Finite Group Databasebeta

← #28346954

Subgroups — #28346954

Order 13696.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812129C211yesmaximal, characteristic
3C22×#13133808C211yesmaximal, characteristic, complemented
4#20293201C211yesmaximal, characteristic, complemented
5#20293198C211yesmaximal, characteristic, complemented
6#20293020C212yesmaximal, complemented
7#20293018C212yesmaximal, complemented
8#20293008C212yesmaximal, complemented
9#20293005C212yesmaximal, complemented
10#20292585C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14#16812142—24no—
15#16812129—24no—
16C2×Dic428C2212yes—
17C2×#13133808—24no—
18C2×#13133808C2212yes—
19C2×#13133808C2212yes—
20C2×#13133808C2211yescharacteristic
21C2×#13133807C2212yes—
22C2×#13133806C2211yescharacteristic
23C2×#13133805C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25C22×C107⋊C8C2211yescharacteristic
26#16812107C2212yes—
27#16812107C2212yes—
28#16812098C2212yes—
29#16812096C2212yes—
30#16812092—22no—
31#16812090—22no—
32#16812088—22no—
33#16812085—22no—
34#16812068C2211yescharacteristic
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×Dic214C2312yes—
49C2×Dic214C2311yescharacteristic
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C22×Dic107—21no—
53C2×C4×D107—21no—
54Dic428—24no—
55#13133808—24no—
56#13133808—24no—
57#13133808—22no—
58#13133807—24no—
59#13133806—22no—
60#13133805—22no—
61#13133803—22no—
62C2×C107⋊C8D414yes—
63C2×C107⋊C8C2311yescharacteristic
64C2×C107⋊C8C2311yescharacteristic
65C4⋊Dic107D412yes—
66Dic107⋊C4—24no—
67Dic107⋊C4—24no—
68C22⋊Dic107—24no—
69C22⋊Dic107D412yes—
70(C2×D107)⋊C4—24no—
71C4×Dic107C2312yes—
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107—21no—
75C2×Q8×C107—22no—
76C2×Q8×C107—22no—
77C2×Q8×C107C2312yes—
78C2×Q8×C107C2311yescharacteristic
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×C107D414yes—
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—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107—42no—
111C2×Dic107C2×D414yes—
112D428—21no—
113D428—21no—
114C4×D107—42no—
115Dic214—21no—
116Dic214—22no—
117Dic214—22no—
118Dic214—21no—
119Dic214—42no—
120C8×C107C2×D414yes—
121C107⋊C8C2×D412yes—
122C107⋊C8C2×D412yes—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107#12712yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—41no—
129C2×D107—42no—
130C4×C107—41no—
131C4×C107—42no—
132C4×C107#12712yes—
133C4×C107C22×D411yescharacteristic
134C4×C107C22×D411yescharacteristic
135Dic107—24no—
136Dic107—42no—
137Dic107—41no—
138C2×C107—22no—
139C2×C107—81no—
140C2×C107C2×#12711yescharacteristic
141C2×C107#57812yes—
142D107—81no—
143#3156—1071nomaximal, Sylow
144C107#315611yescharacteristic, complemented, Sylow
145C2×#137—1071no—
146C2×#137D10711yescharacteristic, complemented
147C2×#135—1071no—
148C22×Q16—1071no—
149#536—1071no—
150#536—1071no—
151#497—1072no—
152#497—1072no—
153#495—1072no—
154#494—1072no—
155#405—1071no—
156C2×#41—1071no—
157C2×#41C2×D10711yescharacteristic
158C22×Q8—1071no—
159C22×Q8C2×D10711yescharacteristic
160#137—24no—
161#137—2144no—
162#135—2144no—
163C2×Q16—1072no—
164C2×Q16—1072no—
165C2×Q16—1071no—
166C2×Q16—1072no—
167C2×Q16—2144no—
168C2×Q16C2×D10712yes—
169C2×SD16—1071no—
170C2×SD16—1071no—
171C2×SD16—1072no—
172C2×SD16C2×D10712yes—
173C2×D8—1071no—
174C2×M16C2×D10711yescharacteristic
175C22×C8—1071no—
176C4⋊Q8—1072no—
177C4⋊Q8—1072no—
178#124—1072no—
179#124—1072no—
180#123—2142no—
181#123—2142no—
182C22⋊Q8—2142no—
183C22⋊Q8—2142no—
184#117—1071no—
185#97—1072no—
186C4×C8—1072no—
187#41—22no—
188#41—22no—
189#41—2142no—
190#41—2142no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1072no—
195C2×Q8—1071no—
196C2×Q8—2142no—
197C2×Q8—2142no—
198C2×Q8—2141no—
199C2×Q8C22×D10712yes—
200C2×Q8C22×D10711yescharacteristic
201C2×D4—21no—
202C2×D4—1071no—
203C2×D4—2141no—
204C2×D4C22×D10711yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4C22×D10711yescharacteristic
210Q16—24no—
211Q16—2144no—
212Q16—2144no—
213Q16—2142no—
214Q16—2144no—
215SD16—24no—
216SD16—2142no—
217SD16—2142no—
218SD16—2144no—
219D8—2142no—
220M16C22⋊D10714yes—
221C2×C8—1074no—
222C2×C8—1071no—
223C2×C8—1071no—
224C2×C8C22×D10712yes—
225C4⋊C4—1072no—
226C4⋊C4—2144no—
227C4⋊C4—2144no—
228C22⋊C4—1072no—
229C22⋊C4—2144no—
230C22⋊C4—2144no—
231C42—1072no—
232C23—21no—
233C23—2141no—
234C23D4×D10711yescharacteristic
235Q8—21no—
236Q8—22no—
237Q8—22no—
238Q8—21no—
239Q8—42no—
240Q8—4282no—
241Q8—2141no—
242Q8—2142no—
243Q8—2142no—
244Q8—2141no—
245D4—21no—
246D4—21no—
247D4—42no—
248D4—4282no—
249D4—2141no—
250D4—2141no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1074no—
256C2×C4—4282no—
257C2×C4—4282no—
258C2×C4—2142no—
259C2×C4—2141no—
260C2×C4D4×D10711yescharacteristic
261C2×C4D4×D10712yes—
262C2×C4C2×C22⋊D10712yes—
263C2×C4C23×D10711yescharacteristic
264C8—1072no—
265C8—1072no—
266C8C2×C22⋊D10714yes—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—4281no—
271C22—4282no—
272C22#1681210512yes—
273C22C2×D4×D10711yescharacteristic
274C4—41no—
275C4—42no—
276C4—4281no—
277C4—4282no—
278C4—2144no—
279C4#1681210512yes—
280C4C2×D4×D10711yescharacteristic
281C4C22×C22⋊D10711yescharacteristic
282C2—22no—
283C2—81no—
284C2—8561no—
285C2C2×#1681210511yescharacteristic
286C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812129C2
hypercenterC2×#137D107
hypocenterC107#3156
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
3. C22×#13133808C2semidirect
4. #20293201C2semidirect
5. #20293198C2semidirect
6. #20293020C2semidirect
7. #20293018C2semidirect
8. #20293008C2semidirect
9. #20293005C2semidirect
10. #20292585C2semidirect
11. C2×#137×C107C2semidirect
144. C107#3156semidirect
146. C2×#137D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121292
C22×#131338082
#202932012
#202931982
#202930202
#202930182
#202930082
#202930052
#202925852
C2×#137×C1072
#3156107