Finite Group Databasebeta

← #28346916

Subgroups — #28346916

Order 13696.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812140C211yesmaximal, characteristic, complemented
3C2×#16812139C211yesmaximal, characteristic, complemented
4#20293173C211yesmaximal, characteristic
5C2×#16812107C211yesmaximal, characteristic, complemented
6#20293020C212yesmaximal, complemented
7#20293018C212yesmaximal, complemented
8#20293005C212yesmaximal, complemented
9#20292998C212yesmaximal, complemented
10M16×Dic107C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14#16812142—24no—
15#16812140—24no—
16#16812139—24no—
17C2×Dic428C2212yes—
18C2×#13133808C2212yes—
19C2×#13133808C2212yes—
20C2×#13133807C2212yes—
21C2×#13133806C2212yes—
22C2×#13133805C2212yes—
23C2×#13133802C2211yescharacteristic
24#16812107—22no—
25#16812107C2212yes—
26#16812107C2212yes—
27#16812107C2211yescharacteristic
28#16812105C2211yescharacteristic
29#16812098C2211yescharacteristic
30#16812096C2211yescharacteristic
31#16812081—22no—
32C4×C22⋊D107—22no—
33C2×Dic107⋊C4—22no—
34C2×C4×Dic107C2211yescharacteristic
35#16812019C2212yes—
36C8×Dic107C2212yes—
37C2×#41×C107C2211yescharacteristic
38C22×Q8×C107C2211yescharacteristic
39#137×C107—24no—
40C2×Q16×C107C2212yes—
41C2×SD16×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—21no—
48C2×Dic214C2311yescharacteristic
49C2×Dic214C2312yes—
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C2×C22⋊D107—22no—
53C22×Dic107—21no—
54C22×Dic107D412yes—
55C2×C4×D107—21no—
56Dic428—24no—
57#13133808—24no—
58#13133808—24no—
59#13133807—24no—
60#13133806—24no—
61#13133805—24no—
62#13133802D414yescomplemented
63C2×C107⋊C8C2312yes—
64Dic107⋊C4—24no—
65Dic107⋊C4—24no—
66Dic107⋊C4—22no—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C4×Dic107C2311yescharacteristic
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107—21no—
75C2×Q8×C107—22no—
76C2×Q8×C107—22no—
77C2×Q8×C107C2311yescharacteristic
78C2×Q8×C107C2312yes—
79C2×D4×C107—21no—
80C2×D4×C107C2311yescharacteristic
81C22×C4×C107—21no—
82C22×C4×C107—21no—
83C22×C4×C107C2311yescharacteristic
84Q16×C107—24no—
85SD16×C107—24no—
86M16×C107D414yescomplemented
87C2×C8×C107C2312yes—
88C23×C107—21no—
89C23×C107C2×D411yescharacteristic
90C22×D107—21no—
91Q8×C107—22no—
92Q8×C107—21no—
93Q8×C107—22no—
94Q8×C107—21no—
95Q8×C107—42no—
96D4×C107—21no—
97D4×C107—21no—
98D4×C107—42no—
99C2×C4×C107—21no—
100C2×C4×C107—22no—
101C2×C4×C107—42no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—42no—
108C22⋊D107—44no—
109C2×Dic107—24no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—42no—
113C2×Dic107C2×D412yes—
114C2×Dic107C2×D412yes—
115D428—21no—
116D428—21no—
117C4×D107—42no—
118Dic214—22no—
119Dic214—22no—
120Dic214—21no—
121Dic214—21no—
122Dic214—42no—
123C8×C107C2×D414yes—
124C107⋊C8C2×D414yes—
125C22×C107—22no—
126C22×C107—41no—
127C22×C107—42no—
128C22×C107#12712yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—41no—
131C2×D107—42no—
132C4×C107—41no—
133C4×C107—42no—
134C4×C107#12712yes—
135C4×C107C22×D411yescharacteristic
136C4×C107C22×D411yescharacteristic
137Dic107—22no—
138Dic107—42no—
139Dic107—41no—
140Dic107#13714yescomplemented
141C2×C107—22no—
142C2×C107—81no—
143C2×C107C2×#12711yescharacteristic
144C2×C107C2×#13712yes—
145D107—81no—
146#3152—1071nomaximal, Sylow
147C107#315211yescharacteristic, complemented, Sylow
