Finite Group Databasebeta

← #29594435

Subgroups — #29594435

Order 15616.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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#21448042C211yesmaximal, characteristic, complemented
2D4×#11758185C211yesmaximal, characteristic, complemented
3#21445218C211yesmaximal, characteristic, complemented
4#21443892C211yesmaximal, characteristic, complemented
5#21443878C211yesmaximal, characteristic, complemented
6#21443865C211yesmaximal, characteristic, complemented
7C61×#1667C211yesmaximal, characteristic
8C22×#11758185—21no—
9#16840167C2211yescharacteristic
10#16840052—21no—
11C4⋊#11758185C2211yescharacteristic
12#16839563—21no—
13#16839563—21no—
14C22⋊#11758185—21no—
15C4×#11758185C2211yescharacteristic
16#16839515C2211yescharacteristic
17#16839510—21no—
18C61×#493C2211yescharacteristic
19C61×#459—21no—
20C8×D4×C61C2211yescharacteristic
21C61×#335C2211yescharacteristic
22C61×#331—21no—
23#14519777—21no—
24C2×#11758185—21no—
25C2×#11758185—42no—
26C2×#11758185D411yescharacteristic
27C2×#11758185D411yescharacteristic
28#14519711—21no—
29#14519711—21no—
30#14519708—41no—
31C4⋊#92418—21no—
32#14519698—41no—
33C2×D8×C61—21no—
34C2×D8×C61—21no—
35C2×D8×C61—41no—
36C22×C8×C61—21no—
37#127×C61—21no—
38C4×D4×C61D411yescharacteristic
39C4⋊C8×C61—21no—
40C4⋊C8×C61D411yescharacteristic
41#102×C61—41no—
42C22⋊C8×C61—21no—
43C4×C8×C61C2311yescharacteristic
44C2×D4×C61—41no—
45C2×D4×C61—41no—
46C2×D4×C61—41no—
47C2×D4×C61D811yescharacteristic
48C22×C4×C61—21no—
49C2×Dic122—41no—
50D8×C61—41no—
51D8×C61—41no—
52D8×C61—81no—
53D8×C61—82no—
54C2×C8×C61—21no—
55C2×C8×C61—41no—
56C2×C8×C61—41no—
57C2×C8×C61—41no—
58C2×C8×C61C2×D411yescharacteristic
59C2×C8×C61#4111yescharacteristic
60C4⋊C4×C61D811yescharacteristic
61C22⋊C4×C61—21no—
62C42×C61C2×D411yescharacteristic
63C4⋊Dic61—41no—
64C4⋊Dic61—41no—
65C2×#92418—41no—
66#11758185—21no—
67#11758185—21no—
68#11758185—41no—
69C23×C61—21no—
70C23×C61—81no—
71D4×C61—21no—
72D4×C61—81no—
73D4×C61—81no—
74D4×C61—81no—
75D4×C61—81no—
76D4×C61—161no—
77D4×C61SD3212yes—
78C2×C4×C61—21no—
79C2×C4×C61—41no—
80C2×C4×C61—41no—
81C2×C4×C61#12111yescharacteristic
82C2×C4×C61C2×D811yescharacteristic
83C2×C4×C61#13511yescharacteristic
84C2×Dic61—81no—
85Dic122—161no—
86C8×C61—21no—
87C8×C61—41no—
88C8×C61—81no—
89C8×C61#13511yescharacteristic
90C8×C61AGL(1,ℤ/8)11yescharacteristic
91#92418—81no—
92#12468—611nomaximal, Sylow
93C22×C61—22no—
94C22×C61—41no—
95C22×C61—41no—
96C22×C61—161no—
97C22×C61—161no—
98C22×C61—161no—
99C22×C61#46611yescharacteristic
100C4×C61—21no—
101C4×C61—41no—
102C4×C61#46311yescharacteristic
103C4×C61#47111yescharacteristic
104C4×C61C2×SD3211yescharacteristic
105C4×C61#50811yescharacteristic
106Dic61—161no—
107#2240—611no—
108D4×C16—611no—
109#1675—611no—
110#1667D6111yescharacteristic
111#1372—611no—
112#1365—611no—
113#1361—611no—
114C2×C61—42no—
115C2×C61—321no—
116C2×C61#168111yescharacteristic
117C2×C61#221711yescharacteristic
118C2×C61#222111yescharacteristic
119C22×C16—1221no—
120#500—611no—
121#493C2×D6111yescharacteristic
122#476—1221no—
123#459—21no—
124C8×D4C2×D6111yescharacteristic
125C4⋊C16—611no—
126#357—1221no—
127#357—1221no—
128C22⋊C16—1221no—
129C4×C16—611no—
130#335—611no—
131#335C2×D6111yescharacteristic
132#333—1221no—
133#331—21no—
134C61#1246811yescharacteristic, complemented, Sylow
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—41no—
138C22×C8—21no—
139C4⋊Q8—1221no—
140#127—21no—
141C4×D4C22⋊D6111yescharacteristic
142C2×C16—611no—
143C2×C16—611no—
144C2×C16—1221no—
145C2×C16—2442no—
146#107—1221no—
147#107—1221no—
148#106—2441no—
149C4⋊C8—21no—
150C4⋊C8—1221no—
151C4⋊C8C22⋊D6111yescharacteristic
152#102—41no—
153#102—2441no—
154C22⋊C8—21no—
155C4×C8C22×D6111yescharacteristic
156C2×Q8—2441no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4#1175819911yescharacteristic
161C22×C4—21no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—82no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—2441no—
171C2×C8D4×D6111yescharacteristic
172C2×C8#1175822411yescharacteristic
173C4⋊C4—2441no—
174C4⋊C4—2441no—
175C4⋊C4#1175819911yescharacteristic
176C22⋊C4—21no—
177C42C2×C22⋊D6111yescharacteristic
178C16—1221no—
179C16—1221no—
180C16—2441no—
181C23—21no—
182C23—81no—
183Q8—9761no—
184D4—21no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—161no—
190D4#1451972112yes—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—4881no—
195C2×C4#1451974911yescharacteristic
196C2×C4C2×#1175819911yescharacteristic
197C2×C4#1451979711yescharacteristic
198C8—21no—
199C8—41no—
200C8—81no—
201C8—4881no—
202C8#1451979611yescharacteristic
203C8#1451980111yescharacteristic
204C22—22no—
205C22—41no—
206C22—41no—
207C22—161no—
208C22—161no—
209C22—161no—
210C22#1683999611yescharacteristic
211C4—21no—
212C4—41no—
213C4—9761no—
214C4#1683997511yescharacteristic
215C4#1684002311yescharacteristic
216C4C2×#1451972111yescharacteristic
217C4#1684019211yescharacteristic
218C2—42no—
219C2—321no—
220C2#2144525411yescharacteristic
221C2#2144791311yescharacteristic
222C2#2144793711yescharacteristic
subgroupG/H
centerC22#16839996
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
Fitting, prime radicalC61×#1667C2
socleC22×C61#466
hypercenter#1667D61
hypocenterC61#12468
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1667

Complements

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

NH ≅ G/Ntype
1. #21448042C2semidirect
2. D4×#11758185C2semidirect
3. #21445218C2semidirect
4. #21443892C2semidirect
5. #21443878C2semidirect
6. #21443865C2semidirect
134. C61#12468semidirect

Maximal subgroups

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

subgroupindex
#214480422
D4×#117581852
#214452182
#214438922
#214438782
#214438652
C61×#16672
#1246861