Finite Group Databasebeta

← #497×D191 #37582956

Subgroups — #497×D191 #37582956

Order 24448.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×Q16×D191C212yesmaximal, complemented
2C2×SD16×D191C212yesmaximal
3C4⋊Q8×D191C211yesmaximal, characteristic, complemented
4#124×D191C211yesmaximal, characteristic
5#27133058C211yesmaximal, characteristic, complemented
6#27133057C211yesmaximal, characteristic, complemented
7#27133056C212yesmaximal, complemented
8#27133055C211yesmaximal, characteristic, complemented
9#27133054C212yesmaximal, complemented
10#97×D191C211yesmaximal, characteristic, complemented
11#497×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2212yes—
13C2×Q8×D191C2211yescharacteristic
14C2×D4×D191C2211yescharacteristic
15C2×Dic764C2212yescomplemented
16C2×#15690277C2212yescomplemented
17C2×#15690277C2212yescomplemented
18Q16×D191—28no—
19C2×#15690276C2212yes—
20C2×#15690275C2212yes—
21C2×#15690274C2212yes—
22SD16×D191—28no—
23C2×C8×D191C2212yes—
24#20259322C2211yescharacteristic, complemented
25#20259321—22no—
26#20259320C2212yescomplemented
27#20259311C2211yescharacteristic
28#20259310C2211yescharacteristic
29#20259309C2211yescharacteristic
30#20259308—22no—
31C4⋊C4×D191—22no—
32C22⋊C4×D191—22no—
33C42×D191C2211yescharacteristic
34#20259233C2211yescharacteristic, complemented
35#20259232C2212yescomplemented
36C2×Q16×C191C2212yescomplemented
37C2×SD16×C191C2212yes—
38C4⋊Q8×C191C2211yescharacteristic, complemented
39#124×C191C2211yescharacteristic
40#97×C191C2211yescharacteristic, complemented
41C23×D191—21no—
42C2×Dic382—21no—
43C2×Dic382—22no—
44C2×Dic382C2312yes—
45C2×Dic382C2311yescharacteristic
46Q8×D191—24no—
47Q8×D191—24no—
48Q8×D191—24no—
49C2×D764C2311yescharacteristic
50C2×C22⋊D191—21no—
51D4×D191—24no—
52C2×C4×D191—21no—
53C2×C4×D191—22no—
54C2×C4×D191D412yes—
55C2×C4×D191C2311yescharacteristic
56Dic764—24no—
57#15690277—24no—
58#15690277—24no—
59#15690276—24no—
60#15690275—24no—
61#15690274—24no—
62C2×C191⋊C8C2312yes—
63C8×D191D418yes—
64C4⋊Dic191—22no—
65Dic191⋊C4—24no—
66C22⋊Dic191—22no—
67(C2×D191)⋊C4—22no—
68(C2×D191)⋊C4—22no—
69C4×Dic191D412yes—
70C4×Dic191C2311yescharacteristic
71C2×Q8×C191C2312yes—
72C2×Q8×C191C2311yescharacteristic
73C2×D4×C191C2311yescharacteristic
74Q16×C191—24no—
75SD16×C191—24no—
76C2×C8×C191C2312yes—
77C4⋊C4×C191—22no—
78C22⋊C4×C191—22no—
79C42×C191C2311yescharacteristic
80C22×D191—21no—
81C22×D191—42no—
82C22×D191—44no—
83C22×D191C2×D411yescharacteristic
84Dic382—22no—
85Dic382—22no—
86Dic382—22no—
87Dic382—44no—
88Dic382—42no—
89D764—22no—
90C22⋊D191—42no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191C2×D412yes—
94C2×Dic191C2×D411yescharacteristic
95C4×D191—24no—
96C4×D191—42no—
97C4×D191—44no—
98C4×D191C2×D414yes—
99C191⋊C8C2×D414yes—
100C23×C191—21no—
101Q8×C191—22no—
102Q8×C191—22no—
103Q8×C191—22no—
104D4×C191—22no—
105C2×C4×C191—21no—
106C2×C4×C191—22no—
107C2×C4×C191C2×D412yes—
108C2×C4×C191C2411yescharacteristic
109C8×C191C2×D414yes—
110C22×C191—41no—
111C22×C191—42no—
112C22×C191C22×D411yescharacteristic
113C2×D191—41no—
114C2×D191—42no—
115C2×D191—84no—
116C2×D191#12712yes—
117C2×D191#13714yes—
118C4×C191—22no—
119C4×C191—41no—
120C4×C191—42no—
121C4×C191C22×D412yes—
122Dic191—22no—
123Dic191—41no—
124Dic191—42no—
125Dic191#12712yes—
126C2×C191—81no—
127C2×C191C2×#12711yescharacteristic
128C2×C191C2×#13712yes—
129D191—81no—
130D191#49714yesdirect product factor
131C191C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1911nomaximal, Sylow
