Finite Group Databasebeta

← #356×D59 #21425775

Subgroups — #356×D59 #21425775

Order 7552.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D59C211yesmaximal, characteristic
2#101×D59C212yesmaximal
3#16837201C211yesmaximal, characteristic, complemented
4#16837200C212yesmaximal, complemented
5C59×#356C211yesmaximal, characteristic, complemented
6C2×Q8×D59C411yescharacteristic
7C2×Q8×D59C2211yescharacteristic
8M16×D59—22no—
9#14273727C411yescharacteristic
10#14273726—22no—
11#14273725C411yescharacteristic
12#14273725C2211yescharacteristic
13C4⋊C4×D59—22no—
14C42×D59C411yescharacteristic
15#14273646C2212yes—
16#14273645—22no—
17C4⋊Q8×C59C2211yescharacteristic
18#101×C59C2212yes—
19C2×Q8×C59C2×C411yescharacteristic
20C2×Q8×C59C2311yescharacteristic
21Q8×D59—41no—
22Q8×D59—41no—
23Q8×D59—41no—
24Q8×D59—41no—
25C2×C4×D59—21no—
26C2×C4×D59—21no—
27C2×C4×D59C2×C411yescharacteristic
28C2×C4×D59D412yes—
29C2×Dic118—21no—
30C2×Dic118C2×C411yescharacteristic
31C2×Dic118C2×C411yescharacteristic
32C2×Dic118D412yes—
33M16×C59—22no—
34C4⋊C4×C59—22no—
35C42×C59C2×C411yescharacteristic
36C4⋊Dic59—22no—
37Dic59⋊C4—22no—
38Dic59⋊C4—22no—
39C4×Dic59—21no—
40C4×Dic59C2×C411yescharacteristic
41#11745664—22no—
42#11745659—42no—
43C8×D59—42no—
44C22×D59C22⋊C411yescharacteristic
45Q8×C59—41no—
46Q8×C59—41no—
47C2×C4×C59—21no—
48C2×C4×C59—21no—
49C2×C4×C59C22×C411yescharacteristic
50C2×C4×C59C2×D412yes—
51C2×Dic59—21no—
52C2×Dic59—21no—
53C2×Dic59C22⋊C411yescharacteristic
54C2×Dic59C22⋊C412yes—
55C4×D59—24no—
56C4×D59—22no—
57C4×D59—22no—
58C4×D59—21no—
59C4×D59—21no—
60C4×D59—41no—
61C4×D59—41no—
62Dic118—41no—
63Dic118—42no—
64Dic118—42no—
65Dic118—41no—
66C8×C59—42no—
67C59⋊C8—42no—
68C22×C59C2×C22⋊C411yescharacteristic
69C2×D59—22no—
70C2×D59#9911yescharacteristic
71C2×D59#9911yescharacteristic
72C4×C59—22no—
73C4×C59—22no—
74C4×C59—21no—
75C4×C59—41no—
76Dic59—22no—
77Dic59—21no—
78Dic59—22no—
79Dic59—41no—
80C2×#356—591nomaximal, Sylow
81C2×C59—21no—
82C2×C59C2×#9911yescharacteristic
83D59—21no—
84D59#35612yesdirect product factor
85C2×C4⋊Q8—591no—
86C2×#101—592no—
87#356—591no—
88#356—592no—
89#356D5911yescharacteristic, direct product factor
90C59C2×#35611yescharacteristic, complemented, Sylow
91C22×Q8—591no—
92C22×Q8—591no—
93C2×M16—1182no—
94C4⋊Q8—591no—
95C4⋊Q8—591no—
96C4⋊Q8—591no—
97C4⋊Q8—1182no—
98C4⋊Q8C2×D5911yescharacteristic
99C2×C4⋊C4—1182no—
100C2×C42—591no—
101#101—592no—
102#101—1182no—
103#101C2×D5912yes—
104C2×Q8—591no—
105C2×Q8—592no—
106C2×Q8—591no—
107C2×Q8—1181no—
108C2×Q8—2361no—
109C2×Q8—2361no—
110C2×Q8—2361no—
111C2×Q8—2361no—
112C2×Q8C4×D5911yescharacteristic
113C2×Q8C22×D5911yescharacteristic
114C22×C4—591no—
115C22×C4—592no—
116C22×C4—1181no—
117C22×C4—1181no—
118M16—22no—
119M16—1182no—
120M16—2362no—
121C2×C8—2362no—
122C4⋊C4—22no—
123C4⋊C4—1182no—
124C4⋊C4—1182no—
125C4⋊C4—1182no—
126C42—591no—
127C42—1181no—
128C42C4×D5911yescharacteristic
129C23—591no—
130Q8—41no—
131Q8—41no—
132Q8—2362no—
133Q8—2361no—
134Q8—2361no—
135Q8—2362no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—591no—
139C2×C4—592no—
140C2×C4—1182no—
141C2×C4—1182no—
142C2×C4—1181no—
143C2×C4—1181no—
144C2×C4—1184no—
145C2×C4—1181no—
146C2×C4—1181no—
147C2×C4—2361no—
148C2×C4—2361no—
149C2×C4C2×C4×D5911yescharacteristic
150C2×C4D4×D5912yes—
151C8—42no—
152C8—2362no—
153C22—591no—
154C22—591no—
155C22—1182no—
156C22C22⋊C4×D5911yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—1182no—
162C4—1181no—
163C4—1182no—
164C4—2361no—
165C2—21no—
166C2—592no—
167C2—1181no—
168C2#99×D5911yescharacteristic
subgroupG/H
centerC2#99×D59
commutatorC2×C4×C59C22×C4
FrattiniC2×Q8C22×D59
FittingC59×#356C2
socleC2×C59C2×#99
hypercenter#356D59
hypocenterC59C2×#356
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #16837201C2semidirect
4. #16837200C2semidirect
5. C59×#356C2semidirect
84. D59#356direct (N × H)
89. #356D59direct (N × H)
90. C59C2×#356semidirect