Finite Group Databasebeta

← #29677275

Subgroups — #29677275

Order 15616.

578 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 131 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#21458638C214yesmaximal, complemented
2#21445356C214yesmaximal, complemented
3#21444957C212yesmaximal, complemented
4#21444876C212yesmaximal
5#21444717C212yesmaximal, complemented
6C61×#1709C211yesmaximal, characteristic
7#16840498C2212yes—
8#16840492C2212yes—
9#16840120—24no—
10#16840054C2214yes—
11#16840040—24no—
12#16839983—24no—
13#16839959C2214yes—
14#16839892C2214yes—
15#16839859—24no—
16C4×Dic244C2212yes—
17#16839515C2212yes—
18#16839511C2212yes—
19#16839506C2212yes—
20#16839494C2214yes—
21C61×#494C2211yescharacteristic
22C61×#493C2211yescharacteristic
23C61×#486C2214yes—
24C82×C61C2211yescharacteristic
25C2×Dic244D412yes—
26C2×#11758200—24no—
27#14519776—24no—
28#14519775—42no—
29#14519764—24no—
30#14519758—42no—
31C4×Dic122C2314yes—
32#14519742—24no—
33D4×Dic61—24no—
34#14519711D412yes—
35#14519710—24no—
36C4⋊#92418C2314yes—
37#14519706—24no—
38#14519702D414yes—
39#14519700—24no—
40#14519698—24no—
41#14519687—24no—
42C2×Q16×C61—41no—
43C2×D8×C61—41no—
44C4⋊Q8×C61C2312yes—
45#127×C61C2312yes—
46#102×C61—24no—
47#102×C61—24no—
48C4×C8×C61C2311yescharacteristic
49C4×C8×C61C2312yes—
50C2×Q8×C61—22no—
51C2×D4×C61—24no—
52C2×D4×C61—22no—
53C22×Dic61—42no—
54C2×Dic122—42no—
55C2×Dic122C2×D414yes—
56Q16×C61—82no—
57D8×C61—82no—
58C2×C8×C61—21no—
59C2×C8×C61C2×D412yes—
60C2×C8×C61C2×D412yes—
61C4⋊C4×C61—24no—
62C42×C61C2411yescharacteristic
63C22⋊Dic61—44no—
64C22⋊Dic61—42no—
65#11758200—48no—
66C4⋊Dic61C2×D414yes—
67#11758196—24no—
68#11758196—44no—
69C4×Dic61—24no—
70C4×Dic61—42no—
71C2×#92418—24no—
72Dic244—42no—
73C23×C61—42no—
74Q8×C61—82no—
75D4×C61—44no—
76D4×C61—44no—
77D4×C61—82no—
78C2×C4×C61—42no—
79C2×C4×C61C22×D412yes—
80C2×C4×C61H(5,𝔽2)11yescharacteristic
81C2×Dic61—24no—
82C2×Dic61—24no—
83C2×Dic61—162no—
84Dic122—24no—
85Dic122—24no—
86Dic122—162no—
87C8×C61—22no—
88C8×C61—22no—
89C8×C61—22no—
90#92418—44no—
91#20319—611nomaximal, Sylow
92C22×C61—84no—
93C22×C61—82no—
94C22×C61D4211yescharacteristic
95C4×C61—21no—
96C4×C61—82no—
97C4×C61#57512yes—
98C4×C61#57812yes—
99Dic61—44no—
100Dic61—44no—
101#3375—614no—
102#1709D6111yescharacteristic
103#1698—614no—
104#1633—612no—
105#1620—612no—
106#1591—612no—
107C2×C61—162no—
108C2×C61#327712yes—
109C2×C61#329211yescharacteristic
110#559—612no—
111Q82—612no—
112#494C2×D6111yescharacteristic
113#493C2×D6111yescharacteristic
114#489—1224no—
115#486C2×D6114yes—
116#477—614no—
117#474—1224no—
118#464—1224no—
119#460—614no—
120#446—614no—
121#440—1224no—
122C4×Q16—612no—
123#335—612no—
124#333—612no—
125#331—612no—
126Q8⋊C8—614no—
127C82C2×D6111yescharacteristic
128C61#2031911yescharacteristic, complemented, Sylow
129C2×Q16—41no—
130C2×Q16—612no—
131C2×SD16—1224no—
132C2×D8—41no—
133C4⋊Q8—1224no—
134C4⋊Q8—2442no—
135C4⋊Q8C22×D6112yes—
136#127C22×D6112yes—
137#124—1224no—
138#123—2442no—
139C4×Q8—614no—
140C4×Q8—1224no—
141C4×D4—1224no—
142#107—612no—
143#107—1224no—
144C4⋊C8—614no—
145C4⋊C8—1224no—
146#103—614no—
147#103—1224no—
148#102—24no—
149#102—24no—
150#102—1224no—
151#97—1224no—
152C4×C8C22×D6111yescharacteristic
153C4×C8C22×D6112yes—
154C2×Q8—22no—
155C2×Q8—614no—
156C2×Q8—2442no—
157C2×D4—24no—
158C2×D4—22no—
159C22×C4—2442no—
160Q16—82no—
161Q16—2442no—
162SD16—2448no—
163D8—82no—
164C2×C8—21no—
165C2×C8—1224no—
166C2×C8D4×D6112yes—
167C2×C8D4×D6112yes—
168C4⋊C4—24no—
169C4⋊C4—614no—
170C4⋊C4—1224no—
171C4⋊C4—2444no—
172C22⋊C4—2444no—
173C22⋊C4—2442no—
174C42—1224no—
175C42—2442no—
176C42C23×D6111yescharacteristic
177C23—42no—
178Q8—82no—
179Q8—1224no—
180Q8—1224no—
181Q8—9762no—
182D4—44no—
183D4—44no—
184D4—82no—
185C2×C4—42no—
186C2×C4—1224no—
187C2×C4—1224no—
188C2×C4—9762no—
189C2×C4C2×D4×D6112yes—
190C2×C4#1451982211yescharacteristic
191C8—22no—
192C8—22no—
193C8—22no—
194C8—2444no—
195C22—84no—
196C22—82no—
197C22#1684039711yescharacteristic
198C4—21no—
199C4—82no—
200C4—2444no—
201C4—2444no—
202C4#1684055912yes—
203C4#1684057612yes—
204C2—162no—
205C2#2145747412yes—
206C2#2145761811yescharacteristic
subgroupG/H
centerC22#16840397
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1709C2
socleC22×C61D42
prime radicalC61×#493C22
hypercenter#1709D61
hypocenterC61#20319
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1709

Complements

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

NH ≅ G/Ntype
1. #21458638C2semidirect
2. #21445356C2semidirect
3. #21444957C2semidirect
5. #21444717C2semidirect
128. C61#20319semidirect

Maximal subgroups

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

subgroupindex
#214586382
#214453562
#214449572
#214448762
#214447172
C61×#17092
#2031961