Finite Group Databasebeta

← #29600361

Subgroups — #29600361

Order 15616.

326 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 79 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#21447484C212yesmaximal
2#21443990C214yesmaximal, complemented
3C61×#3135C211yesmaximal, characteristic
4#16839736—22no—
5#16839557—22no—
6#16839555C2214yes—
7C61×#533C411yescharacteristic, complemented
8C61×#532C2211yescharacteristic
9C61×#491C412yescomplemented
10C61×#490C2212yes—
11C61×#489—22no—
12C61×#416—22no—
13C2×#97×C61C411yescharacteristic, complemented
14#14519736—42no—
15#14519692—24no—
16#14519692—42no—
17C2×#41×C61D412yes—
18C22×C8×C61—21no—
19C4⋊Q8×C61—21no—
20C4⋊Q8×C61C2×C411yescharacteristic
21#127×C61C2311yescharacteristic
22#125×C61C2×C412yes—
23#124×C61D412yes—
24C22⋊Q8×C61—41no—
25#121×C61—41no—
26C4×Q8×C61—41no—
27C4×D4×C61—41no—
28#117×C61—21no—
29C2×C42×C61C2×C411yescharacteristic
30#103×C61—24no—
31#102×C61—24no—
32#97×C61—21no—
33#97×C61C2×C412yes—
34#41×C61—42no—
35#41×C61—42no—
36C2×Q8×C61—41no—
37C2×Q8×C61C22⋊C412yes—
38C2×D4×C61—22no—
39C2×D4×C61—41no—
40C2×D4×C61C2×D412yes—
41C22×C4×C61—21no—
42C22×C4×C61—22no—
43C22×C4×C61C22⋊C411yescharacteristic
44C2×C8×C61—22no—
45C2×C8×C61—41no—
46C2×C8×C61—41no—
47C4⋊C4×C61—22no—
48C4⋊C4×C61—41no—
49C4⋊C4×C61—41no—
50C4⋊C4×C61—42no—
51C22⋊C4×C61—42no—
52C22⋊C4×C61—41no—
53C42×C61—21no—
54C42×C61—41no—
55C42×C61C22⋊C411yescharacteristic
56C42×C61C22×C411yescharacteristic
57C22⋊Dic61—44no—
58C4⋊Dic61—82no—
59C4×Dic61—82no—
60C23×C61—22no—
61C23×C61#9911yescharacteristic
62Q8×C61—42no—
63Q8×C61—82no—
64D4×C61—42no—
65D4×C61—82no—
66D4×C61—82no—
67C2×C4×C61—21no—
68C2×C4×C61—21no—
69C2×C4×C61—21no—
70C2×C4×C61—21no—
71C2×C4×C61—22no—
72C2×C4×C61—42no—
73C2×C4×C61—42no—
74C2×C4×C61—42no—
75C2×C4×C61—81no—
76C2×C4×C61#9911yescharacteristic
77C2×C4×C61C2×C22⋊C411yescharacteristic
78C2×Dic61—84no—
79C8×C61—42no—
80#13594—611nomaximal, Sylow
81C22×C61—21no—
82C22×C61—41no—
83C22×C61—42no—
84C22×C61—82no—
85C22×C61C2×#9911yescharacteristic
86C4×C61—21no—
87C4×C61—41no—
88C4×C61—41no—
89C4×C61—42no—
90C4×C61—82no—
91C4×C61#35411yescharacteristic
92C4×C61#35411yescharacteristic
93Dic61—164no—
94#3135D6111yescharacteristic
95#2126—612no—
96#1412—614no—
97C2×C61—21no—
98C2×C61—41no—
99C2×C61—82no—
100C2×C61C2×#35411yescharacteristic
101#533Dic6111yescharacteristic, complemented
102#532C2×D6111yescharacteristic
103#491Dic6112yescomplemented
104#490C2×D6112yes—
105#489—22no—
106#416—22no—
107#410—1222no—
108C2×#97Dic6111yescharacteristic, complemented
109#354—1222no—
110#353—614no—
111C61#1359411yescharacteristic, complemented, Sylow
112C2×#41C22⋊D6112yes—
113C22×C8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C2×Dic6111yescharacteristic
116#127C22×D6111yescharacteristic
117#125C2×Dic6112yes—
118#124C22⋊D6112yes—
119C22⋊Q8—41no—
120#121—41no—
121C4×Q8—41no—
122C4×D4—41no—
123#117—21no—
124#117—2442no—
125C2×C42C2×Dic6111yescharacteristic
126#103—24no—
127#102—24no—
128#99—1224no—
129#99—2442no—
130#97—21no—
131#97C2×Dic6112yes—
132#41—42no—
133#41—42no—
134C2×Q8—41no—
135C2×Q8C22⋊Dic6112yes—
136C2×D4—22no—
137C2×D4—41no—
138C2×D4C2×C22⋊D6112yes—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4C22⋊Dic6111yescharacteristic
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C4⋊C4—22no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—4882no—
150C22⋊C4—42no—
151C22⋊C4—41no—
152C22⋊C4—2444no—
153C42—21no—
154C42—41no—
155C42—4882no—
156C42C22⋊Dic6111yescharacteristic
157C42C22×Dic6111yescharacteristic
158C23—22no—
159C23#1451969211yescharacteristic
160Q8—42no—
161Q8—82no—
162D4—42no—
163D4—82no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—81no—
174C2×C4—4884no—
175C2×C4#1451969211yescharacteristic
176C2×C4C2×C22⋊Dic6111yescharacteristic
177C8—42no—
178C22—21no—
179C22—41no—
180C22—42no—
181C22—82no—
182C22C2×#1451969211yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—42no—
187C4—82no—
188C4—9764no—
189C4#1683955711yescharacteristic
190C4#1683955711yescharacteristic
191C2—21no—
192C2—41no—
193C2—82no—
194C2C2×#1683955711yescharacteristic
subgroupG/H
centerC4#16839557
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#3135C2
socleC2×C61C2×#354
prime radicalC61×#532C22
cosocle#127×C61C23
hypercenter#3135D61
hypocenterC61#13594
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3135

Complements

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

NH ≅ G/Ntype
2. #21443990C2semidirect
7. C61×#533C4semidirect
9. C61×#491C4semidirect
13. C2×#97×C61C4semidirect
101. #533Dic61semidirect
103. #491Dic61semidirect
108. C2×#97Dic61semidirect
111. C61#13594semidirect

Maximal subgroups

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

subgroupindex
#214474842
#214439902
C61×#31352
#1359461