Finite Group Databasebeta

← #21426109

Subgroups — #21426109

Order 7552.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#16837591C212yesmaximal
2#16837579C211yesmaximal, characteristic
3#16837197C211yesmaximal, characteristic, complemented
4#16837192C212yesmaximal, complemented
5C59×#563C211yesmaximal, characteristic, complemented
6#14273772C2211yescharacteristic
7#14273758—21no—
8#14273710—22no—
9#14273655—21no—
10#14273644C2211yescharacteristic
11#14273642C2212yes—
12#14273641—22no—
13C4⋊Q8×C59—21no—
14#126×C59—22no—
15#125×C59—21no—
16#124×C59C2211yescharacteristic
17#123×C59C2212yes—
18C22⋊Q8×C59—22no—
19C2×Q8×C59—21no—
20C2×Q8×C59D411yescharacteristic
21C2×D4×C59C2311yescharacteristic
22C22×C4×C59D412yes—
23C2×C22⋊D59—22no—
24#11745687—41no—
25D4×D59—41no—
26#11745685—21no—
27C4⋊C4×C59—21no—
28C4⋊C4×C59—22no—
29C4⋊C4×C59—22no—
30C4⋊C4×C59—21no—
31C4⋊C4×C59—22no—
32C22⋊C4×C59—22no—
33C22⋊C4×C59—22no—
34C22⋊C4×C59D412yes—
35C42×C59—21no—
36C42×C59D411yescharacteristic
37C22⋊Dic59—22no—
38#11745672—41no—
39#11745671—41no—
40(C2×D59)⋊C4—42no—
41Dic59⋊C4—42no—
42#11745664—21no—
43C23×C59C2×D412yes—
44C22×D59—22no—
45Q8×C59—41no—
46Q8×C59—81no—
47D4×C59—41no—
48C2×C4×C59—21no—
49C2×C4×C59—22no—
50C2×C4×C59—21no—
51C2×C4×C59—22no—
52C2×C4×C59—42no—
53C2×C4×C59C2×D411yescharacteristic
54C22⋊D59—42no—
55C22⋊D59—41no—
56C2×Dic59—22no—
57C2×Dic59—42no—
58D236—21no—
59C4×D59—41no—
60Dic118—21no—
61C59⋊C8—41no—
62C22×C59—22no—
63C22×C59—42no—
64C22×C59#12011yescharacteristic
65C2×D59—41no—
66C2×D59—82no—
67C4×C59—21no—
68C4×C59—42no—
69C4×C59—41no—
70C4×C59—42no—
71C4×C59—41no—
72Dic59—41no—
73Dic59—82no—
74#2201—591nomaximal, Sylow
75C2×C59—21no—
76C2×C59—42no—
77C2×C59B(4,𝔽2)11yescharacteristic
78D59—81no—
79#563D5911yescharacteristic, complemented
80#458—592no—
81#455—591no—
82#354—591no—
83#352—592no—
84C59#220111yescharacteristic, complemented, Sylow
85H(5,𝔽2)—591no—
86#137—1181no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D5911yescharacteristic
91#123—1182no—
92#123C2×D5912yes—
93C22⋊Q8—22no—
94#104—1181no—
95#100—591no—
96#99—592no—
97#99—1182no—
98#41—1181no—
99#41—2361no—
100C2×Q8—21no—
101C2×Q8C22⋊D5911yescharacteristic
102C2×D4—1182no—
103C2×D4—2361no—
104C2×D4C22×D5911yescharacteristic
105C22×C4C22⋊D5912yes—
106Q16—2361no—
107SD16—2361no—
108M16—1181no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—2362no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—1182no—
118C22⋊C4—2362no—
119C22⋊C4C22⋊D5912yes—
120C42—21no—
121C42C22⋊D5911yescharacteristic
122C23—1182no—
123C23C2×C22⋊D5912yes—
124Q8—41no—
125Q8—81no—
126Q8—1181no—
127D4—41no—
128D4—1181no—
129D4—2362no—
130D4—2361no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—1182no—
137C2×C4—2361no—
138C2×C4—2362no—
139C2×C4C2×C22⋊D5911yescharacteristic
140C8—2361no—
141C22—22no—
142C22—42no—
143C22—4722no—
144C22—2361no—
145C22#1427369611yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—4722no—
152C4—2361no—
153C2—21no—
154C2—42no—
155C2—4721no—
156C2#1683758911yescharacteristic
subgroupG/H
centerC2#16837589
commutatorC2×D4×C59C23
FrattiniC2×D4C22×D59
FittingC59×#563C2
socleC2×C59B(4,𝔽2)
hypercenter#563D59
hypocenterC59#2201
LCSG ▷ C2×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #16837197C2semidirect
4. #16837192C2semidirect
5. C59×#563C2semidirect
79. #563D59semidirect
84. C59#2201semidirect