Finite Group Databasebeta

← #24918546

Subgroups — #24918546

Order 10624.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128313C211yesmaximal, characteristic, complemented
2#19128096C211yesmaximal, characteristic, complemented
3#19128080C211yesmaximal, characteristic, complemented
4#19128069C211yesmaximal, characteristic, complemented
5#19128055C211yesmaximal, characteristic, complemented
6#19127920C212yesmaximal, complemented
7#19127908C212yesmaximal, complemented
8#19127863C212yesmaximal
9#19127857C212yesmaximal, complemented
10#19127762C211yesmaximal, characteristic, complemented
11#536×C83C211yesmaximal, characteristic
12C2×#13050580C2212yes—
13C2×#13050580C2212yes—
14C2×#13050580C2212yes—
15C2×#13050578C2212yes—
16#14581792C2211yescharacteristic
17#14581785—22no—
18#14581784C2211yescharacteristic
19#14581782—22no—
20#14581771C2212yes—
21C4×Dic166C2212yes—
22C4⋊(C83⋊C8)C2212yes—
23#14581716C2212yes—
24#14581716C2212yes—
25#14581716C2212yes—
26#14581713C2212yes—
27C22⋊(C83⋊C8)C2212yes—
28#14581703C2211yescharacteristic
29C4×C83⋊C8C2211yescharacteristic
30C2×#41×C83C2211yescharacteristic
31C22×Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33C4⋊Q8×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#124×C83C2211yescharacteristic
36#123×C83—22no—
37C22⋊Q8×C83—22no—
38#117×C83C2211yescharacteristic
39C2×Dic166C2312yes—
40#13050580—42no—
41#13050580—42no—
42#13050580—42no—
43#13050578—42no—
44C2×C83⋊C8C2312yes—
45C2×C83⋊C8C2312yes—
46C4⋊Dic83—22no—
47C4⋊Dic83C2312yes—
48Dic83⋊C4—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51C22⋊Dic83—22no—
52C4×Dic83—22no—
53#41×C83—42no—
54C2×Q8×C83—21no—
55C2×Q8×C83—42no—
56C2×Q8×C83C2311yescharacteristic
57C2×Q8×C83C2311yescharacteristic
58C2×Q8×C83C2311yescharacteristic
59C2×D4×C83—21no—
60C2×D4×C83C2311yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83C2311yescharacteristic
64C4⋊C4×C83—22no—
65C4⋊C4×C83—22no—
66C4⋊C4×C83D412yes—
67C22⋊C4×C83—22no—
68C22⋊C4×C83—22no—
69C22⋊C4×C83D412yes—
70C42×C83C2311yescharacteristic
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73C23×C83C2×D411yescharacteristic
74Q8×C83—22no—
75Q8×C83—22no—
76Q8×C83—22no—
77Q8×C83—42no—
78D4×C83—22no—
79D4×C83—42no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—42no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C2×Dic83—22no—
91C2×Dic83—22no—
92Dic166—22no—
93Dic166—22no—
94C83⋊C8—22no—
95C83⋊C8—22no—
96C22×C83—21no—
97C22×C83—22no—
98C22×C83—41no—
99C22×C83—42no—
100C22×C83C22×D411yescharacteristic
101C4×C83—21no—
102C4×C83—22no—
103C4×C83—22no—
104C4×C83—41no—
105C4×C83—41no—
106C4×C83—41no—
107C4×C83H(5,𝔽2)12yes—
108Dic83—42no—
109Dic83—42no—
110C2×C83—41no—
111C2×C83—81no—
112C2×C83#53511yescharacteristic
113C2×C83#57812yes—
114#3223—831nomaximal, Sylow
115C83#322311yescharacteristic, complemented, Sylow
116#541—831no—
117#536D8311yescharacteristic
118#497—831no—
119#494—831no—
120#491—831no—
121#488—831no—
122#464—832no—
123#462—832no—
124#452—832no—
125#451—832no—
126#433—831no—
127C2×#41C2×D8311yescharacteristic
128C22×Q8C2×D8311yescharacteristic
129C2×Q16—832no—
130C2×Q16—832no—
131C2×Q16—832no—
132C2×SD16—832no—
133C4⋊Q8—831no—
134C4⋊Q8C2×D8311yescharacteristic
135C4⋊Q8C2×D8311yescharacteristic
136#126—1662no—
137#125—831no—
138#125—1662no—
139#124C2×D8311yescharacteristic
140#124C2×D8311yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—832no—
144C4×Q8—832no—
145#117C2×D8311yescharacteristic
146C4⋊C8—832no—
147#103—832no—
148#103—832no—
149#103—832no—
150#102—832no—
151C22⋊C8—832no—
152#97—831no—
153C4×C8—831no—
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—832no—
158C2×Q8C22×D8311yescharacteristic
159C2×Q8C22×D8311yescharacteristic
160C2×Q8C22×D8311yescharacteristic
161C2×D4—21no—
162C2×D4C22×D8311yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D8311yescharacteristic
166Q16—3322no—
167Q16—3322no—
168Q16—3322no—
169SD16—3322no—
170C2×C8—832no—
171C2×C8—832no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—832no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4C22⋊D8312yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—1662no—
183C22⋊C4C22⋊D8312yes—
184C42—1662no—
185C42C22×D8311yescharacteristic
186C42C22×D8311yescharacteristic
187C23—21no—
188C23C2×C22⋊D8311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—1662no—
194Q8—1662no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4C2×C22⋊D8312yes—
206C2×C4C2×C22⋊D8312yes—
207C2×C4C2×C22⋊D8311yescharacteristic
208C2×C4C23×D8311yescharacteristic
209C8—1662no—
210C8—1662no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D8311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—3322no—
223C4—3322no—
224C4#1458183712yes—
225C2—41no—
226C2—81no—
227C2#1912826811yescharacteristic
228C2#1912851612yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#536×C83C2
socleC22×C83C22×D4
prime radicalC2×#41×C83C22
hypercenter#536D83
hypocenterC83#3223
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #19128313C2semidirect
2. #19128096C2semidirect
3. #19128080C2semidirect
4. #19128069C2semidirect
5. #19128055C2semidirect
6. #19127920C2semidirect
7. #19127908C2semidirect
9. #19127857C2semidirect
10. #19127762C2semidirect
115. C83#3223semidirect

Maximal subgroups

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

subgroupindex
#191283132
#191280962
#191280802
#191280692
#191280552
#191279202
#191279082
#191278632
#191278572
#191277622
#536×C832
#322383