148C2×#137—1071no—
149C2×#137—1071no—
150C2×#137—1071no—
151C2×#137D10711yescharacteristic, complemented
152#532—1071no—
153C2×C4⋊Q8—1071no—
154#497—1072no—
155#497—1072no—
156#494—1072no—
157#492—1072no—
158C4×M16—1071no—
159C2×#41—1071no—
160C2×#41C2×D10711yescharacteristic
161C22×Q8—1071no—
162C22×Q8C2×D10711yescharacteristic
163#137—24no—
164#137—2144no—
165#137—2144no—
166#137—2144no—
167C2×Q16—1072no—
168C2×Q16—1072no—
169C2×Q16—1072no—
170C2×Q16C2×D10712yes—
171C2×SD16—1072no—
172C2×SD16—1072no—
173C2×SD16—1072no—
174C2×SD16C2×D10712yes—
175C2×M16—1071no—
176C2×M16C2×D10711yescharacteristic
177C4⋊Q8—1072no—
178C4⋊Q8—1072no—
179C4⋊Q8—1071no—
180C4⋊Q8—2142no—
181#127—1071no—
182#124—1071no—
183#124—1071no—
184#121—2142no—
185C4×D4—2142no—
186C2×C4⋊C4—2142no—
187C2×C42—1071no—
188#97—1072no—
189C4×C8—1072no—
190#41—22no—
191#41—22no—
192#41—2142no—
193#41—2142no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1071no—
198C2×Q8—1072no—
199C2×Q8—2142no—
200C2×Q8—2142no—
201C2×Q8—2141no—
202C2×Q8C22×D10711yescharacteristic
203C2×Q8C22×D10712yes—
204C2×D4—21no—
205C2×D4—1071no—
206C2×D4—2142no—
207C2×D4—2141no—
208C2×D4C22×D10711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1072no—
212C22×C4—2141no—
213C22×C4—2141no—
214C22×C4C22×D10711yescharacteristic
215Q16—24no—
216Q16—2144no—
217Q16—2144no—
218Q16—2144no—
219SD16—24no—
220SD16—2144no—
221SD16—2144no—
222SD16—2144no—
223M16—1074no—
224M16C22⋊D10714yescomplemented
225C2×C8—1072no—
226C2×C8C22×D10712yes—
227C4⋊C4—2144no—
228C4⋊C4—2144no—
229C4⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C42—1072no—
233C42—1071no—
234C42—1071no—
235C23—21no—
236C23—2141no—
237C23D4×D10711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—4282no—
244Q8—2142no—
245Q8—2141no—
246Q8—2142no—
247Q8—2141no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—4284no—
252D4—4282no—
253D4—2141no—
254D4—2141no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1072no—
260C2×C4—1072no—
261C2×C4—4282no—
262C2×C4—4282no—
263C2×C4—2144no—
264C2×C4—2141no—
265C2×C4—2142no—
266C2×C4D4×D10711yescharacteristic
267C2×C4D4×D10712yes—
268C2×C4C2×C22⋊D10712yes—
269C2×C4C23×D10711yescharacteristic
270C8—1074no—
271C8C2×C22⋊D10714yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—4282no—
276C22—4281no—
277C22#1681210512yes—
278C22C2×D4×D10711yescharacteristic
279C4—41no—
280C4—42no—
281C4—1074no—
282C4—4281no—
283C4—4282no—
284C4—2142no—
285C4#1681210512yes—
286C4C2×D4×D10711yescharacteristic
287C4C22×C22⋊D10711yescharacteristic
288C2—22no—
289C2—81no—
290C2—8561no—
291C2C2×#1681210511yescharacteristic
292C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293173C2
hypercenterC2×#137D107
hypocenterC107#3152
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×#16812140C2semidirect
3. C2×#16812139C2semidirect
5. C2×#16812107C2semidirect
6. #20293020C2semidirect
7. #20293018C2semidirect
8. #20293005C2semidirect
9. #20292998C2semidirect
10. M16×Dic107C2semidirect
11. C2×#137×C107C2semidirect
62. #13133802D4semidirect
86. M16×C107D4semidirect
140. Dic107#137semidirect
147. C107#3152semidirect
151. C2×#137D107semidirect
224. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121402
C2×#168121392
#202931732
C2×#168121072
#202930202
#202930182
#202930052
#202929982
M16×Dic1072
C2×#137×C1072
#3152107