Finite Group Databasebeta

← #29594231

Subgroups — #29594231

Order 15616.

252 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 71 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#21448076C211yesmaximal, characteristic, complemented
2D4×#11758185C211yesmaximal, characteristic, complemented
3#21445174C211yesmaximal, characteristic
4#21443892C211yesmaximal, characteristic, complemented
5#21443868C211yesmaximal, characteristic, complemented
6#21443864C211yesmaximal, characteristic, complemented
7C61×#1669C211yesmaximal, characteristic, complemented
8C2×#14519720—21no—
9C2×#14519720—21no—
10C22×#11758185—21no—
11#16840139C2211yescharacteristic
12#16839957—21no—
13C4⋊#11758185C2211yescharacteristic
14C22⋊#11758185—21no—
15C4×#11758185C2211yescharacteristic
16#16839515C2211yescharacteristic
17#16839505—21no—
18C61×#494C2211yescharacteristic
19C61×#474—21no—
20C8×D4×C61C2211yescharacteristic
21C61×#335C2211yescharacteristic
22C61×#332—21no—
23C2×D488—21no—
24C2×D488—21no—
25C2×#11758199—41no—
26#14519774—21no—
27#14519720—42no—
28#14519720—42no—
29C2×#11758185—21no—
30C2×#11758185—42no—
31C2×#11758185D411yescharacteristic
32C2×#11758185D411yescharacteristic
33C4⋊#92418—21no—
34#14519697—41no—
35C2×Q16×C61—21no—
36C2×Q16×C61—21no—
37C22×C8×C61—21no—
38C4⋊Q8×C61—21no—
39C4×D4×C61D411yescharacteristic
40#107×C61—41no—
41C4⋊C8×C61—21no—
42C4⋊C8×C61D411yescharacteristic
43#102×C61—41no—
44C22⋊C8×C61—21no—
45C4×C8×C61C2311yescharacteristic
46C2×Q8×C61—41no—
47C2×D4×C61D811yescharacteristic
48C22×C4×C61—21no—
49C2×D244—41no—
50C2×D244—41no—
51C2×D244—41no—
52Q16×C61—41no—
53Q16×C61—41no—
54Q16×C61—81no—
55C2×C8×C61—21no—
56C2×C8×C61—41no—
57C2×C8×C61—41no—
58C2×C8×C61—41no—
59C2×C8×C61C2×D411yescharacteristic
60C2×C8×C61#4111yescharacteristic
61C4⋊C4×C61—41no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61D811yescharacteristic
64C22⋊C4×C61—21no—
65C42×C61C2×D411yescharacteristic
66#11758199—82no—
67C2×#92418—41no—
68D488—41no—
69D488—41no—
70D488—81no—
71#11758185—21no—
72#11758185—41no—
73#11758185D812yescomplemented
74C23×C61—21no—
75C22×D61—81no—
76Q8×C61—81no—
77Q8×C61—81no—
78D4×C61—21no—
79D4×C61SD3212yescomplemented
80C2×C4×C61—21no—
81C2×C4×C61—41no—
82C2×C4×C61—41no—
83C2×C4×C61—81no—
84C2×C4×C61#12111yescharacteristic
85C2×C4×C61C2×D811yescharacteristic
86C2×C4×C61#13511yescharacteristic
87D244—81no—
88D244—81no—
89D244—81no—
90D244—81no—
91D244—161no—
92C8×C61—21no—
93C8×C61—41no—
94C8×C61—81no—
95C8×C61C2×D811yescharacteristic
96C8×C61AGL(1,ℤ/8)11yescharacteristic
97#92418—81no—
98#12432—611nomaximal, Sylow
99C22×C61—22no—
100C22×C61—41no—
101C22×C61—41no—
102C22×C61#46611yescharacteristic
103C2×D61—161no—
104C2×D61—161no—
105C2×D61—161no—
106C4×C61—21no—
107C4×C61—41no—
108C4×C61—161no—
109C4×C61#45911yescharacteristic
110C4×C61#47211yescharacteristic
111C4×C61C2×SD3211yescharacteristic
112C4×C61#50811yescharacteristic
113#2248—611no—
114D4×C16—611no—
115#1669D6111yescharacteristic, complemented
116#1667—611no—
117#1372—611no—
118#1362—611no—
119#1361—611no—
120C2×C61—42no—
