Finite Group Databasebeta

← #497×D109 #28373468

Subgroups — #497×D109 #28373468

Order 13952.

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×D109C212yesmaximal, complemented
2C2×SD16×D109C212yesmaximal
3C4⋊Q8×D109C211yesmaximal, characteristic, complemented
4#124×D109C211yesmaximal, characteristic
5#20295421C211yesmaximal, characteristic, complemented
6#20295420C211yesmaximal, characteristic, complemented
7#20295419C212yesmaximal, complemented
8#20295418C211yesmaximal, characteristic, complemented
9#20295417C212yesmaximal, complemented
10#97×D109C211yesmaximal, characteristic, complemented
11#497×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2211yescharacteristic
13C2×Q8×D109C2212yes—
14C2×D4×D109C2211yescharacteristic
15C2×Dic436C2212yescomplemented
16C2×#13182019C2212yescomplemented
17C2×#13182019C2212yescomplemented
18Q16×D109—28no—
19C2×#13182018C2212yes—
20C2×#13182017C2212yes—
21C2×#13182016C2212yes—
22SD16×D109—28no—
23C2×C8×D109C2212yes—
24#16812699C2211yescharacteristic, complemented
25#16812698—22no—
26#16812697C2212yescomplemented
27#16812688C2211yescharacteristic
28#16812687C2211yescharacteristic
29#16812686C2211yescharacteristic
30#16812685—22no—
31C4⋊C4×D109—22no—
32C22⋊C4×D109—22no—
33C42×D109C2211yescharacteristic
34#16812610C2211yescharacteristic, complemented
35#16812609C2212yescomplemented
36C2×Q16×C109C2212yescomplemented
37C2×SD16×C109C2212yes—
38C4⋊Q8×C109C2211yescharacteristic, complemented
39#124×C109C2211yescharacteristic
40#97×C109C2211yescharacteristic, complemented
41C23×D109—21no—
42C2×Dic218—21no—
43C2×Dic218—22no—
44C2×Dic218C2311yescharacteristic
45C2×Dic218C2312yes—
46Q8×D109—24no—
47Q8×D109—24no—
48Q8×D109—24no—
49C2×D436C2311yescharacteristic
50C2×C22⋊D109—21no—
51D4×D109—24no—
52C2×C4×D109—21no—
53C2×C4×D109—22no—
54C2×C4×D109D412yes—
55C2×C4×D109C2311yescharacteristic
56Dic436—24no—
57#13182019—24no—
58#13182019—24no—
59#13182018—24no—
60#13182017—24no—
61#13182016—24no—
62C2×#11724784C2312yes—
63C8×D109D418yes—
64C4⋊Dic109—22no—
65#13182008—24no—
66C22⋊Dic109—22no—
67#13182006—22no—
68#13182006—22no—
69C4×Dic109D412yes—
70C4×Dic109C2311yescharacteristic
71C2×Q8×C109C2311yescharacteristic
72C2×Q8×C109C2312yes—
73C2×D4×C109C2311yescharacteristic
74Q16×C109—24no—
75SD16×C109—24no—
76C2×C8×C109C2312yes—
77C4⋊C4×C109—22no—
78C22⋊C4×C109—22no—
79C42×C109C2311yescharacteristic
80C23×C109—21no—
81C22×D109—21no—
82C22×D109—42no—
83C22×D109—44no—
84C22×D109C2×D411yescharacteristic
85Q8×C109—22no—
86Q8×C109—22no—
87Q8×C109—22no—
88D4×C109—22no—
89C2×C4×C109—21no—
90C2×C4×C109—22no—
91C2×C4×C109C2×D412yes—
92C2×C4×C109C2411yescharacteristic
93C22⋊D109—42no—
94C2×Dic109—21no—
95C2×Dic109—22no—
96C2×Dic109C2×D412yes—
97C2×Dic109C2×D411yescharacteristic
98D436—22no—
99C4×D109—24no—
100C4×D109—42no—
101C4×D109—44no—
102C4×D109C2×D414yes—
103Dic218—22no—
104Dic218—22no—
105Dic218—22no—
106Dic218—44no—
107Dic218—42no—
108C8×C109C2×D414yes—
109#11724784C2×D414yes—
110C22×C109—41no—
111C22×C109—42no—
112C22×C109C22×D411yescharacteristic
113C2×D109—41no—
114C2×D109—42no—
115C2×D109—84no—
116C2×D109#12712yes—
117C2×D109#13714yes—
118C4×C109—22no—
119C4×C109—41no—
120C4×C109—42no—
121C4×C109C22×D412yes—
122Dic109—22no—
123Dic109—42no—
124Dic109—41no—
125Dic109#12712yes—
126C2×C109—81no—
127C2×C109C2×#12711yescharacteristic
128C2×C109C2×#13712yes—
129D109—81no—
130D109#49714yesdirect product factor
131C2×#497—1091nomaximal, Sylow
