Finite Group Databasebeta

← #36374895

Subgroups — #36374895

Order 22336.

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

normal subgroups only

This group has 196 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#26013638C211yesmaximal, characteristic, complemented
2#26013637C211yesmaximal, characteristic, complemented
3#26013634C211yesmaximal, characteristic, complemented
4#26013634C211yesmaximal, characteristic, complemented
5#26013632C211yesmaximal, characteristic, complemented
6#26013632C211yesmaximal, characteristic, complemented
7#26013630C211yesmaximal, characteristic, complemented
8#26013629C211yesmaximal, characteristic, complemented
9#26013625C211yesmaximal, characteristic
10#26013622C211yesmaximal, characteristic
11#26013621C211yesmaximal, characteristic
12#26013618C211yesmaximal, characteristic, complemented
13#26013617C211yesmaximal, characteristic, complemented
14#26013615C211yesmaximal, characteristic, complemented
15C22⋊Q8×C349C211yesmaximal, characteristic, complemented
16C2×Dic698C2211yescharacteristic
17C2×Dic698C2211yescharacteristic
18C2×C22⋊D349C2211yescharacteristic
19C2×C4×D349C2211yescharacteristic
20C4⋊Dic349C2211yescharacteristic
21C4⋊Dic349C2211yescharacteristic
22C4⋊Dic349C2211yescharacteristic
23C4⋊Dic349C2211yescharacteristic
24#19150212C2211yescharacteristic
25#19150212C2211yescharacteristic
26#19150212C2211yescharacteristic
27#19150212C2211yescharacteristic
28#19150212C2211yescharacteristic
29#19150212C2211yescharacteristic
30#19150212C2211yescharacteristic
31#19150212C2211yescharacteristic
32#19150212C2211yescharacteristic
33C22⋊Dic349C2211yescharacteristic
34C22⋊Dic349C2211yescharacteristic
35C22⋊Dic349C2211yescharacteristic
36#19150210C2211yescharacteristic
37#19150210C2211yescharacteristic
38#19150210C2211yescharacteristic
39#19150210C2211yescharacteristic
40#19150210C2211yescharacteristic
41C4×Dic349C2211yescharacteristic
42C4×Dic349C2211yescharacteristic
43C4×Dic349C2211yescharacteristic
44C2×Q8×C349C2211yescharacteristic
45C22×C4×C349C2211yescharacteristic
46C4⋊C4×C349C2211yescharacteristic
47C4⋊C4×C349C2211yescharacteristic
48C4⋊C4×C349C2211yescharacteristic
49C22⋊C4×C349C2211yescharacteristic
50C22⋊C4×C349C2211yescharacteristic
51C22×D349C2311yescharacteristic
52Dic698—41no—
53Dic698—41no—
54C22⋊D349—41no—
55C2×Dic349C2311yescharacteristic
56C2×Dic349C2311yescharacteristic
57C2×Dic349C2311yescharacteristic
58C2×Dic349C2311yescharacteristic
59C2×Dic349C2311yescharacteristic
60C2×Dic349C2311yescharacteristic
61C2×Dic349C2311yescharacteristic
62C4×D349—41no—
63C23×C349C2311yescharacteristic
64Q8×C349—41no—
65C2×C4×C349—41no—
66C2×C4×C349C2311yescharacteristic
67C2×C4×C349C2311yescharacteristic
68C2×C4×C349C2311yescharacteristic
69C2×C4×C349C2311yescharacteristic
70C2×C4×C349C2311yescharacteristic
71C2×C4×C349C2311yescharacteristic
72C22×C349—21no—
73C22×C349—21no—
74C22×C349—21no—
75C22×C349C2411yescharacteristic
76C2×D349—21no—
77C2×D349—21no—
78C2×D349—21no—
79C4×C349—21no—
80C4×C349—21no—
