Finite Group Databasebeta

← #36374654

Subgroups — #36374654

Order 22336.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#19150232C211yesmaximal, characteristic, complemented
2C22×C4×D349C211yesmaximal, characteristic, complemented
3#26013621C212yesmaximal, complemented
4#26013619C211yesmaximal, characteristic, complemented
5#26013618C211yesmaximal, characteristic, complemented
6#26013610C211yesmaximal, characteristic, complemented
7#26013607C212yesmaximal, complemented
8D4×Dic349C212yesmaximal, complemented
9#26013599C212yesmaximal, complemented
10#26013597C211yesmaximal, characteristic, complemented
11#121×C349C211yesmaximal, characteristic, complemented
12C23×D349C2211yescharacteristic
13#19150232—24no—
14C2×Dic698C2211yescharacteristic, complemented
15C2×C22⋊D349C2212yes—
16C2×C22⋊D349C2212yes—
17C22×Dic349C2211yescharacteristic
18C22×Dic349C2212yescomplemented
19C2×C4×D349—24no—
20C2×C4×D349C2211yescharacteristic
21C2×C4×D349C2211yescharacteristic
22C2×C4×D349C2211yescharacteristic
23C2×C4×D349C2211yescharacteristic
24C4⋊Dic349C2211yescharacteristic, complemented
25C4⋊Dic349C2212yescomplemented
26#19150212C2212yescomplemented
27C22⋊Dic349C2212yes—
28C22⋊Dic349C2212yescomplemented
29#19150210C2212yes—
30#19150210C2212yes—
31C4×Dic349C2212yescomplemented
32C2×D4×C349C2212yes—
33C2×D4×C349C2211yescharacteristic, complemented
34C22×C4×C349C2211yescharacteristic
35C4⋊C4×C349C2211yescharacteristic, complemented
36C22⋊C4×C349C2212yescomplemented
37C22×D349—22no—
38C22×D349—22no—
39C22×D349—22no—
40C22×D349C2311yescharacteristic
41C22×D349C2311yescharacteristic
42Dic698—22no—
43C22⋊D349—24no—
44C22⋊D349—24no—
45C2×Dic349—22no—
46C2×Dic349—24no—
47C2×Dic349C2312yes—
48C2×Dic349C2312yes—
49C2×Dic349C2311yescharacteristic
50C2×Dic349C2311yescharacteristic
51C4×D349—22no—
52C4×D349—22no—
53C4×D349—22no—
54C4×D349D414yescomplemented
55C23×C349C2312yes—
56C23×C349C2311yescharacteristic
57D4×C349—24no—
58D4×C349—22no—
59C2×C4×C349—22no—
60C2×C4×C349C2311yescharacteristic
61C2×C4×C349C2312yes—
62C2×C4×C349C2311yescharacteristic
63C22×C349—21no—
64C22×C349—21no—
65C22×C349—22no—
66C22×C349—22no—
67C22×C349—22no—
68C22×C349#4112yescomplemented
69C22×C349C2411yescharacteristic
70C2×D349—28no—
71C2×D349—21no—
72C2×D349—21no—
73C2×D349—21no—
74C2×D349—21no—
75C2×D349C2×D412yes—
76C2×D349#4112yescomplemented
77C4×C349—21no—
78C4×C349—22no—
79C4×C349C2×D412yes—
80Dic349—21no—
81Dic349—22no—
82Dic349—22no—
83Dic349C2×D412yes—
84C2×C349—22no—
85C2×C349—42no—
86C2×C349C22×D411yescharacteristic
87C2×C349C2×#4111yescharacteristic
88C2×C349C2×#4111yescharacteristic
89D349—22no—
90D349—22no—
91C349#52511yescharacteristic, complemented, Sylow
92#525—3491nomaximal, Sylow
93C2×#41—3491no—
94C23×C4—3491no—
95#123—3492no—
96C22⋊Q8—3491no—