132C109C2×#49711yescharacteristic, complemented, Sylow
133C22×Q16—1092no—
134C22×SD16—1092no—
135C2×C4⋊Q8—1091no—
136C2×#124—1091no—
137#497—1092no—
138#497—1092no—
139#497—1091no—
140#497—1091no—
141#497—1091no—
142#497D10911yescharacteristic, direct product factor
143C2×#97—1091no—
144C22×Q8—1091no—
145C22×Q8—1092no—
146C22×D4—1091no—
147C2×Q16—1092no—
148C2×Q16—1092no—
149C2×Q16—1092no—
150C2×Q16—2188no—
151C2×Q16C2×D10912yescomplemented
152C2×SD16—1092no—
153C2×SD16—1092no—
154C2×SD16—1092no—
155C2×SD16—2188no—
156C2×SD16C2×D10912yes—
157C22×C8—1092no—
158C4⋊Q8—1092no—
159C4⋊Q8—1091no—
160C4⋊Q8—2182no—
161C4⋊Q8C2×D10911yescharacteristic, complemented
162#124—1091no—
163#124—1091no—
164#124—1091no—
165#124—2182no—
166#124C2×D10911yescharacteristic
167C2×C4⋊C4—2182no—
168C2×C22⋊C4—2182no—
169C2×C42—1091no—
170#97—1092no—
171#97—1091no—
172#97C2×D10911yescharacteristic, complemented
173C24—2181no—
174C2×Q8—1091no—
175C2×Q8—1092no—
176C2×Q8—2181no—
177C2×Q8—2182no—
178C2×Q8—2184no—
179C2×Q8—2184no—
180C2×Q8—2184no—
181C2×Q8C22×D10911yescharacteristic
182C2×Q8C22×D10912yes—
183C2×D4—1091no—
184C2×D4—2184no—
185C2×D4—2181no—
186C2×D4C22×D10911yescharacteristic
187C22×C4—1091no—
188C22×C4—1092no—
189C22×C4—2181no—
190C22×C4—2182no—
191Q16—24no—
192Q16—2184no—
193Q16—2184no—
194Q16—2184no—
195SD16—24no—
196SD16—2184no—
197SD16—2184no—
198SD16—2184no—
199C2×C8—1092no—
200C2×C8—1098no—
201C2×C8C22×D10912yes—
202C4⋊C4—22no—
203C4⋊C4—2184no—
204C4⋊C4—2182no—
205C22⋊C4—22no—
206C22⋊C4—2182no—
207C22⋊C4—2182no—
208C22⋊C4—2182no—
209C42—1092no—
210C42—1091no—
211C42C22×D10911yescharacteristic
212C23—21no—
213C23—1091no—
214C23—4362no—
215C23—4364no—
216C23—2181no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—4362no—
221Q8—4364no—
222Q8—2182no—
223Q8—2182no—
224Q8—2182no—
225D4—22no—
226D4—4362no—
227D4—2182no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—1091no—
231C2×C4—1094no—
232C2×C4—1092no—
233C2×C4—4362no—
234C2×C4—4364no—
235C2×C4—2184no—
236C2×C4—2181no—
237C2×C4—2182no—
238C2×C4D4×D10912yes—
239C2×C4C23×D10911yescharacteristic
240C8—1094no—
241C8D4×D10914yes—
242C22—42no—
243C22—41no—
244C22—8724no—
245C22—1094no—
246C22—1092no—
247C22—4362no—
248C22—4361no—
249C22C2×D4×D10911yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—1092no—
254C4—4361no—
255C4—4362no—
256C4—2182no—
257C4C2×D4×D10912yes—
258C2—81no—
259C2—8721no—
260C2—1094no—
261C2#127×D10911yescharacteristic
262C2#137×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#497×C109C2
socleC22×C109C22×D4
prime radicalC2×D4×D109C22
hypercenter#497D109
hypocenterC109C2×#497
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D109C2semidirect
3. C4⋊Q8×D109C2semidirect
5. #20295421C2semidirect
6. #20295420C2semidirect
7. #20295419C2semidirect
8. #20295418C2semidirect
9. #20295417C2semidirect
10. #97×D109C2semidirect
11. #497×C109C2semidirect
15. C2×Dic436C22semidirect
16. C2×#13182019C22semidirect
17. C2×#13182019C22semidirect
24. #16812699C22semidirect
26. #16812697C22semidirect
34. #16812610C22semidirect
35. #16812609C22semidirect
36. C2×Q16×C109C22semidirect
38. C4⋊Q8×C109C22semidirect
40. #97×C109C22semidirect
130. D109#497direct (N × H)
132. C109C2×#497semidirect
142. #497D109direct (N × H)
151. C2×Q16C2×D109semidirect
161. C4⋊Q8C2×D109semidirect
172. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1092
C2×SD16×D1092
C4⋊Q8×D1092
#124×D1092
#202954212
#202954202
#202954192
#202954182
#202954172
#97×D1092
#497×C1092
C2×#497109