Finite Group Databasebeta

← #36374891

Subgroups — #36374891

Order 22336.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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#26013630C211yesmaximal, characteristic, complemented
2#26013625C211yesmaximal, characteristic, complemented
3#26013625C211yesmaximal, characteristic, complemented
4#26013624C211yesmaximal, characteristic, complemented
5#26013623C211yesmaximal, characteristic, complemented
6#26013621C211yesmaximal, characteristic, complemented
7#26013619C211yesmaximal, characteristic, complemented
8#26013618C211yesmaximal, characteristic, complemented
9#26013617C211yesmaximal, characteristic, complemented
10#26013616C211yesmaximal, characteristic, complemented
11#26013613C211yesmaximal, characteristic, complemented
12#26013612C211yesmaximal, characteristic, complemented
13#26013611C211yesmaximal, characteristic, complemented
14#26013610C211yesmaximal, characteristic, complemented
15#123×C349C211yesmaximal, characteristic, complemented
16C2×Dic698C2211yescharacteristic
17C2×Dic698C2211yescharacteristic
18C2×D1396C2211yescharacteristic
19C2×C22⋊D349C2211yescharacteristic
20C2×C22⋊D349C2211yescharacteristic
21C2×C22⋊D349C2211yescharacteristic
22C2×C22⋊D349C2211yescharacteristic
23C22×Dic349C2211yescharacteristic
24C2×C4×D349C2211yescharacteristic
25C2×C4×D349C2211yescharacteristic
26C4⋊Dic349C2211yescharacteristic
27C4⋊Dic349C2211yescharacteristic
28C4⋊Dic349C2211yescharacteristic
29C4⋊Dic349C2211yescharacteristic
30#19150212C2211yescharacteristic
31#19150212C2211yescharacteristic
32#19150212C2211yescharacteristic
33#19150212C2211yescharacteristic
34C22⋊Dic349C2211yescharacteristic
35C22⋊Dic349C2211yescharacteristic
36C22⋊Dic349C2211yescharacteristic
37#19150210C2211yescharacteristic
38#19150210C2211yescharacteristic
39#19150210C2211yescharacteristic
40#19150210C2211yescharacteristic
41#19150210C2211yescharacteristic
42#19150210C2211yescharacteristic
43C4×Dic349C2211yescharacteristic
44C2×D4×C349C2211yescharacteristic
45C22×C4×C349C2211yescharacteristic
46C4⋊C4×C349C2211yescharacteristic
47C4⋊C4×C349C2211yescharacteristic
48C22⋊C4×C349C2211yescharacteristic
49C22⋊C4×C349C2211yescharacteristic
50C22⋊C4×C349C2211yescharacteristic
51C22×D349C2311yescharacteristic
52C22×D349C2311yescharacteristic
53Dic698—41no—
54Dic698—41no—
55D1396—41no—
56C22⋊D349—41no—
57C22⋊D349—41no—
58C22⋊D349—41no—
59C22⋊D349—41no—
60C2×Dic349—41no—
61C2×Dic349C2311yescharacteristic
62C2×Dic349C2311yescharacteristic
63C2×Dic349C2311yescharacteristic
64C2×Dic349C2311yescharacteristic
65C2×Dic349C2311yescharacteristic
66C2×Dic349C2311yescharacteristic
67C4×D349—41no—
68C4×D349—41no—
69C23×C349C2311yescharacteristic
70C23×C349C2311yescharacteristic
71D4×C349—41no—
72C2×C4×C349—41no—
73C2×C4×C349C2311yescharacteristic
74C2×C4×C349C2311yescharacteristic
75C2×C4×C349C2311yescharacteristic
76C2×C4×C349C2311yescharacteristic
77C2×C4×C349C2311yescharacteristic
78C22×C349—21no—
79C22×C349—21no—
80C22×C349—21no—
81C22×C349—21no—
82C22×C349—21no—
83C22×C349—21no—
84C22×C349C2411yescharacteristic
85C2×D349—21no—
86C2×D349—21no—
87C2×D349—21no—
88C2×D349—21no—
89C2×D349—21no—
90C2×D349—21no—
91C4×C349—21no—
92C4×C349—21no—
