Finite Group Databasebeta

← #49121780

Subgroups — #49121780

Order 36224.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#31827231C211yesmaximal, characteristic
2#31827039C211yesmaximal, characteristic
3#31826474C211yesmaximal, characteristic, complemented
4#31826438C211yesmaximal, characteristic
5#31826222C211yesmaximal, characteristic, complemented
6#31826221C211yesmaximal, characteristic, complemented
7#419×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9#22583618—22no—
10#22583616—22no—
11C2×#17985376C2211yescharacteristic
12M16×D283—22no—
13#22583608C412yescomplemented
14#22583599C412yescomplemented
15#22583593—22no—
16#22583586—22no—
17#22583569C2211yescharacteristic
18#22583569C2211yescharacteristic
19#22583534—22no—
20#22583524—22no—
21C22×Q8×C283C2211yescharacteristic
22C2×M16×C283C2211yescharacteristic
23#117×C283C2211yescharacteristic
24#103×C283—22no—
25#17985392—21no—
26#17985392—21no—
27#17985392—21no—
28#17985392—21no—
29C2×Dic566C2×C411yescharacteristic
30C2×D1132C2×C411yescharacteristic
31C2×C22⋊D283D412yes—
32C2×C4×D283D412yes—
33#17985376—21no—
34#17985376—21no—
35#17985375—24no—
36C2×C283⋊C8D412yes—
37C8×D283—24no—
38C4⋊Dic283—21no—
39C4⋊Dic283—21no—
40Dic283⋊C4—24no—
41C22⋊Dic283—21no—
42C22⋊Dic283—21no—
43(C2×D283)⋊C4—24no—
44C4×Dic283—21no—
45C4×Dic283C2×C412yes—
46C2×Q8×C283—21no—
47C2×Q8×C283—21no—
48C2×Q8×C283—21no—
49C2×Q8×C283—21no—
50C2×Q8×C283—21no—
51C2×Q8×C283C2×C412yes—
52C22×C4×C283—21no—
53C22×C4×C283C2311yescharacteristic
54M16×C283—21no—
55M16×C283—21no—
56C2×C8×C283D412yes—
57C4⋊C4×C283—21no—
58C22⋊C4×C283—21no—
59C42×C283—21no—
60C22×D283C22⋊C412yes—
61Dic566—21no—
62Dic566—21no—
63D1132—21no—
64D1132—21no—
65C22⋊D283—22no—
66C22⋊D283—22no—
67C2×Dic283—22no—
68C2×Dic283—41no—
69C2×Dic283C22⋊C412yes—
70C4×D283—22no—
71C4×D283—22no—
72C283⋊C8—22no—
73C23×C283C2×D411yescharacteristic
74Q8×C283—22no—
75Q8×C283—22no—
76Q8×C283—41no—
77Q8×C283—41no—
78C2×C4×C283—22no—
79C2×C4×C283—41no—
80C2×C4×C283—41no—
81C2×C4×C283—41no—
82C2×C4×C283C22×C411yescharacteristic
83C2×C4×C283C2×D411yescharacteristic
84C2×C4×C283C2×D411yescharacteristic
85C2×C4×C283C2×D411yescharacteristic
86C2×C4×C283#4111yescharacteristic
87C2×C4×C283#4111yescharacteristic
88C8×C283—22no—
89C22×C283—41no—
90C22×C283C2×C22⋊C411yescharacteristic
91C22×C283#12111yescharacteristic
92C22×C283#12111yescharacteristic
93C2×D283—22no—
94C2×D283—42no—
95C4×C283—42no—
96C4×C283—81no—
97C4×C283C4×D411yescharacteristic
98C4×C283C4×D411yescharacteristic
99C4×C283#12011yescharacteristic
100C4×C283#12311yescharacteristic
101Dic283—22no—
102Dic283—42no—
103Dic283—81no—
104C2×C283—21no—
105C2×C283—21no—
106C2×C283—21no—
107C2×C283#38611yescharacteristic
108D283—42no—
109C283#190911yescharacteristic, complemented, Sylow
110#1909—2831nomaximal, Sylow
111#568—2831no—
112#536—2831no—
113#419—2831no—
114#419D28311yescharacteristic, complemented
115#410—2831no—
116H(3,ℤ/4)—2831no—
117H(3,ℤ/4)—2831no—
118C2×#41—2831no—
119C22×Q8C2×D28311yescharacteristic
120#131—5662no—
121#131—5662no—
122C2×M16—2831no—
123C2×M16—5662no—
124C2×M16C2×D28311yescharacteristic
125C4⋊Q8—2832no—
126#124—2832no—
127#123—5662no—
128C22⋊Q8—5662no—
129#117—2831no—
130#117—2831no—
131#117C2×D28311yescharacteristic
132#103—22no—
133#103—5662no—
134#99—5662no—
135#41—5661no—
136#41—5661no—
137#41—5661no—
138#41—5661no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—2831no—
145C2×Q8C4×D28312yes—
146C2×D4—2832no—
147C2×D4—2831no—
148C22×C4—21no—
149C22×C4—2832no—
150C22×C4C22×D28311yescharacteristic
151M16—21no—
152M16—21no—
153M16—5664no—
154M16—5661no—
155M16—5661no—
156C2×C8—2832no—
157C2×C8—5664no—
158C2×C8C22⋊D28312yes—
159C4⋊C4—21no—
160C4⋊C4—5664no—
161C4⋊C4—5661no—
162C4⋊C4—5661no—
163C22⋊C4—21no—
164C22⋊C4—5664no—
165C22⋊C4—5661no—
166C22⋊C4—5661no—
167C42—21no—
168C42—2832no—
169C42—5661no—
170C23—2832no—
171C23D4×D28311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—5661no—
177Q8—5661no—
178D4—5662no—
179D4—5662no—
180D4—5661no—
181D4—5661no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2832no—
187C2×C4—5662no—
188C2×C4—5662no—
189C2×C4—5662no—
190C2×C4—11321no—
191C2×C4C2×C4×D28311yescharacteristic
192C2×C4D4×D28311yescharacteristic
193C2×C4D4×D28311yescharacteristic
194C2×C4C2×C22⋊D28311yescharacteristic
195C2×C4#1798539111yescharacteristic
196C2×C4#1798539211yescharacteristic
197C8—22no—
198C8—5662no—
199C22—41no—
200C22—5662no—
201C22—11322no—
202C22C22⋊C4×D28311yescharacteristic
203C22#2258358111yescharacteristic
204C22#2258358311yescharacteristic
205C4—42no—
206C4—81no—
207C4—5662no—
208C4—11322no—
209C4—22641no—
210C4(C22⋊D283)⋊C411yescharacteristic
211C4C4×C22⋊D28311yescharacteristic
212C4#2258357811yescharacteristic
213C4#2258359211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—11322no—
218C2#3182634111yescharacteristic
subgroupG/H
centerC2#31826341
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting#419×C283C2
socleC2×C283#386
prime radicalC2×#17985392C22
cosocleC22×C4×C283C23
hypercenter#419D283
hypocenterC283#1909
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #31826474C2semidirect
5. #31826222C2semidirect
6. #31826221C2semidirect
7. #419×C283C2semidirect
13. #22583608C4semidirect
14. #22583599C4semidirect
109. C283#1909semidirect
114. #419D283semidirect

Maximal subgroups

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

subgroupindex
#318272312
#318270392
#318264742
#318264382
#318262222
#318262212
#419×C2832
#1909283