Finite Group Databasebeta

← #36374677

Subgroups — #36374677

Order 22336.

234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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#26013633C211yesmaximal, characteristic, complemented
2#26013629C211yesmaximal, characteristic, complemented
3#26013625C211yesmaximal, characteristic, complemented
4#26013624C211yesmaximal, characteristic, complemented
5#26013621C211yesmaximal, characteristic, complemented
6#26013617C211yesmaximal, characteristic, complemented
7#26013612C211yesmaximal, characteristic
8#26013604C211yesmaximal, characteristic, complemented
9#26013601C211yesmaximal, characteristic, complemented
10#26013599C211yesmaximal, characteristic, complemented
11#26013599C211yesmaximal, characteristic, complemented
12#26013597C211yesmaximal, characteristic, complemented
13#26013595C211yesmaximal, characteristic, complemented
14C42×D349C211yesmaximal, characteristic, complemented
15#126×C349C211yesmaximal, characteristic, complemented
16C2×Dic698C2211yescharacteristic
17C2×D1396C2211yescharacteristic
18C2×C22⋊D349C2211yescharacteristic
19C2×C22⋊D349C2211yescharacteristic
20C22×Dic349C2211yescharacteristic
21C2×C4×D349C2211yescharacteristic
22C2×C4×D349C2211yescharacteristic
23C2×C4×D349C2211yescharacteristic
24C2×C4×D349C2211yescharacteristic
25C4⋊Dic349C2211yescharacteristic
26#19150212C2211yescharacteristic
27#19150212C2211yescharacteristic
28#19150212C2211yescharacteristic
29#19150212C2211yescharacteristic
30#19150212C2211yescharacteristic
31#19150212C2211yescharacteristic
32C22⋊Dic349C2211yescharacteristic
33#19150210C2211yescharacteristic
34#19150210C2211yescharacteristic
35#19150210C2211yescharacteristic
36#19150210C2211yescharacteristic
37#19150210C2211yescharacteristic
38#19150210C2211yescharacteristic
39C4×Dic349C2211yescharacteristic
40C4×Dic349C2211yescharacteristic
41C4×Dic349C2211yescharacteristic
42C4×Dic349C2211yescharacteristic
43C4×Dic349C2211yescharacteristic
44C4⋊C4×C349C2211yescharacteristic
45C4⋊C4×C349C2211yescharacteristic
46C4⋊C4×C349C2211yescharacteristic
47C22⋊C4×C349C2211yescharacteristic
48C22⋊C4×C349C2211yescharacteristic
49C22⋊C4×C349C2211yescharacteristic
50C42×C349C2211yescharacteristic
51C22×D349C2311yescharacteristic
52C22×D349C2311yescharacteristic
53Dic698—22no—
54D1396—22no—
55C22⋊D349—22no—
56C22⋊D349—22no—
57C2×Dic349—22no—
58C2×Dic349C2311yescharacteristic
59C2×Dic349C2311yescharacteristic
60C2×Dic349C2311yescharacteristic
61C2×Dic349C2311yescharacteristic
62C2×Dic349C2311yescharacteristic
63C2×Dic349C2311yescharacteristic
64C4×D349—22no—
65C4×D349—22no—
66C4×D349—22no—
67C4×D349—22no—
68C23×C349C2311yescharacteristic
69C2×C4×C349C2311yescharacteristic
70C2×C4×C349C2311yescharacteristic
71C2×C4×C349C2311yescharacteristic
72C2×C4×C349C2311yescharacteristic
73C2×C4×C349C2311yescharacteristic
74C2×C4×C349C2311yescharacteristic
75C22×C349—21no—
76C22×C349—21no—
77C22×C349—21no—
78C22×C349C2411yescharacteristic
79C2×D349—21no—
80C2×D349—21no—
81C2×D349—21no—
82C2×D349—21no—
83C2×D349—21no—
84C2×D349#4112yes—
85C4×C349—21no—
86C4×C349—21no—
87C4×C349—21no—
88C4×C349—21no—
