
13M16×Dic59C411yescharacteristic, complemented
14C2×C8×Dic59C411yescharacteristic, complemented
15#16837163—22no—
16#16837159—22no—
17C59×#413C2212yes—
18C59×#408C2211yescharacteristic
19C2×#11745664C2×C411yescharacteristic
20C22×C59⋊C8C2×C411yescharacteristic
21Q8×Dic59—21no—
22D4×Dic59—22no—
23D4×Dic59—21no—
24#14273686—22no—
25#14273686C2×C411yescharacteristic
26#14273686D412yes—
27C2×C4×Dic59C2×C411yescharacteristic
28C2×C4×Dic59D412yes—
29Dic59⋊C8—22no—
30Dic59⋊C8—22no—
31#14273642—22no—
32#14273642—22no—
33C22⋊(C59⋊C8)—22no—
34#14273637—21no—
35C8×Dic59—21no—
36C8×Dic59—21no—
37C8×Dic59—21no—
38C2×#41×C59C2311yescharacteristic
39C2×M16×C59—22no—
40C2×M16×C59C2×C411yescharacteristic
41C22×C8×C59C2×C411yescharacteristic
42#101×C59—22no—
43#100×C59—22no—
44C22⋊C8×C59—22no—
45#41×C59—41no—
46#41×C59—41no—
47C2×Q8×C59#4111yescharacteristic
48C2×D4×C59#4111yescharacteristic
49C2×D4×C59#4112yes—
50C22×C4×C59C22×C411yescharacteristic
51C22×C4×C59C2×D412yes—
52C22×Dic59—21no—
53C22×Dic59—22no—
54M16×C59—41no—
55M16×C59—42no—
56M16×C59—42no—
57C2×C8×C59—21no—
58C2×C8×C59—21no—
59C2×C8×C59—41no—
60C2×C8×C59—42no—
61C22⋊Dic59—21no—
62C22⋊Dic59—22no—
63C22⋊Dic59—42no—
64C4⋊Dic59—21no—
65C4⋊Dic59—22no—
66C4⋊Dic59—42no—
67C4×Dic59—21no—
68C4×Dic59—22no—
69C4×Dic59—42no—
70C4×Dic59C22⋊C411yescharacteristic
71C4×Dic59C22⋊C411yescharacteristic
72C4×Dic59C22⋊C412yes—
73#11745664—41no—
74C2×C59⋊C8—21no—
75C2×C59⋊C8—21no—
76C2×C59⋊C8—41no—
77C23×C59#11711yescharacteristic
78C23×C59#12312yes—
79Q8×C59—41no—
80D4×C59—42no—
81D4×C59—41no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59C2×C22⋊C411yescharacteristic
87C2×C4×C59#11711yescharacteristic
88C2×C4×C59#12312yes—
89C2×Dic59—21no—
90C2×Dic59—21no—
91C2×Dic59—22no—
92C2×Dic59—42no—
93C2×Dic59—44no—
94C2×Dic59—82no—
95C8×C59—41no—
96C8×C59—41no—
97C8×C59—82no—
98C59⋊C8—41no—
99C59⋊C8—41no—
100#12864—591nomaximal, Sylow
101C22×C59—22no—
102C22×C59—21no—
103C22×C59—41no—
104C22×C59—42no—
105C22×C59#38111yescharacteristic
106C4×C59—22no—
107C4×C59—21no—
108C4×C59#41011yescharacteristic
109C4×C59#41011yescharacteristic
110Dic59—22no—
111Dic59—41no—
112Dic59—42no—
113Dic59—162no—
114#2922—591no—
115#1827—592no—
116#1395—592no—
117#1392D5911yescharacteristic
118#1391—591no—
119C2×C59—21no—
120C2×C59—42no—
121C2×C59—41no—
122C2×C59#182011yescharacteristic
123C4×#41—591no—
124#433—1182no—
125#431—1182no—
126#424—591no—
127#424—591no—
128#413C2×D5912yes—
129#410—592no—
130#408—591no—
131#408C2×D5911yescharacteristic
132C4×M16—591no—
133C2×C4×C8—591no—
134#339—1182no—
135H(3,ℤ/4)—1182no—
136C59#1286411yescharacteristic, complemented, Sylow
137C2×#41C22×D5911yescharacteristic
138C2×M16—22no—
139C2×M16—591no—
140C2×M16C4×D5911yescharacteristic
141C22×C8—591no—
142C22×C8C4×D5911yescharacteristic
143C4×Q8—1181no—
144C4×D4—1182no—
145C4×D4—1181no—
146#117—591no—
147#117—592no—
148#117—1182no—
149C2×C42—591no—
150C2×C42—592no—
151C4⋊C8—1182no—
152C4⋊C8—1182no—
153#101—22no—
154#100—22no—
155#99—1182no—
156#99—1182no—
157C22⋊C8—22no—
158C22⋊C8—1182no—
159#97—1181no—
160C4×C8—1181no—
161C4×C8—1181no—
162C4×C8—1181no—
163#41—41no—
164#41—41no—
165C2×Q8#1174568711yescharacteristic
166C2×D4#1174568512yes—
167C2×D4#1174568911yescharacteristic
168C22×C4—1181no—
169C22×C4—1182no—
170C22×C4C2×C4×D5911yescharacteristic
171C22×C4D4×D5912yes—
172M16—41no—
173M16—42no—
174M16—42no—
175M16—2361no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—1181no—
181C2×C8—1181no—
182C2×C8—2361no—
183C4⋊C4—1181no—
184C4⋊C4—1182no—
185C4⋊C4—2362no—
186C22⋊C4—1181no—
187C22⋊C4—1182no—
188C22⋊C4—2362no—
189C42—591no—
190C42—591no—
191C42—592no—
192C42—1181no—
193C42—1182no—
194C42—2362no—
195C23#1427368411yescharacteristic
196C23#1427371212yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—1181no—
205C2×C4—1181no—
206C2×C4—1182no—
207C2×C4—4722no—
208C2×C4—2362no—
209C2×C4—2364no—
210C2×C4C22⋊C4×D5911yescharacteristic
211C2×C4#1427368511yescharacteristic
212C2×C4#1427370912yes—
213C8—41no—
214C8—41no—
215C8—82no—
216C8—2361no—
217C8—2361no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1683726411yescharacteristic
223C4—22no—
224C4—21no—
225C4—1182no—
226C4—9442no—
227C4—2361no—
228C4—2362no—
229C4#1683737411yescharacteristic
230C4#1683737411yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2142417211yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#1392C2
socleC2×C59#1820
prime radical, cosocleC2×#41×C59C23
hypercenter#1392D59
hypocenterC59#12864
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
9. #16837430C4semidirect
10. #16837428C4semidirect
13. M16×Dic59C4semidirect
14. C2×C8×Dic59C4semidirect
136. C59#12864semidirect

Maximal subgroups

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

subgroupindex
#214327362
#214242012
#214222262
#214222172
C59×#13922
#1286459