93C4×C349—21no—
94C4×C349—21no—
95C4×C349—21no—
96Dic349—21no—
97Dic349—21no—
98Dic349—21no—
99Dic349—21no—
100Dic349—21no—
101Dic349—21no—
102C2×C349—41no—
103C2×C349—41no—
104C2×C349H(5,𝔽2)11yescharacteristic
105C2×C349H(5,𝔽2)11yescharacteristic
106C2×C3492−1+411yescharacteristic
107D349—41no—
108D349—41no—
109C349#56211yescharacteristic, complemented, Sylow
110#562—3491nomaximal, Sylow
111#125—3491no—
112#124—3491no—
113#124—3491no—
114#123—3491no—
115#123—3491no—
116#123—3491no—
117#123D34911yescharacteristic, complemented
118C22⋊Q8—3491no—
119C22⋊Q8—3491no—
120C22⋊Q8—3491no—
121C22⋊Q8—3491no—
122#121—3491no—
123#121—3491no—
124#121—3491no—
125#121—3491no—
126C2×Q8—3491no—
127C2×Q8—3491no—
128C2×D4—3491no—
129C2×D4—3491no—
130C2×D4—3491no—
131C2×D4—3491no—
132C2×D4—3491no—
133C2×D4C2×D34911yescharacteristic
134C22×C4—3491no—
135C22×C4—3491no—
136C22×C4—3491no—
137C22×C4C2×D34911yescharacteristic
138C4⋊C4—3491no—
139C4⋊C4—3491no—
140C4⋊C4—3491no—
141C4⋊C4—3491no—
142C4⋊C4—3491no—
143C4⋊C4—3491no—
144C4⋊C4—3491no—
145C4⋊C4—3491no—
146C4⋊C4C2×D34911yescharacteristic
147C4⋊C4C2×D34911yescharacteristic
148C22⋊C4—3491no—
149C22⋊C4—3491no—
150C22⋊C4—3491no—
151C22⋊C4—3491no—
152C22⋊C4—3491no—
153C22⋊C4—3491no—
154C22⋊C4—3491no—
155C22⋊C4—3491no—
156C22⋊C4—3491no—
157C22⋊C4C2×D34911yescharacteristic
158C22⋊C4C2×D34911yescharacteristic
159C22⋊C4C2×D34911yescharacteristic
160C42—3491no—
161C23—3491no—
162C23—3491no—
163C23C22×D34911yescharacteristic
164C23C22×D34911yescharacteristic
165Q8—13961no—
166Q8—13961no—
167D4—41no—
168D4—13961no—
169D4—13961no—
170D4—13961no—
171D4—13961no—
172D4—13961no—
173C2×C4—41no—
174C2×C4—3491no—
175C2×C4—3491no—
176C2×C4—3491no—
177C2×C4—3491no—
178C2×C4—3491no—
179C2×C4—3491no—
180C2×C4—13961no—
181C2×C4—13961no—
182C2×C4—13961no—
183C2×C4C22×D34911yescharacteristic
184C2×C4C22×D34911yescharacteristic
185C2×C4C22×D34911yescharacteristic
186C2×C4C22×D34911yescharacteristic
187C2×C4C22×D34911yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6981no—
195C22—6981no—
196C22—6981no—
197C22—6981no—
198C22—6981no—
199C22—6981no—
200C22C23×D34911yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—6981no—
207C4—6981no—
208C4—6981no—
209C4—6981no—
210C4—6981no—
211C4—6981no—
212C2—41no—
213C2—41no—
214C2—13961no—
215C2—13961no—
216C2#2601368411yescharacteristic
217C2#2601368511yescharacteristic
218C2#2601368611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#123×C349C2
hypercenter#123D349
hypocenterC349#562
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26013630C2semidirect
2. #26013625C2semidirect
3. #26013625C2semidirect
4. #26013624C2semidirect
5. #26013623C2semidirect
6. #26013621C2semidirect
7. #26013619C2semidirect
8. #26013618C2semidirect
9. #26013617C2semidirect
10. #26013616C2semidirect
11. #26013613C2semidirect
12. #26013612C2semidirect
13. #26013611C2semidirect
14. #26013610C2semidirect
15. #123×C349C2semidirect
109. C349#562semidirect
117. #123D349semidirect

Maximal subgroups

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

subgroupindex
#260136302
#260136252
#260136242
#260136232
#260136212
#260136192
#260136182
#260136172
#260136162
#260136132
#260136122
#260136112
#260136102
#123×C3492
#562349