97C22⋊Q8—3491no—
98#121—3491no—
99#121D34911yescharacteristic, complemented
100#120—3492no—
101C4×D4—3492no—
102C4×D4—3492no—
103#117—3491no—
104C24—3491no—
105#41—6984no—
106C2×Q8—3491no—
107C2×D4—3492no—
108C2×D4—3492no—
109C2×D4C2×D34912yes—
110C2×D4C2×D34911yescharacteristic, complemented
111C22×C4—3491no—
112C22×C4—3491no—
113C22×C4—3492no—
114C22×C4—3491no—
115C22×C4—3491no—
116C22×C4—3491no—
117C22×C4—6984no—
118C22×C4C2×D34911yescharacteristic
119C4⋊C4—3491no—
120C4⋊C4—3492no—
121C4⋊C4—3492no—
122C4⋊C4C2×D34911yescharacteristic, complemented
123C22⋊C4—3492no—
124C22⋊C4—3492no—
125C22⋊C4—3492no—
126C22⋊C4—3492no—
127C22⋊C4C2×D34912yescomplemented
128C42—3492no—
129C23—3491no—
130C23—3491no—
131C23—6982no—
132C23—6982no—
133C23—6982no—
134C23C22×D34912yes—
135C23C22×D34911yescharacteristic
136Q8—6982no—
137D4—24no—
138D4—22no—
139D4—6984no—
140D4—6984no—
141C2×C4—22no—
142C2×C4—3491no—
143C2×C4—3494no—
144C2×C4—3492no—
145C2×C4—3491no—
146C2×C4—3492no—
147C2×C4—6982no—
148C2×C4—6982no—
149C2×C4—6982no—
150C2×C4—6984no—
151C2×C4—6982no—
152C2×C4C22×D34912yes—
153C2×C4C22×D34911yescharacteristic
154C2×C4C22×D34911yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3492no—
161C22—3492no—
162C22—6988no—
163C22—6981no—
164C22—6981no—
165C22—6981no—
166C22—6981no—
167C22#1915023212yescomplemented
168C22C23×D34911yescharacteristic
169C4—21no—
170C4—22no—
171C4—3492no—
172C4—6981no—
173C4—6982no—
174C4—6982no—
175C4D4×D34912yes—
176C2—22no—
177C2—42no—
178C2—6982no—
179C2—6982no—
180C2C2×D4×D34911yescharacteristic
181C2#41×D34911yescharacteristic
182C2C2×#1915023211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#121×C349C2
hypercenter#121D349
hypocenterC349#525
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19150232C2semidirect
2. C22×C4×D349C2semidirect
3. #26013621C2semidirect
4. #26013619C2semidirect
5. #26013618C2semidirect
6. #26013610C2semidirect
7. #26013607C2semidirect
8. D4×Dic349C2semidirect
9. #26013599C2semidirect
10. #26013597C2semidirect
11. #121×C349C2semidirect
14. C2×Dic698C22semidirect
18. C22×Dic349C22semidirect
24. C4⋊Dic349C22semidirect
25. C4⋊Dic349C22semidirect
26. #19150212C22semidirect
28. C22⋊Dic349C22semidirect
31. C4×Dic349C22semidirect
33. C2×D4×C349C22semidirect
35. C4⋊C4×C349C22semidirect
36. C22⋊C4×C349C22semidirect
54. C4×D349D4semidirect
68. C22×C349#41semidirect
76. C2×D349#41semidirect
91. C349#525semidirect
99. #121D349semidirect
110. C2×D4C2×D349semidirect
122. C4⋊C4C2×D349semidirect
127. C22⋊C4C2×D349semidirect
167. C22#19150232semidirect

Maximal subgroups

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

subgroupindex
C2×#191502322
C22×C4×D3492
#260136212
#260136192
#260136182
#260136102
#260136072
D4×Dic3492
#260135992
#260135972
#121×C3492
#525349