Finite Group Databasebeta

← #24909115

Subgroups — #24909115

Order 10624.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#19128273C211yesmaximal, characteristic
2#19127682C211yesmaximal, characteristic
3#19127673C211yesmaximal, characteristic
4#19127492C211yesmaximal, characteristic, complemented
5#19127491C211yesmaximal, characteristic, complemented
6#19127490C211yesmaximal, characteristic, complemented
7#352×C83C211yesmaximal, characteristic, complemented
8C2×#13050589C2211yescharacteristic
9C22×Dic166C411yescharacteristic, complemented
10M16×D83—21no—
11#14581791—21no—
12#14581778—21no—
13#14581776C411yescharacteristic, complemented
14#14581775C411yescharacteristic, complemented
15#14581773C2211yescharacteristic
16#14581770—21no—
17#14581769—21no—
18#14581748—21no—
19#14581748C411yescharacteristic, complemented
20#14581710—21no—
21#14581709C2211yescharacteristic
22#14581707C2211yescharacteristic
23#14581706—21no—
24#123×C83C2211yescharacteristic
25#100×C83C2211yescharacteristic
26#99×C83C2211yescharacteristic
27#13050589—41no—
28#13050589—41no—
29C2×Dic166—21no—
30C2×Dic166—21no—
31C2×Dic166—41no—
32C2×Dic166—41no—
33C2×Dic166D411yescharacteristic
34C2×C22⋊D83C2×C411yescharacteristic
35C2×C22⋊D83D411yescharacteristic
36C22×Dic83C2×C411yescharacteristic
37C22×Dic83C2×C411yescharacteristic
38C22×Dic83D411yescharacteristic
39C2×C4×D83D411yescharacteristic
40#13050574—21no—
41#13050573—41no—
42C8×D83—41no—
43C4⋊Dic83—21no—
44Dic83⋊C4—21no—
45Dic83⋊C4—21no—
46Dic83⋊C4—21no—
47Dic83⋊C4—41no—
48C22⋊Dic83—21no—
49C22⋊Dic83—21no—
50C22⋊Dic83C2×C411yescharacteristic
51(C2×D83)⋊C4—21no—
52(C2×D83)⋊C4—21no—
53C4×Dic83—21no—
54C4×Dic83—41no—
55C2×D4×C83C2311yescharacteristic
56C22×C4×C83C2×C411yescharacteristic
57M16×C83—21no—
58C4⋊C4×C83—21no—
59C22⋊C4×C83—21no—
60C22⋊C4×C83—21no—
61C22⋊C4×C83C2×C411yescharacteristic
62C23×C83C22×C411yescharacteristic
63C23×C83C2×D411yescharacteristic
64C22×D83C22⋊C411yescharacteristic
65D4×C83—41no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—41no—
69C2×C4×C83—41no—
70C2×C4×C83C2×D411yescharacteristic
71C22⋊D83—41no—
72C22⋊D83—41no—
73C2×Dic83—21no—
74C2×Dic83—21no—
75C2×Dic83—21no—
76C2×Dic83—21no—
77C2×Dic83—21no—
78C2×Dic83—21no—
79C2×Dic83—41no—
80C2×Dic83—41no—
81C2×Dic83C22⋊C411yescharacteristic
82C2×Dic83C22⋊C411yescharacteristic
83C2×Dic83C22⋊C411yescharacteristic
84C4×D83—21no—
85C4×D83—21no—
86Dic166—41no—
87Dic166—42no—
88Dic166—41no—
89Dic166—41no—
90C8×C83—41no—
91C83⋊C8—41no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83—41no—
95C22×C83—41no—
96C22×C83C2×C22⋊C411yescharacteristic
97C2×D83—21no—
98C2×D83—41no—
99C4×C83—21no—
100C4×C83—41no—
101C4×C83—41no—
102C4×C83—81no—
103Dic83—21no—
104Dic83—21no—
105Dic83—41no—
106Dic83—41no—
107Dic83—81no—
108Dic83#9911yescharacteristic
109Dic83#9911yescharacteristic
110C2×C83—21no—
111C2×C83—41no—
112C2×C83—41no—
113C2×C83C2×#9911yescharacteristic
114D83—41no—
115#2123—831nomaximal, Sylow
116C83#212311yescharacteristic, complemented, Sylow
117#536—831no—
118#413—831no—
119#410—831no—
120#352—831no—
121#352—831no—
122#352—831no—
123#352D8311yescharacteristic, complemented
124C2×#41—831no—
125C22×Q8—831no—
126C2×M16—1661no—
127C4⋊Q8—1661no—
128#124—1661no—
129#123—831no—
130#123—831no—
131#123—831no—
132#123C2×D8311yescharacteristic
133C22⋊Q8—1661no—
134C22⋊Q8—1661no—
135#117—831no—
136#117—1661no—
137#101—1661no—
138#100—831no—
139#100C2×D8311yescharacteristic
140#99—831no—
141#99—1661no—
142#99C2×D8311yescharacteristic
143#41—3321no—
144#41—3321no—
145C2×Q8—3321no—
146C2×Q8—3321no—
147C2×Q8—831no—
148C2×Q8—1661no—
149C2×Q8—1661no—
150C2×D4—831no—
151C2×D4—831no—
152C2×D4C22×D8311yescharacteristic
153C22×C4—831no—
154C22×C4—831no—
155C22×C4—831no—
156C22×C4—831no—
157C22×C4C4×D8311yescharacteristic
158M16—21no—
159M16—3321no—
160M16—1661no—
161C2×C8—3321no—
162C4⋊C4—21no—
163C4⋊C4—3321no—
164C4⋊C4—1661no—
165C4⋊C4—1661no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—831no—
171C22⋊C4—1661no—
172C22⋊C4—1661no—
173C22⋊C4—1661no—
174C22⋊C4—1661no—
175C22⋊C4C4×D8311yescharacteristic
176C42—3321no—
177C42—1661no—
178C23—831no—
179C23C2×C4×D8311yescharacteristic
180C23D4×D8311yescharacteristic
181Q8—3321no—
182Q8—3321no—
183Q8—3321no—
184Q8—3322no—
185D4—41no—
186D4—3321no—
187D4—3321no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3321no—
193C2×C4—3321no—
194C2×C4—831no—
195C2×C4—831no—
196C2×C4—831no—
197C2×C4—1661no—
198C2×C4—1661no—
199C2×C4—1661no—
200C2×C4—1661no—
201C2×C4—1661no—
202C2×C4—1661no—
203C2×C4—1661no—
204C2×C4—1661no—
205C2×C4D4×D8311yescharacteristic
206C8—41no—
207C8—3321no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3321no—
213C22—1661no—
214C22C22⋊C4×D8311yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—3321no—
220C4—3321no—
221C4—831no—
222C4—831no—
223C4—6641no—
224C4—1661no—
225C4—1661no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—3321no—
230C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC23×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#352×C83C2
socleC2×C83C2×#99
prime radicalC2×#13050589C22
cosocleC2×D4×C83C23
hypercenter#352D83
hypocenterC83#2123
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #19127492C2semidirect
5. #19127491C2semidirect
6. #19127490C2semidirect
7. #352×C83C2semidirect
9. C22×Dic166C4semidirect
13. #14581776C4semidirect
14. #14581775C4semidirect
19. #14581748C4semidirect
116. C83#2123semidirect
123. #352D83semidirect

Maximal subgroups

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

subgroupindex
#191282732
#191276822
#191276732
#191274922
#191274912
#191274902
#352×C832
#212383