Finite Group Databasebeta

← #28346111

Subgroups — #28346111

Order 13696.

460 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 127 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×D107C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic
3#20293200C212yesmaximal
4#20292884C212yesmaximal, complemented
5#20292883C212yesmaximal, complemented
6#20292869C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292865C211yesmaximal, characteristic, complemented
9#20292664C211yesmaximal, characteristic, complemented
10#20292634C212yesmaximal, complemented
11#467×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×Q8×D107C2212yes—
14C2×Dic428C2211yescharacteristic
15C2×#13133808C2212yes—
16Q16×D107—24no—
17#16812115—24no—
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C2×#13133801C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812108—22no—
24#16812095—22no—
25#16812094—22no—
26#16812091—22no—
27#16812089C2212yes—
28#16812088C2212yes—
29#16812085C2212yes—
30#16812067C2212yes—
31#16812043C2211yescharacteristic
32#16812040C2212yes—
33#16812034C2212yes—
34#16812033C2212yes—
35#16812032C2212yes—
36C2×Q16×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#107×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214D411yescharacteristic
45C2×Dic214C2312yes—
46Q8×D107—24no—
47Q8×D107—24no—
48C2×D428D411yescharacteristic
49C2×C22⋊D107D411yescharacteristic
50C2×C22⋊D107D411yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107C2311yescharacteristic
53C2×C4×D107C2311yescharacteristic
54Dic428—22no—
55#13133808—24no—
56#13133802—22no—
57#13133801—22no—
58#13133801—22no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—22no—
62C8×D107D414yes—
63C4⋊Dic107—22no—
64C4⋊Dic107C2312yes—
65Dic107⋊C4—22no—
66Dic107⋊C4—22no—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—22no—
72C2×Q8×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74Q16×C107—22no—
75C2×C8×C107—22no—
76C2×C8×C107C2311yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C4⋊C4×C107—22no—
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107C2×D411yescharacteristic
83C22×D107C2×D411yescharacteristic
84Q8×C107—22no—
85Q8×C107—22no—
86C2×C4×C107—21no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—22no—
93C22⋊D107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D411yescharacteristic
98D428—22no—
99C4×D107—21no—
100C4×D107—21no—
101C4×D107—44no—
102C4×D107C2×D412yes—
103C4×D107#4112yes—
104Dic214—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—44no—
108C8×C107—21no—
109C8×C107C2×D412yes—
110C107⋊C8—21no—
111C107⋊C8C2×D412yes—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107C22×D411yescharacteristic
116C2×D107—21no—
117C2×D107—21no—
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C2×D107#12112yes—
122C4×C107—21no—
123C4×C107—42no—
124C4×C107—42no—
125C4×C107C22×D411yescharacteristic
126C4×C107C2×#4111yescharacteristic
127Dic107—21no—
128Dic107—42no—
129Dic107—42no—
130Dic107#12112yes—
131C2×C107—41no—
132C2×C107C2×#12111yescharacteristic
133C2×C107#57811yescharacteristic
134C2×C107#57811yescharacteristic
135D107—22no—
136D107—41no—
137#3059—1071nomaximal, Sylow
138C107#305911yescharacteristic, complemented, Sylow
139C22×Q16—1071no—
140C2×#131—1071no—
141#536—1072no—
142#470—1072no—
143#470—1072no—
144#467—1071no—
145#467—1071no—
146#467—1071no—
147#467D10711yescharacteristic, complemented
148#427—1071no—
149#419—1072no—
150C2×#41—1071no—
151C22×Q8—1072no—
152C2×Q16—1072no—
153C2×Q16—1071no—
154C2×Q16—2144no—
155C2×Q16C2×D10711yescharacteristic
156#131—2144no—
157C2×M16—1071no—
158C2×M16—1071no—
159C2×M16—1071no—
160C22×C8—1071no—
161C22×C8—1071no—
162C22×C8C2×D10711yescharacteristic
163C4⋊Q8—2142no—
164#124—2142no—
165#123—2142no—
166#123—2142no—
167C22⋊Q8—1072no—
168C22⋊Q8—1072no—
169C22⋊Q8—1072no—
170C22⋊Q8C2×D10712yes—
171#117—1072no—
172#107—1071no—
173#107C2×D10711yescharacteristic
174#106—1072no—
175#103—1072no—
176#103—1072no—
177#103—1072no—
178#103C2×D10712yes—
179#41—2142no—
180#41—2142no—
181C2×Q8—1071no—
182C2×Q8—1072no—
183C2×Q8—2142no—
184C2×Q8—2144no—
185C2×Q8—2144no—
186C2×Q8C22×D10712yes—
187C2×D4—1071no—
188C2×D4—1071no—
189C2×D4—1071no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—2142no—
193C22×C4C22×D10711yescharacteristic
194Q16—22no—
195Q16—2144no—
196Q16—2142no—
197M16—2142no—
198M16—2142no—
199M16—2142no—
200C2×C8—22no—
201C2×C8—1071no—
202C2×C8—1074no—
203C2×C8—1071no—
204C2×C8—2142no—
205C2×C8C22×D10711yescharacteristic
206C2×C8C22×D10711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—1072no—
209C4⋊C4—2142no—
210C4⋊C4—2142no—
211C4⋊C4—2142no—
212C4⋊C4C22×D10712yes—
213C22⋊C4—22no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C22⋊C4—2142no—
217C22⋊C4—2142no—
218C42—2142no—
219C23—1071no—
220C23—1071no—
221C23D4×D10711yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—4284no—
225Q8—2142no—
226Q8—2142no—
227Q8—2142no—
228D4—2142no—
229D4—2142no—
230D4—2142no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—1071no—
236C2×C4—1072no—
237C2×C4—1072no—
238C2×C4—1071no—
239C2×C4—4284no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4—2141no—
244C2×C4D4×D10711yescharacteristic
245C2×C4C23×D10711yescharacteristic
246C8—21no—
247C8—1072no—
248C8—2141no—
249C8D4×D10712yes—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—1072no—
254C22—2141no—
255C22—2141no—
256C22—2141no—
257C22—2141no—
258C22—2141no—
259C22C2×D4×D10711yescharacteristic
260C4—21no—
261C4—42no—
262C4—42no—
263C4—1072no—
264C4—4282no—
265C4—4282no—
266C4—2141no—
267C4C2×D4×D10711yescharacteristic
268C4#41×D10711yescharacteristic
269C2—41no—
270C2—4281no—
271C2—2142no—
272C2#121×D10711yescharacteristic
273C2#2029344011yescharacteristic
274C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#467D107
hypocenterC107#3059
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D107C2semidirect
4. #20292884C2semidirect
5. #20292883C2semidirect
6. #20292869C2semidirect
7. #20292868C2semidirect
8. #20292865C2semidirect
9. #20292664C2semidirect
10. #20292634C2semidirect
11. #467×C107C2semidirect
138. C107#3059semidirect
147. #467D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
C2×#168121152
#202932002
#202928842
#202928832
#202928692
#202928682
#202928652
#202926642
#202926342
#467×C1072
#3059107