Finite Group Databasebeta

← #28345976

Subgroups — #28345976

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×#16812134C211yesmaximal, characteristic
2C22×#13133807C211yesmaximal, characteristic, complemented
3#20293198C211yesmaximal, characteristic, complemented
4#20293101C211yesmaximal, characteristic, complemented
5#20292841C214yesmaximal, complemented
6#20292831C214yesmaximal, complemented
7#20292653C211yesmaximal, characteristic, complemented
8#20292635C211yesmaximal, characteristic, complemented
9#416×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812134—24no—
13C2×#13133807—24no—
14C2×#13133807C2214yes—
15C2×#13133805C2212yes—
16C2×#13133803C2212yes—
17C2×#13133802C2211yescharacteristic
18#16812107C2212yescomplemented
19#16812096C2212yescomplemented
20#16812090—22no—
21#16812088—22no—
22C4×D428—22no—
23#16812071C2214yes—
24#16812068C2211yescharacteristic
25C22⋊C4×D107—22no—
26Dic107⋊C8C2214yescomplemented
27#16812032C2214yescomplemented
28C2×#41×C107C2211yescharacteristic
29C22×C8×C107C2211yescharacteristic
30#117×C107C2211yescharacteristic
31#103×C107C2212yescomplemented
32#102×C107C2212yescomplemented
33C23×D107—21no—
34C2×Dic214—22no—
35C2×Dic214—22no—
36C2×Dic214—21no—
37C2×Dic214C2312yes—
38C2×D428—21no—
39C2×D428—22no—
40C2×D428D414yes—
41C2×D428C2312yes—
42C22×Dic107—21no—
43C2×C4×D107—24no—
44#13133807—28no—
45#13133805—24no—
46#13133803—24no—
47#13133802—22no—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107D412yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—24no—
52C22⋊Dic107D412yes—
53(C2×D107)⋊C4—24no—
54C4×Dic107C2312yes—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107C2311yescharacteristic
58C2×D4×C107—21no—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62C2×C8×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69C22×D107—22no—
70C22×D107—24no—
71C22×D107—82no—
72Q8×C107—21no—
73Q8×C107—21no—
74D4×C107—21no—
75D4×C107—21no—
76D4×C107—42no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—42no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D412yes—
82C2×C4×C107#4112yes—
83C2×C4×C107C2411yescharacteristic
84C2×Dic107—22no—
85C2×Dic107—42no—
86C2×Dic107C2×D414yes—
87D428—24no—
88D428—22no—
89D428C2×D414yes—
90C4×D107—84no—
91Dic214—22no—
92Dic214—22no—
93Dic214—42no—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107#12112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—24no—
102C2×D107—44no—
103C2×D107—84no—
104C4×C107—41no—
105C4×C107—42no—
106C4×C107#12112yes—
107C4×C107C22×D411yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—24no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—81no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—44no—
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#419—1071no—
126#416D10711yescharacteristic, complemented
127C2×#41C2×D10711yescharacteristic
128C22×Q8—1071no—
129C22×D4—1071no—
130AGL(1,ℤ/8)—2144no—
131C2×SD16—1072no—
132C2×SD16—1074no—
133C2×SD16—2144no—
134C2×D8—1072no—
135C2×M16—1071no—
136C22×C8C2×D10711yescharacteristic
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—1074no—
148#103C2×D10712yescomplemented
149#102C2×D10712yescomplemented
150C24—2141no—
151#41—22no—
152#41—22no—
153C2×Q8—1072no—
154C2×Q8—2142no—
155C2×Q8—2142no—
156C2×Q8—2141no—
157C2×Q8C22×D10711yescharacteristic
158C2×D4—21no—
159C2×D4—1074no—
160C2×D4—1072no—
161C2×D4—2141no—
162C2×D4—2142no—
163C2×D4C22×D10711yescharacteristic
164C22×C4—21no—
165C22×C4—2144no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168SD16—2148no—
169SD16—2144no—
170D8—2144no—
171M16—2142no—
172C2×C8—22no—
173C2×C8—1072no—
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—21no—
185C23—8562no—
186C23—2144no—
187C23—2142no—
188C23D4×D10711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—4282no—
192Q8—2142no—
193Q8—2142no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—1074no—
198D4—2144no—
199D4—2142no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—8564no—
204C2×C4—1074no—
205C2×C4—4282no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4D4×D10712yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—8564no—
217C22—4284no—
218C22—2144no—
219C22#1681208212yes—
220C22C2×D4×D10711yescharacteristic
221C4—42no—
222C4—41no—
223C4—4282no—
224C4—2144no—
225C4#1681208212yes—
226C4C2×D4×D10711yescharacteristic
227C4C2×#1313381811yescharacteristic
228C2—22no—
229C2—81no—
230C2—4284no—
231C2C2×#1681208211yescharacteristic
232C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812134C2
hypercenter#416D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×#13133807C2semidirect
3. #20293198C2semidirect
4. #20293101C2semidirect
5. #20292841C2semidirect
6. #20292831C2semidirect
7. #20292653C2semidirect
8. #20292635C2semidirect
9. #416×C107C2semidirect
18. #16812107C22semidirect
19. #16812096C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812032C22semidirect
31. #103×C107C22semidirect
32. #102×C107C22semidirect
117. C107#3042semidirect
126. #416D107semidirect
148. #103C2×D107semidirect
149. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C22×#131338072
#202931982
#202931012
#202928412
#202928312
#202926532
#202926352
#416×C1072
#3042107