Finite Group Databasebeta

← #24918539

Subgroups — #24918539

Order 10624.

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

normal subgroups only

This group has 364 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
1#19128312C211yesmaximal, characteristic, complemented
2#19128276C211yesmaximal, characteristic
3#19128093C211yesmaximal, characteristic, complemented
4#19128069C211yesmaximal, characteristic, complemented
5#19128056C211yesmaximal, characteristic, complemented
6#19127925C212yesmaximal, complemented
7#19127909C212yesmaximal, complemented
8#19127862C212yesmaximal
9#19127855C212yesmaximal, complemented
10#19127760C211yesmaximal, characteristic, complemented
11#494×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×Q8×D83C2211yescharacteristic
14C2×#13050580C2212yes—
15C2×#13050577C2212yes—
16#14581790C2211yescharacteristic
17#14581787—22no—
18#14581784C2211yescharacteristic
19#14581782—22no—
20#14581781C2211yescharacteristic
21#14581780C2211yescharacteristic
22#14581777—22no—
23#14581770—22no—
24#14581768C2212yes—
25Q8×Dic83C2212yes—
26#14581749C2211yescharacteristic
27Dic83⋊C8C2212yes—
28#14581717C2212yes—
29#14581716C2212yes—
30#14581716C2212yes—
31#14581714C2212yes—
32(C2×D83)⋊C8C2212yes—
33#14581703C2211yescharacteristic
34C2×Q16×C83C2212yes—
35C2×Q16×C83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38C4×C8×C83C2211yescharacteristic
39#13050591—42no—
40C2×Dic166—21no—
41C2×Dic166C2311yescharacteristic
42Q8×D83—42no—
43C2×D332—21no—
44C2×D332C2311yescharacteristic
45C2×C4×D83—21no—
46C2×C4×D83—21no—
47C2×C4×D83C2311yescharacteristic
48#13050580—42no—
49#13050577—42no—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—22no—
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56Dic83⋊C4D412yes—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4D412yes—
61C4×Dic83—22no—
62C4×Dic83C2311yescharacteristic
63C2×Q8×C83C2312yes—
64C2×Q8×C83C2311yescharacteristic
65C2×Q8×C83C2311yescharacteristic
66Q16×C83—42no—
67Q16×C83—42no—
68C2×C8×C83C2312yes—
69C4⋊C4×C83—22no—
70C4⋊C4×C83—22no—
71C42×C83C2311yescharacteristic
72C22×D83—21no—
73C22×D83C2×D411yescharacteristic
74Q8×C83—22no—
75Q8×C83—22no—
76Q8×C83—22no—
77Q8×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2411yescharacteristic
83C2×Dic83—22no—
84C2×Dic83—21no—
85C2×Dic83C2×D412yes—
86C2×Dic83C2×D411yescharacteristic
87D332—22no—
88D332—42no—
89C4×D83—22no—
90C4×D83—42no—
91C4×D83—42no—
92Dic166—22no—
93Dic166—42no—
94C8×C83—22no—
95C83⋊C8—22no—
96C22×C83C22×D411yescharacteristic
97C2×D83—21no—
98C2×D83—22no—
99C2×D83—41no—
100C2×D83—42no—
101C4×C83—22no—
102C4×C83—41no—
103C4×C83—41no—
104C4×C83—42no—
105C4×C83H(5,𝔽2)12yes—
106Dic83—21no—
107Dic83—22no—
108Dic83—41no—
109Dic83—42no—
110C2×C83#53511yescharacteristic
111C2×C83#57812yes—
112D83—41no—
113D83—81no—
114#3223—831nomaximal, Sylow
115C83#322311yescharacteristic, complemented, Sylow
116#541—831no—
117#536—831no—
118#497—831no—
119#494D8311yescharacteristic, complemented
120#491—831no—
121#488—831no—
122#464—832no—
123#462—832no—
124#452—832no—
125#451—832no—
126#433—831no—
127C2×#41—831no—
128C22×Q8—831no—
129C2×Q16—832no—
130C2×Q16C2×D8312yes—
131C2×Q16C2×D8312yes—
132C2×SD16—832no—
133C4⋊Q8—831no—
134C4⋊Q8C2×D8311yescharacteristic
135C4⋊Q8C2×D8311yescharacteristic
136#126—1662no—
137#125—831no—
138#125—1662no—
139#124—831no—
140#124—831no—
141#123—1662no—
142C22⋊Q8—832no—
143C22⋊Q8—1662no—
144C4×Q8—832no—
145#117—831no—
146C4⋊C8—832no—
147#103—832no—
148#103—832no—
149#103—832no—
150#102—832no—
151C22⋊C8—832no—
152#97—831no—
153C4×C8C2×D8311yescharacteristic
154#41—3322no—
155C2×Q8—3322no—
156C2×Q8—831no—
157C2×Q8—1661no—
158C2×Q8C22×D8311yescharacteristic
159C2×Q8C22×D8311yescharacteristic
160C2×Q8C22×D8312yes—
161C2×D4—831no—
162C2×D4—1661no—
163C22×C4—831no—
164C22×C4—1661no—
165C22×C4—1661no—
166Q16—42no—
167Q16—42no—
168Q16—3322no—
169SD16—3322no—
170C2×C8—832no—
171C2×C8C22×D8312yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—832no—
175C4⋊C4—832no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C22⋊C4—832no—
181C22⋊C4—1662no—
182C22⋊C4—1662no—
183C22⋊C4—1662no—
184C42—831no—
185C42—1662no—
186C42C22×D8311yescharacteristic
187C23—831no—
188C23—1661no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—3322no—
194Q8—1662no—
195D4—3322no—
196D4—1662no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—3322no—
201C2×C4—3322no—
202C2×C4—832no—
203C2×C4—831no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4—1662no—
207C2×C4D4×D8312yes—
208C2×C4C23×D8311yescharacteristic
209C8—22no—
210C8—1662no—
211C22—3322no—
212C22—3321no—
213C22—1662no—
214C22—1661no—
215C22C2×D4×D8311yescharacteristic
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—3322no—
221C4—3321no—
222C4—1662no—
223C4—1661no—
224C4#1458183712yes—
225C2—3321no—
226C2—6641no—
227C2#1912826411yescharacteristic
228C2#1912851412yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#494×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050591C22
hypercenter#494D83
hypocenterC83#3223
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #19128312C2semidirect
3. #19128093C2semidirect
4. #19128069C2semidirect
5. #19128056C2semidirect
6. #19127925C2semidirect
7. #19127909C2semidirect
9. #19127855C2semidirect
10. #19127760C2semidirect
11. #494×C83C2semidirect
115. C83#3223semidirect
119. #494D83semidirect

Maximal subgroups

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

subgroupindex
#191283122
#191282762
#191280932
#191280692
#191280562
#191279252
#191279092
#191278622
#191278552
#191277602
#494×C832
#322383