81C4×C349—21no—
82C4×C349—21no—
83C4×C349—21no—
84C4×C349—21no—
85Dic349—21no—
86Dic349—21no—
87Dic349—21no—
88Dic349—21no—
89Dic349—21no—
90Dic349—21no—
91Dic349—21no—
92C2×C349—41no—
93C2×C349H(5,𝔽2)11yescharacteristic
94C2×C3492−1+411yescharacteristic
95C2×C3492−1+411yescharacteristic
96D349—41no—
97C349#56311yescharacteristic, complemented, Sylow
98#563—3491nomaximal, Sylow
99C4⋊Q8—3491no—
100C4⋊Q8—3491no—
101#126—3491no—
102#126—3491no—
103#126—3491no—
104#126—3491no—
105#125—3491no—
106#125—3491no—
107#124—3491no—
108#123—3491no—
109#123—3491no—
110C22⋊Q8—3491no—
111C22⋊Q8—3491no—
112C22⋊Q8—3491no—
113C22⋊Q8D34911yescharacteristic, complemented
114C2×Q8—3491no—
115C2×Q8—3491no—
116C2×Q8C2×D34911yescharacteristic
117C2×D4—3491no—
118C22×C4—3491no—
119C22×C4C2×D34911yescharacteristic
120C4⋊C4—3491no—
121C4⋊C4—3491no—
122C4⋊C4—3491no—
123C4⋊C4—3491no—
124C4⋊C4—3491no—
125C4⋊C4—3491no—
126C4⋊C4—3491no—
127C4⋊C4—3491no—
128C4⋊C4—3491no—
129C4⋊C4—3491no—
130C4⋊C4—3491no—
131C4⋊C4—3491no—
132C4⋊C4—3491no—
133C4⋊C4C2×D34911yescharacteristic
134C4⋊C4C2×D34911yescharacteristic
135C4⋊C4C2×D34911yescharacteristic
136C22⋊C4—3491no—
137C22⋊C4—3491no—
138C22⋊C4—3491no—
139C22⋊C4—3491no—
140C22⋊C4—3491no—
141C22⋊C4—3491no—
142C22⋊C4—3491no—
143C22⋊C4—3491no—
144C22⋊C4C2×D34911yescharacteristic
145C22⋊C4C2×D34911yescharacteristic
146C42—3491no—
147C42—3491no—
148C42—3491no—
149C23—3491no—
150C23C22×D34911yescharacteristic
151Q8—41no—
152Q8—13961no—
153Q8—13961no—
154D4—13961no—
155C2×C4—41no—
156C2×C4—3491no—
157C2×C4—3491no—
158C2×C4—3491no—
159C2×C4—3491no—
160C2×C4—3491no—
161C2×C4—3491no—
162C2×C4—3491no—
163C2×C4—13961no—
164C2×C4C22×D34911yescharacteristic
165C2×C4C22×D34911yescharacteristic
166C2×C4C22×D34911yescharacteristic
167C2×C4C22×D34911yescharacteristic
168C2×C4C22×D34911yescharacteristic
169C2×C4C22×D34911yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—6981no—
174C22—6981no—
175C22—6981no—
176C22C23×D34911yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—6981no—
184C4—6981no—
185C4—6981no—
186C4—6981no—
187C4—6981no—
188C4—6981no—
189C4—6981no—
190C2—41no—
191C2—13961no—
192C2#2601368411yescharacteristic
193C2#2601368611yescharacteristic
194C2#2601368711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
FittingC22⋊Q8×C349C2
prime radicalC2×C22⋊D349C22
hypercenterC22⋊Q8D349
hypocenterC349#563
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26013638C2semidirect
2. #26013637C2semidirect
3. #26013634C2semidirect
4. #26013634C2semidirect
5. #26013632C2semidirect
6. #26013632C2semidirect
7. #26013630C2semidirect
8. #26013629C2semidirect
12. #26013618C2semidirect
13. #26013617C2semidirect
14. #26013615C2semidirect
15. C22⋊Q8×C349C2semidirect
97. C349#563semidirect
113. C22⋊Q8D349semidirect

Maximal subgroups

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

subgroupindex
#260136382
#260136372
#260136342
#260136322
#260136302
#260136292
#260136252
#260136222
#260136212
#260136182
#260136172
#260136152
C22⋊Q8×C3492
#563349