Finite Group Databasebeta

← #28345972

Subgroups — #28345972

Order 13696.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812137C211yesmaximal, characteristic
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293201C211yesmaximal, characteristic, complemented
4#20293101C211yesmaximal, characteristic, complemented
5#20292845C214yesmaximal, complemented
6#20292831C214yesmaximal, complemented
7#20292655C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812137—24no—
13C2×#13133807C2212yes—
14C2×#13133805—24no—
15C2×#13133805C2214yes—
16C2×D856C2212yes—
17C22×C107⋊C8C2211yescharacteristic
18#16812107C2212yescomplemented
19#16812098C2212yescomplemented
20#16812092—22no—
21#16812085—22no—
22C4×D428—22no—
23#16812071C2214yes—
24#16812068C2211yescharacteristic
25C22⋊C4×D107—22no—
26Dic107⋊C8C2214yescomplemented
27#16812032C2212yescomplemented
28#16812029C2212yescomplemented
29C22×Q8×C107C2211yescharacteristic
30C2×M16×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32#103×C107C2214yescomplemented
33C23×D107—21no—
34#13133818—22no—
35#13133818—22no—
36C2×Dic214C2311yescharacteristic
37C2×D428—21no—
38C2×D428—22no—
39C2×D428D414yes—
40C2×D428C2311yescharacteristic
41C2×D428C2312yes—
42C2×C22⋊D107—21no—
43C2×C4×D107—21no—
44C2×C4×D107—24no—
45#13133807—24no—
46#13133805—28no—
47D856—24no—
48C2×C107⋊C8—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107D412yes—
51Dic107⋊C4—24no—
52C22⋊Dic107D412yes—
53(C2×D107)⋊C4—24no—
54(C2×D107)⋊C4—24no—
55C4×Dic107C2312yes—
56C2×Q8×C107—21no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107C2312yes—
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107C2×D411yescharacteristic
68C22×D107—22no—
69C22×D107—24no—
70C22×D107—21no—
71C22×D107—82no—
72Q8×C107—22no—
73Q8×C107—22no—
74Q8×C107—42no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—42no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D412yes—
80C2×C4×C107#4112yes—
81C2×C4×C107C2411yescharacteristic
82C22⋊D107—42no—
83C2×Dic107—21no—
84C2×Dic107C2×D414yes—
85D428—24no—
86D428—21no—
87D428—22no—
88D428—21no—
89D428C2×D414yes—
90C4×D107—42no—
91C4×D107—84no—
92Dic214—21no—
93Dic214—21no—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C2×D107—24no—
100C2×D107—44no—
101C2×D107—41no—
102C2×D107—42no—
103C2×D107—84no—
104C4×C107—42no—
105C4×C107—42no—
106C4×C107#12112yes—
107C4×C107C22×D411yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—24no—
110Dic107—41no—
111C2×C107—22no—
112C2×C107C2×#12111yescharacteristic
113C2×C107#57712yes—
114D107—44no—
115D107—81no—
116#3042—1071nomaximal, Sylow
117C107#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1071no—
119C22×SD16—1071no—
120#536—1071no—
121#518—1071no—
122#463—1074no—
123#461—1074no—
124#424—1071no—
125#419D10711yescharacteristic, complemented
126#416—1071no—
127C2×#41—1071no—
128C22×Q8C2×D10711yescharacteristic
129C22×D4—1071no—
130AGL(1,ℤ/8)—2144no—
131C2×SD16—1074no—
132C2×SD16—1072no—
133C2×SD16—2144no—
134C2×D8—1072no—
135C2×M16C2×D10711yescharacteristic
136C22×C8—1071no—
137C4⋊Q8—1072no—
138#124—1072no—
139#123—2142no—
140C22⋊Q8—2142no—
141C4×D4—1074no—
142C4×D4—2142no—
143#117—1071no—
144#117C2×D10711yescharacteristic
145C2×C22⋊C4—2142no—
146C4⋊C8—1074no—
147#103—1072no—
148#103C2×D10714yescomplemented
149#102—1072no—
150C24—2141no—
151#41—2142no—
152#41—2142no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1071no—
157C2×Q8C22×D10712yes—
158C2×D4—1074no—
159C2×D4—1072no—
160C2×D4—1071no—
161C2×D4—2141no—
162C2×D4—2141no—
163C2×D4—2142no—
164C22×C4—21no—
165C22×C4—2144no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168SD16—2144no—
169SD16—2148no—
170D8—2144no—
171M16—22no—
172C2×C8—1072no—
173C2×C8—2142no—
174C2×C8C22×D10712yes—
175C4⋊C4—1072no—
176C4⋊C4—2144no—
177C4⋊C4C22×D10712yes—
178C22⋊C4—21no—
179C22⋊C4—1072no—
180C22⋊C4—2144no—
181C22⋊C4—2144no—
182C42—21no—
183C42—1072no—
184C23—8562no—
185C23—2144no—
186C23—2141no—
187C23—2142no—
188C23D4×D10711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2141no—
193Q8—2141no—
194D4—1074no—
195D4—4282no—
196D4—2144no—
197D4—2141no—
198D4—2141no—
199D4—2142no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—8564no—
204C2×C4—1074no—
205C2×C4—4282no—
206C2×C4—2141no—
207C2×C4D4×D10711yescharacteristic
208C2×C4D4×D10712yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—8564no—
215C22—4282no—
216C22—4281no—
217C22—4284no—
218C22—2144no—
219C22#1681208212yes—
220C22C2×D4×D10711yescharacteristic
221C4—42no—
222C4—42no—
223C4—4281no—
224C4—2144no—
225C4#1681208212yes—
226C4C2×D4×D10711yescharacteristic
227C4C2×#1313381811yescharacteristic
228C2—22no—
229C2—8561no—
230C2—4284no—
231C2C2×#1681208211yescharacteristic
232C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812137C2
hypercenter#419D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#13133805C2semidirect
3. #20293201C2semidirect
4. #20293101C2semidirect
5. #20292845C2semidirect
6. #20292831C2semidirect
7. #20292655C2semidirect
8. #20292621C2semidirect
9. #419×C107C2semidirect
18. #16812107C22semidirect
19. #16812098C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812032C22semidirect
28. #16812029C22semidirect
32. #103×C107C22semidirect
117. C107#3042semidirect
125. #419D107semidirect
148. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C22×#131338052
#202932012
#202931012
#202928452
#202928312
#202926552
#202926212
#419×C1072
#3042107