Finite Group Databasebeta

← #28500660

Subgroups — #28500660

Order 15104.

218 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 67 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#21425962C211yesmaximal, characteristic
2#21423009C211yesmaximal, characteristic
3#21422914C211yesmaximal, characteristic, complemented
4#21422065C211yesmaximal, characteristic, complemented
5#21422062C211yesmaximal, characteristic
6#21422018C211yesmaximal, characteristic, complemented
7C59×#1368C211yesmaximal, characteristic, complemented
8C2×#14273672C411yescharacteristic
9#16837526—21no—
10C4×Dic236C2211yescharacteristic
11C4×D472C2211yescharacteristic
12#16837479—21no—
13#16837228C411yescharacteristic
14#16837207C411yescharacteristic
15#16837204C411yescharacteristic
16#16837178C2211yescharacteristic
17#16837132—21no—
18#16837129—21no—
19#16837123C2211yescharacteristic
20C59×#346C2211yescharacteristic
21C59×#334C2211yescharacteristic
22C59×#334C2211yescharacteristic
23C2×Dic236C2×C411yescharacteristic
24C2×D472C2×C411yescharacteristic
25C2×#11745659—41no—
26C4×Dic118—21no—
27C4×D236—21no—
28#14273672—22no—
29C2×C59⋊C16—21no—
30#14273661C2×C411yescharacteristic
31#14273661C2×C411yescharacteristic
32Dic59⋊C8—41no—
33#14273652—21no—
34#14273649—21no—
35(C2×D59)⋊C8—41no—
36C4×C59⋊C8D411yescharacteristic
37C4×C59⋊C8D411yescharacteristic
38C8×Dic59—41no—
39C2×C16×C59C2×C411yescharacteristic
40C2×C16×C59C2×C411yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D236—21no—
45C2×C4×D59—41no—
46C2×Dic118—21no—
47C2×C8×C59—41no—
48C2×C8×C59—41no—
49C2×C8×C59C22×C411yescharacteristic
50C2×C8×C59#4111yescharacteristic
51C42×C59C2×D411yescharacteristic
52(C2×D59)⋊C4—41no—
53C4⋊Dic59—21no—
54C4⋊Dic59—21no—
55Dic59⋊C4—41no—
56C4×Dic59—41no—
57C2×C59⋊C8—21no—
58C2×C59⋊C8—21no—
59Dic236—21no—
60Dic236M1612yescomplemented
61D472—21no—
62D472—21no—
63#11745659—82no—
64C16×C59—21no—
65C16×C59—21no—
66C59⋊C16—22no—
67C22×D59—41no—
68C2×C4×C59C4×D411yescharacteristic
69C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
70C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
71C2×Dic59—41no—
72C2×Dic59—41no—
73C2×Dic59—41no—
74D236—41no—
75D236—41no—
76C4×D59—81no—
77C4×D59—81no—
78Dic118—22no—
79Dic118—41no—
80C8×C59—21no—
81C8×C59—81no—
82C8×C59—81no—
83C8×C59C2×M1611yescharacteristic
84C8×C59#13111yescharacteristic
85C59⋊C8—41no—
86C59⋊C8—41no—
87#12080—591nomaximal, Sylow
88C22×C59#44211yescharacteristic
89C2×D59—81no—
90C2×D59—81no—
91C2×D59—81no—
92C4×C59—21no—
93C4×C59—21no—
94C4×C59#43511yescharacteristic
95C4×C59#44311yescharacteristic
96Dic59—42no—
97Dic59—81no—
98Dic59—81no—
99#2175—591no—
100#1594—591no—
101#1579—591no—
102#1368D5911yescharacteristic, complemented
103#1334—591no—
104#1333—591no—
105#1313—591no—
106C2×C59#158611yescharacteristic
107C2×C59#218011yescharacteristic
108C2×C59AGL(1,ℤ/16)11yescharacteristic
109D59—161no—
110C2×SD32—591no—
111C8×Q8—1181no—
112C4×Q16—591no—
113C4×D8—591no—
114#436—1181no—
115#366—591no—
116#358—591no—
117#357—591no—
118#346—591no—
119#346C2×D5911yescharacteristic
120#334C2×D5911yescharacteristic
121#334C2×D5911yescharacteristic
122Q8⋊C8—1181no—
123D4⋊C8—1181no—
124#322—591no—
125C59#1208011yescharacteristic, complemented, Sylow
126C2×Q16—591no—
127C2×D8—591no—
128C2×M16—2361no—
129C4×Q8—1181no—
130C4×D4—1181no—
131SD32—1182no—
132C2×C16—1181no—
133C2×C16C4×D5911yescharacteristic
134C2×C16C4×D5911yescharacteristic
135#107—591no—
136#107—591no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—2361no—
140#103—1181no—
141#102—1181no—
142C22⋊C8—2361no—
143C4×C8—591no—
144C4×C8—591no—
145C4×C8—2361no—
146C4×C8C22×D5911yescharacteristic
147C2×Q8—1181no—
148C2×D4—1181no—
149C22×C4—2361no—
150Q16—592no—
151Q16—1181no—
152D8—1181no—
153D8—1181no—
154M16—4722no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—1181no—
158C2×C8—1181no—
159C2×C8C2×C4×D5911yescharacteristic
160C2×C8#1174568911yescharacteristic
161C4⋊C4—1181no—
162C4⋊C4—1181no—
163C4⋊C4—2361no—
164C22⋊C4—2361no—
165C42—2361no—
166C42D4×D5911yescharacteristic
167C16—21no—
168C16—21no—
169C16—1182no—
170C23—2361no—
171Q8—1182no—
172Q8—2361no—
173D4—2361no—
174D4—2361no—
175C2×C4—4721no—
176C2×C4—4721no—
177C2×C4—2361no—
178C2×C4—2361no—
179C2×C4—2361no—
180C2×C4#1427368911yescharacteristic
181C2×C4#1427375411yescharacteristic
182C2×C4#1427375411yescharacteristic
183C8—21no—
184C8—81no—
185C8—81no—
186C8—2361no—
187C8—2361no—
188C8M16×D5911yescharacteristic
189C8#1427373311yescharacteristic
190C22—4721no—
191C22—4721no—
192C22—4721no—
193C22#1683751311yescharacteristic
194C4—21no—
195C4—21no—
196C4—4721no—
197C4—4721no—
198C4—2362no—
199C4#1683747211yescharacteristic
200C4#1683751811yescharacteristic
201C2—9441no—
202C2#2142295911yescharacteristic
203C2#2142598611yescharacteristic
204C2#2142599311yescharacteristic
subgroupG/H
centerC22#16837513
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1368C2
socleC22×C59#442
prime radicalC2×D472C2×C4
cosocleC4×C8×C59C23
hypercenter#1368D59
hypocenterC59#12080
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1368

Complements

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

NH ≅ G/Ntype
3. #21422914C2semidirect
4. #21422065C2semidirect
6. #21422018C2semidirect
7. C59×#1368C2semidirect
60. Dic236M16semidirect
102. #1368D59semidirect
125. C59#12080semidirect

Maximal subgroups

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

subgroupindex
#214259622
#214230092
#214229142
#214220652
#214220622
#214220182
C59×#13682
#1208059