121C2×C61#166711yescharacteristic
122C2×C61#221711yescharacteristic
123C2×C61#222311yescharacteristic
124D61—321no—
125C2×SD32—1221no—
126C2×SD32—1221no—
127C22×C16—1221no—
128#494C2×D6111yescharacteristic
129#493—611no—
130#474—21no—
131#459—1221no—
132C8×D4C2×D6111yescharacteristic
133C4⋊C16—611no—
134C22⋊C16—1221no—
135C4×C16—611no—
136#335—611no—
137#335C2×D6111yescharacteristic
138#332—21no—
139#331—1221no—
140C61#1243211yescharacteristic, complemented, Sylow
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—1221no—
144C2×D8—1221no—
145C2×D8—2441no—
146C22×C8—21no—
147C4⋊Q8—21no—
148#127—1221no—
149C4×D4C22⋊D6111yescharacteristic
150SD32—2442no—
151SD32—2442no—
152C2×C16—611no—
153C2×C16—611no—
154C2×C16—1221no—
155C2×C16—2442no—
156#107—41no—
157C4⋊C8—21no—
158C4⋊C8—1221no—
159C4⋊C8C22⋊D6111yescharacteristic
160#102—41no—
161#102—2441no—
162C22⋊C8—21no—
163C4×C8C22×D6111yescharacteristic
164C2×Q8—41no—
165C2×D4—2441no—
166C2×D4—2441no—
167C2×D4—2441no—
168C2×D4#1175819911yescharacteristic
169C22×C4—21no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—4881no—
174D8—4882no—
175D8—2441no—
176D8—2441no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—2441no—
182C2×C8D4×D6111yescharacteristic
183C2×C8#1175822411yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4#1175819911yescharacteristic
187C22⋊C4—21no—
188C42C2×C22⋊D6111yescharacteristic
189C16—612no—
190C16—1221no—
191C16—2441no—
192C23—21no—
193C23—4881no—
194Q8—81no—
195Q8—81no—
196D4—21no—
197D4—9761no—
198D4—4881no—
199D4—4881no—
200D4—4881no—
201D4—4881no—
202D4#1451972012yescomplemented
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4#1451974911yescharacteristic
208C2×C4C2×#1175819911yescharacteristic
209C2×C4#1451979711yescharacteristic
210C8—21no—
211C8—41no—
212C8—81no—
213C8—4881no—
214C8D8×D6111yescharacteristic
215C8#1451980111yescharacteristic
216C22—22no—
217C22—41no—
218C22—41no—
219C22—9761no—
220C22—9761no—
221C22—9761no—
222C22#1683999611yescharacteristic
223C4—21no—
224C4—41no—
225C4—161no—
226C4#1683995511yescharacteristic
227C4#1684003011yescharacteristic
228C4C2×#1451972011yescharacteristic
229C4#1684019211yescharacteristic
230C2—42no—
231C2—19521no—
232C2#2144517211yescharacteristic
233C2#2144791711yescharacteristic
234C2#2144795111yescharacteristic
subgroupG/H
centerC22#16839996
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1669C2
socleC22×C61#466
prime radical#21445174C2
hypercenter#1669D61
hypocenterC61#12432
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1669

Complements

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

NH ≅ G/Ntype
1. #21448076C2semidirect
2. D4×#11758185C2semidirect
4. #21443892C2semidirect
5. #21443868C2semidirect
6. #21443864C2semidirect
7. C61×#1669C2semidirect
73. #11758185D8semidirect
79. D4×C61SD32semidirect
115. #1669D61semidirect
140. C61#12432semidirect
202. D4#14519720semidirect

Maximal subgroups

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

subgroupindex
#214480762
D4×#117581852
#214451742
#214438922
#214438682
#214438642
C61×#16692
#1243261