Finite Group Databasebeta

← #24918353

Subgroups — #24918353

Order 10624.

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

normal subgroups only

This group has 378 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#19128274C212yesmaximal
2#19128044C212yesmaximal, complemented
3#19128041C212yesmaximal, complemented
4#19127957C212yesmaximal, complemented
5#19127942C211yesmaximal, characteristic, complemented
6#19127858C212yesmaximal, complemented
7#19127855C212yesmaximal, complemented
8#19127667C211yesmaximal, characteristic
9#467×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C2×Q8×D83C2212yes—
12C2×Dic332C2211yescharacteristic
13C2×#13050580C2212yes—
14C2×#13050574C2211yescharacteristic
15#14581791—22no—
16#14581778—22no—
17#14581777—22no—
18#14581774—22no—
19#14581772C2212yes—
20#14581771C2212yes—
21#14581768C2212yes—
22#14581750C2212yes—
23#14581726C2211yescharacteristic
24#14581723C2212yes—
25#14581717C2212yes—
26#14581716C2212yes—
27#14581715C2212yes—
28(C2×D83)⋊C8C2212yes—
29(C2×D83)⋊C8C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#107×C83C2211yescharacteristic
34#103×C83C2212yes—
35#13050590—41no—
36#13050590—41no—
37C2×Dic166—22no—
38C2×Dic166D411yescharacteristic
39C2×Dic166C2312yes—
40Q8×D83—42no—
41Q8×D83—42no—
42C2×D332D411yescharacteristic
43C2×C22⋊D83D412yes—
44C2×C4×D83—22no—
45C2×C4×D83C2312yes—
46Dic332—41no—
47#13050580—42no—
48#13050574—41no—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—22no—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58C4×Dic83—22no—
59C2×Q8×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61Q16×C83—41no—
62C2×C8×C83—41no—
63C2×C8×C83C2311yescharacteristic
64C2×C8×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C23×C83C2×D411yescharacteristic
69C22×D83C2×D412yes—
70Q8×C83—22no—
71Q8×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—42no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83C2×D412yes—
82D332—41no—
83C4×D83—22no—
84C4×D83—22no—
85C4×D83—44no—
86Dic166—22no—
87Dic166—22no—
88Dic166—41no—
89Dic166—44no—
90C8×C83—21no—
91C8×C83—21no—
92C83⋊C8—22no—
93C22×C83—21no—
94C22×C83—21no—
95C22×C83—21no—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—22no—
99C2×D83—22no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83—42no—
103C4×C83H(5,𝔽2)11yescharacteristic
104C4×C83H(5,𝔽2)11yescharacteristic
105Dic83—22no—
106Dic83—42no—
107Dic83—42no—
108C2×C83—41no—
109C2×C83#53411yescharacteristic
110C2×C83#57811yescharacteristic
111C2×C83#57811yescharacteristic
112D83—42no—
113#3202—831nomaximal, Sylow
114C83#320211yescharacteristic, complemented, Sylow
115#536—832no—
116#485—832no—
117#485—832no—
118#470—832no—
119#467—831no—
120#467D8311yescharacteristic, complemented
121#451—832no—
122#451—832no—
123#408—831no—
124C2×#41—831no—
125C22×Q8—832no—
126C2×Q16—832no—
127C2×Q16—831no—
128C2×Q16C2×D8311yescharacteristic
129C2×M16—831no—
130C22×C8C2×D8311yescharacteristic
131C4⋊Q8—1662no—
132#124—1662no—
133#123—1662no—
134#123—1662no—
135C22⋊Q8—832no—
136C22⋊Q8—832no—
137C22⋊Q8—832no—
138C22⋊Q8C2×D8312yes—
139#117—832no—
140#107—831no—
141#107C2×D8311yescharacteristic
142#106—832no—
143#103—832no—
144#103—832no—
145#103—832no—
146#103C2×D8312yes—
147C22⋊C8—832no—
148C22⋊C8—832no—
149#41—3321no—
150#41—3321no—
151C2×Q8—3322no—
152C2×Q8—3322no—
153C2×Q8—831no—
154C2×Q8—832no—
155C2×Q8—1662no—
156C2×Q8C22×D8312yes—
157C2×D4—832no—
158C2×D4—831no—
159C22×C4—832no—
160C22×C4—1662no—
161C22×C4C22×D8311yescharacteristic
162Q16—41no—
163Q16—3321no—
164Q16—3322no—
165M16—3321no—
166C2×C8—41no—
167C2×C8—832no—
168C2×C8C22×D8311yescharacteristic
169C2×C8C22×D8311yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—832no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4C22×D8312yes—
176C22⋊C4—22no—
177C22⋊C4—1662no—
178C22⋊C4—1662no—
179C22⋊C4—1662no—
180C22⋊C4—1662no—
181C42—1662no—
182C23—832no—
183C23D4×D8311yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—3321no—
187Q8—3324no—
188Q8—1662no—
189Q8—1662no—
190D4—3321no—
191D4—3322no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3324no—
197C2×C4—832no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4D4×D8311yescharacteristic
203C2×C4C23×D8311yescharacteristic
204C8—21no—
205C8—21no—
206C8—1662no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—1662no—
211C22—1662no—
212C22—1662no—
213C22C2×D4×D8311yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—3322no—
218C4—3322no—
219C4—1662no—
220C4#1458183711yescharacteristic
221C4#1458183811yescharacteristic
222C2—41no—
223C2—3322no—
224C2#1912825911yescharacteristic
225C2#1912851411yescharacteristic
226C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#467×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenter#467D83
hypocenterC83#3202
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #19128044C2semidirect
3. #19128041C2semidirect
4. #19127957C2semidirect
5. #19127942C2semidirect
6. #19127858C2semidirect
7. #19127855C2semidirect
9. #467×C83C2semidirect
114. C83#3202semidirect
120. #467D83semidirect

Maximal subgroups

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

subgroupindex
#191282742
#191280442
#191280412
#191279572
#191279422
#191278582
#191278552
#191276672
#467×C832
#320283