Finite Group Databasebeta

← #37583446

Subgroups — #37583446

Order 24448.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133214C211yesmaximal, characteristic
2C4⋊Q8×D191C211yesmaximal, characteristic, complemented
3#27133037C211yesmaximal, characteristic, complemented
4#27133036C211yesmaximal, characteristic, complemented
5#27133015C211yesmaximal, characteristic, complemented
6#27132867C214yesmaximal, complemented
7#27132803C214yesmaximal, complemented
8#27132714C211yesmaximal, characteristic, complemented
9#487×C191C211yesmaximal, characteristic, complemented
10C2×#15690288—21no—
11C2×Q8×D191C2212yes—
12C2×#15690276C2214yes—
13C2×#15690274C2214yes—
14#20259322C2211yescharacteristic
15#20259321—21no—
16#20259321—21no—
17#20259320C2212yes—
18#20259319C2211yescharacteristic
19#20259311—21no—
20#20259297—21no—
21#20259297C2212yes—
22#20259284—21no—
23#20259284C2212yes—
24C4⋊C4×D191—21no—
25C4⋊C4×D191—21no—
26C42×D191C2211yescharacteristic
27Dic191⋊C8C2212yes—
28#20259245C2214yes—
29(C2×D191)⋊C8C2212yes—
30C4×C191⋊C8C2211yescharacteristic
31C4⋊Q8×C191C2211yescharacteristic
32C4⋊Q8×C191C2211yescharacteristic
33#103×C191C2214yes—
34C4×C8×C191C2211yescharacteristic
35#15690288—44no—
36C2×Dic382—22no—
37C2×Dic382C2312yes—
38Q8×D191—44no—
39C2×D764—22no—
40C2×D764—21no—
41C2×D764—21no—
42C2×D764C2312yes—
43C2×C4×D191—22no—
44C2×C4×D191—22no—
45C2×C4×D191D411yescharacteristic
46C2×C4×D191D411yescharacteristic
47C2×C4×D191C2311yescharacteristic
48#15690276—44no—
49#15690274—44no—
50C2×C191⋊C8C2312yes—
51C4⋊Dic191—21no—
52C4⋊Dic191—21no—
53Dic191⋊C4—22no—
54Dic191⋊C4—22no—
55(C2×D191)⋊C4—22no—
56(C2×D191)⋊C4—22no—
57C4×Dic191D411yescharacteristic
58C4×Dic191D411yescharacteristic
59C4×Dic191C2311yescharacteristic
60C2×Q8×C191—21no—
61C2×Q8×C191C2312yes—
62C2×C8×C191C2312yes—
63C4⋊C4×C191—21no—
64C4⋊C4×C191—21no—
65C4⋊C4×C191—21no—
66C4⋊C4×C191C2312yes—
67C42×C191C2311yescharacteristic
68C22×D191—22no—
69C22×D191C2×D411yescharacteristic
70Dic382—24no—
71Dic382—44no—
72D764—24no—
73D764—44no—
74D764—42no—
75D764—42no—
76C2×Dic191—22no—
77C2×Dic191C2×D411yescharacteristic
78C2×Dic191C2×D411yescharacteristic
79C2×Dic191C2×D411yescharacteristic
80C4×D191—22no—
81C4×D191—22no—
82C4×D191—44no—
83C4×D191—44no—
84C4×D191SD1614yescomplemented
85C191⋊C8—22no—
86Q8×C191—24no—
87Q8×C191—42no—
88C2×C4×C191—22no—
89C2×C4×C191—22no—
90C2×C4×C191C2×D411yescharacteristic
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2411yescharacteristic
93C8×C191—22no—
94C22×C191C22×D411yescharacteristic
95C2×D191—21no—
96C2×D191—21no—
97C2×D191—42no—
98C2×D191—42no—
99C2×D191—42no—
100C2×D191C2×SD1612yes—
101C4×C191—42no—
102C4×C191—42no—
103C4×C191C2×SD1612yes—
104C4×C191#13712yes—
105C4×C191H(5,𝔽2)12yes—
106Dic191—21no—
107Dic191—21no—
108Dic191—42no—
109Dic191C2×SD1612yes—
110C2×C191#53511yescharacteristic
111C2×C191C22×SD1611yescharacteristic
112C2×C191C2×#13711yescharacteristic
