Finite Group Databasebeta

← #28509050

Subgroups — #28509050

Order 15104.

312 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 59 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#21434475C211yesmaximal, characteristic
2#356×D59C212yesmaximal
3#21422269C211yesmaximal, characteristic, complemented
4#21422268C212yesmaximal, complemented
5C59×#1413C211yesmaximal, characteristic, complemented
6#16837958C411yescharacteristic
7C4⋊Q8×D59C2211yescharacteristic
8#16837772C411yescharacteristic
9#16837771C411yescharacteristic
10#16837770C2211yescharacteristic
11#16837751—22no—
12#16837409—22no—
13#101×D59—22no—
14#16837354C411yescharacteristic
15#16837201C2212yes—
16#16837200—22no—
17C59×#491C2211yescharacteristic
18C59×#356C2212yes—
19C2×Q8×D59D412yes—
20C2×#11745659—21no—
21M16×D59—42no—
22#14273727C2×C411yescharacteristic
23#14273726—21no—
24#14273726—41no—
25#14273725D412yes—
26#14273719C2×C411yescharacteristic
27#14273705—41no—
28#14273692—41no—
29#14273685—21no—
30C4⋊C4×D59—41no—
31C42×D59C2×C411yescharacteristic
32#14273652—22no—
33#14273651—22no—
34#14273650—22no—
35#14273646—22no—
36#14273645—42no—
37#14273638C2×C411yescharacteristic
38C8×Dic59—21no—
39C4⋊Q8×C59C2311yescharacteristic
40#125×C59C2×C411yescharacteristic
41#103×C59—22no—
42#101×C59—22no—
43#97×C59C2×C411yescharacteristic
44C2×Q8×C59C2×D412yes—
45Q8×D59—42no—
46Q8×D59—42no—
47C2×C4×D59—21no—
48C2×C4×D59—22no—
49C2×C4×D59C22⋊C411yescharacteristic
50C2×Dic118—22no—
51C2×Dic118—41no—
52C2×Dic118C22⋊C412yes—
53M16×C59—42no—
54C2×C8×C59—21no—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—41no—
57C4⋊C4×C59—41no—
58C42×C59C22×C411yescharacteristic
59(C2×D59)⋊C4—41no—
60C4⋊Dic59—21no—
61C4⋊Dic59—41no—
62Dic59⋊C4—42no—
63Dic59⋊C4—41no—
64Dic59⋊C4—41no—
65C4×Dic59—21no—
66C4×Dic59—41no—
67C4×Dic59C22⋊C411yescharacteristic
68#11745664—42no—
69C2×C59⋊C8—21no—
70#11745659—42no—
71#11745659—82no—
72C8×D59—82no—
73C22×D59#9911yescharacteristic
74Q8×C59—42no—
75C2×C4×C59—22no—
76C2×C4×C59—21no—
77C2×C4×C59—41no—
78C2×C4×C59C2×C22⋊C411yescharacteristic
79C2×Dic59—21no—
80C2×Dic59—22no—
81C2×Dic59—41no—
82C2×Dic59#9911yescharacteristic
83C4×D59—22no—
84C4×D59—42no—
85C4×D59—42no—
86C4×D59—42no—
87Dic118—42no—
88Dic118—82no—
89Dic118—82no—
90C8×C59—41no—
91C8×C59—82no—
92C59⋊C8—41no—
93C59⋊C8—82no—
94#13599—591nomaximal, Sylow
95C22×C59C2×#9911yescharacteristic
96C2×D59—41no—
97C2×D59#35412yes—
98C4×C59—21no—
99C4×C59—41no—
100C4×C59—42no—
101C4×C59—81no—
102Dic59—22no—
103Dic59—21no—
104Dic59—42no—
105Dic59—81no—
106#3139—591no—
107C2×#356—592no—
108#1413—591no—
109#1413—592no—
110#1413D5911yescharacteristic, complemented
111C2×C59—21no—
112C2×C59C2×#35411yescharacteristic
113D59—22no—
114#533—591no—
115C2×C4⋊Q8—591no—
116#491—591no—
117#491—591no—
118#491—591no—
119#491C2×D5911yescharacteristic
120#487—1182no—
121#419—1182no—
122C2×#101—1182no—
123C4×M16—591no—
124#356—592no—
125#356—1182no—
126#356C2×D5912yes—
127C59#1359911yescharacteristic, complemented, Sylow
128C22×Q8—592no—
129C2×M16—1181no—
130C2×M16—2362no—
131C4⋊Q8—592no—
132C4⋊Q8—591no—
133C4⋊Q8—1181no—
134C4⋊Q8—2361no—
135C4⋊Q8C22×D5911yescharacteristic
136#125—591no—
137#125C4×D5911yescharacteristic
138C22⋊Q8—2361no—
139C4×Q8—2361no—
140#117—1181no—
141C2×C4⋊C4—2361no—
142C2×C42—591no—
143#103—22no—
144#103—1182no—
145#103—1182no—
146#103—1182no—
147#101—22no—
148#101—1182no—
149#101—2362no—
150#97—591no—
151#97C4×D5911yescharacteristic
152C4×C8—1181no—
153C2×Q8—592no—
154C2×Q8—1182no—
155C2×Q8—2361no—
156C2×Q8—2362no—
157C2×Q8—2362no—
158C2×Q8D4×D5912yes—
159C22×C4—591no—
160C22×C4—1181no—
161C22×C4—1182no—
162M16—42no—
163M16—4722no—
164M16—2362no—
165M16—2362no—
166C2×C8—21no—
167C2×C8—1181no—
168C2×C8—4722no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1181no—
173C4⋊C4—2362no—
174C4⋊C4—2361no—
175C4⋊C4—2361no—
176C4⋊C4—2361no—
177C22⋊C4—2361no—
178C42—591no—
179C42—1181no—
180C42—2361no—
181C42C2×C4×D5911yescharacteristic
182C23—591no—
183Q8—42no—
184Q8—4722no—
185Q8—4722no—
186Q8—2362no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—591no—
191C2×C4—1182no—
192C2×C4—1181no—
193C2×C4—1182no—
194C2×C4—2362no—
195C2×C4—2362no—
196C2×C4—2362no—
197C2×C4—2361no—
198C2×C4C22⋊C4×D5911yescharacteristic
199C8—41no—
200C8—82no—
201C8—4722no—
202C8—2361no—
203C22—592no—
204C22—2361no—
205C22#99×D5911yescharacteristic
206C4—21no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—1182no—
211C4—1181no—
212C4—4721no—
213C4—2362no—
214C2—21no—
215C2—1182no—
216C2#354×D5911yescharacteristic
subgroupG/H
centerC2#354×D59
commutatorC42×C59C22×C4
FrattiniC4⋊Q8C22×D59
FittingC59×#1413C2
socleC2×C59C2×#354
prime radicalC22×D59#99
cosocleC4⋊Q8×C59C23
hypercenter#1413D59
hypocenterC59#13599
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #21422269C2semidirect
4. #21422268C2semidirect
5. C59×#1413C2semidirect
110. #1413D59semidirect
127. C59#13599semidirect

Maximal subgroups

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

subgroupindex
#214344752
#356×D592
#214222692
#214222682
C59×#14132
#1359959