89C4×C349—21no—
90C4×C349—21no—
91Dic349—21no—
92Dic349—21no—
93Dic349—21no—
94Dic349—21no—
95Dic349#4112yes—
96Dic349#4112yes—
97C2×C349—41no—
98C2×C349C2×#4111yescharacteristic
99C2×C349C2×#4111yescharacteristic
100C2×C349C2×#4111yescharacteristic
101D349—22no—
102D349—41no—
103C349#52911yescharacteristic, complemented, Sylow
104#529—3491nomaximal, Sylow
105#126—3491no—
106#126D34911yescharacteristic, complemented
107#125—3491no—
108#124—3491no—
109#123—3491no—
110#123—3491no—
111C22⋊Q8—3491no—
112#121—3491no—
113C4×Q8—3491no—
114C4×D4—3491no—
115C4×D4—3491no—
116C4×D4—3491no—
117#117—3491no—
118#117—3491no—
119C2×C42—3491no—
120C2×Q8—3491no—
121C2×D4—3491no—
122C2×D4—3491no—
123C2×D4—3491no—
124C22×C4—3491no—
125C22×C4—3491no—
126C22×C4—3491no—
127C22×C4—3491no—
128C22×C4—3491no—
129C4⋊C4—3491no—
130C4⋊C4—3491no—
131C4⋊C4—3491no—
132C4⋊C4—3491no—
133C4⋊C4—3491no—
134C4⋊C4—3491no—
135C4⋊C4—3491no—
136C4⋊C4C2×D34911yescharacteristic
137C4⋊C4C2×D34911yescharacteristic
138C4⋊C4C2×D34911yescharacteristic
139C22⋊C4—3491no—
140C22⋊C4—3491no—
141C22⋊C4—3491no—
142C22⋊C4—3491no—
143C22⋊C4—3491no—
144C22⋊C4—3491no—
145C22⋊C4—3491no—
146C22⋊C4C2×D34911yescharacteristic
147C22⋊C4C2×D34911yescharacteristic
148C22⋊C4C2×D34911yescharacteristic
149C42—3491no—
150C42—3491no—
151C42—3491no—
152C42—3491no—
153C42—3491no—
154C42C2×D34911yescharacteristic
155C23—3491no—
156C23—3491no—
157C23C22×D34911yescharacteristic
158Q8—6982no—
159D4—6982no—
160D4—6982no—
161D4—6982no—
162C2×C4—3491no—
163C2×C4—3491no—
164C2×C4—3491no—
165C2×C4—3491no—
166C2×C4—3491no—
167C2×C4—3491no—
168C2×C4—6982no—
169C2×C4—6982no—
170C2×C4—6982no—
171C2×C4—6982no—
172C2×C4—6982no—
173C2×C4C22×D34911yescharacteristic
174C2×C4C22×D34911yescharacteristic
175C2×C4C22×D34911yescharacteristic
176C2×C4C22×D34911yescharacteristic
177C2×C4C22×D34911yescharacteristic
178C2×C4C22×D34911yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—3492no—
183C22—6981no—
184C22—6981no—
185C22—6981no—
186C22—6981no—
187C22—6981no—
188C22C23×D34911yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—3492no—
196C4—3492no—
197C4—6981no—
198C4—6981no—
199C4—6981no—
200C4—6981no—
201C2—41no—
202C2—13961no—
203C2—6982no—
204C2#41×D34911yescharacteristic
205C2#41×D34911yescharacteristic
206C2#41×D34911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#126×C349C2
prime radical#26013612C2
hypercenter#126D349
hypocenterC349#529
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #26013633C2semidirect
2. #26013629C2semidirect
3. #26013625C2semidirect
4. #26013624C2semidirect
5. #26013621C2semidirect
6. #26013617C2semidirect
8. #26013604C2semidirect
9. #26013601C2semidirect
10. #26013599C2semidirect
11. #26013599C2semidirect
12. #26013597C2semidirect
13. #26013595C2semidirect
14. C42×D349C2semidirect
15. #126×C349C2semidirect
103. C349#529semidirect
106. #126D349semidirect

Maximal subgroups

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

subgroupindex
#260136332
#260136292
#260136252
#260136242
#260136212
#260136172
#260136122
#260136042
#260136012
#260135992
#260135972
#260135952
C42×D3492
#126×C3492
#529349