133C22×Q16—1912no—
134C22×SD16—1912no—
135C2×C4⋊Q8—1911no—
136C2×#124—1911no—
137#497—1912no—
138#497—1912no—
139#497—1911no—
140#497—1911no—
141#497—1911no—
142#497D19111yescharacteristic, direct product factor
143C2×#97—1911no—
144C22×Q8—1911no—
145C22×Q8—1912no—
146C22×D4—1911no—
147C2×Q16—3828no—
148C2×Q16—1912no—
149C2×Q16—1912no—
150C2×Q16—1912no—
151C2×Q16C2×D19112yescomplemented
152C2×SD16—3828no—
153C2×SD16—1912no—
154C2×SD16—1912no—
155C2×SD16—1912no—
156C2×SD16C2×D19112yes—
157C22×C8—1912no—
158C4⋊Q8—3822no—
159C4⋊Q8—1912no—
160C4⋊Q8—1911no—
161C4⋊Q8C2×D19111yescharacteristic, complemented
162#124—3822no—
163#124—1911no—
164#124—1911no—
165#124—1911no—
166#124C2×D19111yescharacteristic
167C2×C4⋊C4—3822no—
168C2×C22⋊C4—3822no—
169C2×C42—1911no—
170#97—1912no—
171#97—1911no—
172#97C2×D19111yescharacteristic, complemented
173C24—3821no—
174C2×Q8—3821no—
175C2×Q8—3822no—
176C2×Q8—3824no—
177C2×Q8—3824no—
178C2×Q8—3824no—
179C2×Q8—1911no—
180C2×Q8—1912no—
181C2×Q8C22×D19111yescharacteristic
182C2×Q8C22×D19112yes—
183C2×D4—3824no—
184C2×D4—3821no—
185C2×D4—1911no—
186C2×D4C22×D19111yescharacteristic
187C22×C4—3821no—
188C22×C4—3822no—
189C22×C4—1912no—
190C22×C4—1911no—
191Q16—24no—
192Q16—3824no—
193Q16—3824no—
194Q16—3824no—
195SD16—24no—
196SD16—3824no—
197SD16—3824no—
198SD16—3824no—
199C2×C8—1912no—
200C2×C8—1918no—
201C2×C8C22×D19112yes—
202C4⋊C4—22no—
203C4⋊C4—3824no—
204C4⋊C4—3822no—
205C22⋊C4—22no—
206C22⋊C4—3822no—
207C22⋊C4—3822no—
208C22⋊C4—3822no—
209C42—1912no—
210C42—1911no—
211C42C22×D19111yescharacteristic
212C23—21no—
213C23—3821no—
214C23—1911no—
215C23—7642no—
216C23—7644no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—3822no—
221Q8—3822no—
222Q8—3822no—
223Q8—7642no—
224Q8—7644no—
225D4—22no—
226D4—3822no—
227D4—7642no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—3824no—
231C2×C4—3821no—
232C2×C4—3822no—
233C2×C4—1912no—
234C2×C4—1911no—
235C2×C4—1914no—
236C2×C4—7642no—
237C2×C4—7644no—
238C2×C4D4×D19112yes—
239C2×C4C23×D19111yescharacteristic
240C8—1914no—
241C8D4×D19114yes—
242C22—42no—
243C22—41no—
244C22—1914no—
245C22—1912no—
246C22—15284no—
247C22—7642no—
248C22—7641no—
249C22C2×D4×D19111yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—3822no—
254C4—1912no—
255C4—7641no—
256C4—7642no—
257C4C2×D4×D19112yes—
258C2—81no—
259C2—1914no—
260C2—15281no—
261C2#127×D19111yescharacteristic
262C2#137×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#497×C191C2
socleC22×C191C22×D4
prime radicalC2×D4×D191C22
hypercenter#497D191
hypocenterC191C2×#497
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D191C2semidirect
3. C4⋊Q8×D191C2semidirect
5. #27133058C2semidirect
6. #27133057C2semidirect
7. #27133056C2semidirect
8. #27133055C2semidirect
9. #27133054C2semidirect
10. #97×D191C2semidirect
11. #497×C191C2semidirect
15. C2×Dic764C22semidirect
16. C2×#15690277C22semidirect
17. C2×#15690277C22semidirect
24. #20259322C22semidirect
26. #20259320C22semidirect
34. #20259233C22semidirect
35. #20259232C22semidirect
36. C2×Q16×C191C22semidirect
38. C4⋊Q8×C191C22semidirect
40. #97×C191C22semidirect
130. D191#497direct (N × H)
131. C191C2×#497semidirect
142. #497D191direct (N × H)
151. C2×Q16C2×D191semidirect
161. C4⋊Q8C2×D191semidirect
172. #97C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1912
C2×SD16×D1912
C4⋊Q8×D1912
#124×D1912
#271330582
#271330572
#271330562
#271330552
#271330542
#97×D1912
#497×C1912
C2×#497191