113D191—22no—
114D191—82no—
115C191#320611yescharacteristic, complemented, Sylow
116#3206—1911nomaximal, Sylow
117#532—1911no—
118C2×C4⋊Q8—1911no—
119#492—1911no—
120#492—1911no—
121#487—1911no—
122#487D19111yescharacteristic, complemented
123#461—1914no—
124#448—1914no—
125#431—1911no—
126C2×#41—3821no—
127C22×Q8—1912no—
128C2×SD16—1914no—
129C2×SD16—1914no—
130C4⋊Q8—3821no—
131C4⋊Q8—3821no—
132C4⋊Q8—1912no—
133C4⋊Q8—1911no—
134C4⋊Q8C2×D19111yescharacteristic
135C4⋊Q8C2×D19111yescharacteristic
136#127—1911no—
137#124—3821no—
138#121—3821no—
139#121—1912no—
140C4×D4—3821no—
141C4×D4—1912no—
142C2×C4⋊C4—3821no—
143C2×C4⋊C4—3821no—
144C2×C42—1911no—
145C4⋊C8—1912no—
146#103—1914no—
147#103C2×D19114yes—
148C22⋊C8—1912no—
149C4×C8—1911no—
150C4×C8C2×D19111yescharacteristic
151#41—7644no—
152C2×Q8—21no—
153C2×Q8—3822no—
154C2×Q8—1912no—
155C2×Q8—7644no—
156C2×Q8C22×D19112yes—
157C2×D4—3822no—
158C2×D4—3821no—
159C2×D4—3821no—
160C2×D4—1912no—
161C22×C4—3822no—
162C22×C4—3822no—
163C22×C4—1911no—
164C22×C4—1911no—
165C22×C4—1911no—
166SD16—7644no—
167SD16—7644no—
168C2×C8—1912no—
169C2×C8C22×D19112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—3822no—
174C4⋊C4—3822no—
175C4⋊C4—3821no—
176C4⋊C4—3821no—
177C4⋊C4C22×D19112yes—
178C22⋊C4—3822no—
179C22⋊C4—3822no—
180C42—1911no—
181C42—1911no—
182C42—1911no—
183C42C22×D19111yescharacteristic
184C23—3822no—
185C23—1911no—
186Q8—24no—
187Q8—42no—
188Q8—3824no—
189Q8—7644no—
190D4—3824no—
191D4—7644no—
192D4—7642no—
193D4—7642no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3822no—
197C2×C4—3822no—
198C2×C4—3822no—
199C2×C4—1911no—
200C2×C4—1911no—
201C2×C4—1911no—
202C2×C4—1914no—
203C2×C4—7644no—
204C2×C4—7644no—
205C2×C4D4×D19111yescharacteristic
206C2×C4D4×D19111yescharacteristic
207C2×C4C23×D19111yescharacteristic
208C8—22no—
209C8—3822no—
210C22—3821no—
211C22—3821no—
212C22—1912no—
213C22—7642no—
214C22—7642no—
215C22—7642no—
216C22C2×D4×D19111yescharacteristic
217C4—42no—
218C4—42no—
219C4—3821no—
220C4—3821no—
221C4—1912no—
222C4—7642no—
223C4SD16×D19112yes—
224C4#2025935412yes—
225C4#2025936812yes—
226C2—3822no—
227C2—15282no—
228C2#2713323111yescharacteristic
229C2C2×SD16×D19111yescharacteristic
230C2C2×#2025935411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#487×C191C2
socleC22×C191C22×D4
prime radical#27133214C2
hypercenter#487D191
hypocenterC191#3206
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D191C2semidirect
3. #27133037C2semidirect
4. #27133036C2semidirect
5. #27133015C2semidirect
6. #27132867C2semidirect
7. #27132803C2semidirect
8. #27132714C2semidirect
9. #487×C191C2semidirect
84. C4×D191SD16semidirect
115. C191#3206semidirect
122. #487D191semidirect

Maximal subgroups

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

subgroupindex
#271332142
C4⋊Q8×D1912
#271330372
#271330362
#271330152
#271328672
#271328032
#271327142
#487×C1912
#3206191