Finite Group Databasebeta

← #28336435

Subgroups — #28336435

Order 13696.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16812156C211yesmaximal, characteristic
2#20292644C211yesmaximal, characteristic, complemented
3#20292637C211yesmaximal, characteristic, complemented
4#20292635C211yesmaximal, characteristic, complemented
5#20292609C211yesmaximal, characteristic, complemented
6#20292600C211yesmaximal, characteristic, complemented
7#421×C107C211yesmaximal, characteristic, complemented
8#16812156—22no—
9#16812156—22no—
10#16812156—22no—
11#16812156—22no—
12C2×#13133818—21no—
13C2×#13133818C2211yescharacteristic
14C2×#13133817—22no—
15C2×#13133817C412yescomplemented
16C22×Dic214—21no—
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107C412yescomplemented
19C2×#13133802C2211yescharacteristic
20#16812068C2211yescharacteristic
21#16812036—22no—
22#16812034—22no—
23#16812032—22no—
24#16812027—21no—
25#16812025—21no—
26#16812023—22no—
27C2×#41×C107C2211yescharacteristic
28C2×M16×C107C2211yescharacteristic
29#117×C107C2211yescharacteristic
30#104×C107—22no—
31#13133818—24no—
32#13133818—24no—
33#13133818—21no—
34#13133818—21no—
35#13133818—21no—
36#13133818—21no—
37#13133817—28no—
38#13133817—28no—
39#13133817—24no—
40#13133817—24no—
41C2×Dic214—24no—
42C2×Dic214—24no—
43C2×Dic214—21no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×Dic214—22no—
47C2×Dic214C2×C412yes—
48C2×Dic214D411yescharacteristic
49C2×Dic214D412yes—
50C2×Dic214D412yes—
51Q8×D107—24no—
52Q8×D107—24no—
53C2×D428D411yescharacteristic
54C2×C22⋊D107—22no—
55C2×C22⋊D107D412yes—
56C22×Dic107—21no—
57C22×Dic107—22no—
58C2×C4×D107—22no—
59C2×C4×D107C2×C412yes—
60#13133802—21no—
61#13133802—21no—
62C2×C107⋊C8—21no—
63C4⋊Dic107—21no—
64C22⋊Dic107—21no—
65C4×Dic107—21no—
66#41×C107—21no—
67#41×C107—21no—
68#41×C107D412yes—
69#41×C107D412yes—
70C2×Q8×C107C2×C411yescharacteristic
71C2×D4×C107—21no—
72C2×D4×C107C2×C411yescharacteristic
73C22×C4×C107—21no—
74C22×C4×C107C2311yescharacteristic
75M16×C107—21no—
76M16×C107—21no—
77C2×C8×C107—21no—
78C4⋊C4×C107—21no—
79C22⋊C4×C107—21no—
80C42×C107—21no—
81C23×C107—21no—
82C23×C107C2×D411yescharacteristic
83C22×D107C22⋊C412yes—
84Q8×C107—21no—
85Q8×C107C22⋊C412yes—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—21no—
89D4×C107C22⋊C412yes—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—41no—
94C2×C4×C107C22×C411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C22⋊D107—28no—
101C22⋊D107—22no—
102C22⋊D107—22no—
103C2×Dic107—28no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—41no—
108C2×Dic107C22⋊C412yes—
109D428—21no—
110D428—21no—
111C4×D107—28no—
112C4×D107—22no—
113C4×D107—22no—
114Dic214—28no—
115Dic214—22no—
116Dic214—21no—
117Dic214—22no—
118Dic214—21no—
119Dic214—22no—
120Dic214C22⋊C414yes—
121C8×C107—41no—
122C107⋊C8—41no—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107C2×C22⋊C411yescharacteristic
127C22×C107#12011yescharacteristic
128C22×C107#12011yescharacteristic
129C2×D107—22no—
130C2×D107—42no—
131C4×C107—22no—
132C4×C107—81no—
133C4×C107C2×C22⋊C411yescharacteristic
134C4×C107C2×C22⋊C411yescharacteristic
135C4×C107#12011yescharacteristic
136C4×C107#12011yescharacteristic
137Dic107—22no—
138Dic107—24no—
139Dic107—81no—
140C2×C107—21no—
141C2×C107—21no—
142C2×C107—21no—
143C2×C107—42no—
144C2×C107#37911yescharacteristic
145D107—42no—
146#1796—1071nomaximal, Sylow
147C107#179611yescharacteristic, complemented, Sylow
148C2×2−1+4—1071no—
149#421—1071no—
150#421D10711yescharacteristic, complemented
151#419—1071no—
152#419—1071no—
153#413—1071no—
154#410—1071no—
1552−1+4—2142no—
1562−1+4—2142no—
1572−1+4—2142no—
1582−1+4—2142no—
159C2×#41—1071no—
160C2×#41—1072no—
161C2×#41—2141no—
162C2×#41—2142no—
163C2×#41C2×D10711yescharacteristic
164C22×Q8—1072no—
165C22×Q8—1071no—
166C22×Q8—2141no—
167C2×M16—1071no—
168C2×M16C2×D10711yescharacteristic
169#117—1071no—
170#117C2×D10711yescharacteristic
171#104—22no—
172#104—2142no—
173#103—2142no—
174#103—2142no—
175#101—2141no—
176#100—2141no—
177#99—2142no—
178#41—21no—
179#41—21no—
180#41—2144no—
181#41—2144no—
182#41—2144no—
183#41—2148no—
184#41—2148no—
185#41—2144no—
186#41—2141no—
187#41—2141no—
188#41—2141no—
189#41—2141no—
190#41D42812yes—
191#41C22⋊D10712yes—
192C2×Q8—1071no—
193C2×Q8—1072no—
194C2×Q8—1072no—
195C2×Q8—1072no—
196C2×Q8—2142no—
197C2×Q8—2144no—
198C2×Q8—2144no—
199C2×Q8—2144no—
200C2×Q8—2144no—
201C2×Q8—2141no—
202C2×Q8—2141no—
203C2×Q8—2141no—
204C2×Q8C4×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—1072no—
207C2×D4—1071no—
208C2×D4—2142no—
209C2×D4C4×D10711yescharacteristic
210C22×C4—21no—
211C22×C4—1072no—
212C22×C4—2142no—
213C22×C4—2142no—
214C22×C4—2141no—
215C22×C4C22×D10711yescharacteristic
216M16—21no—
217M16—21no—
218M16—2141no—
219M16—2141no—
220C2×C8—21no—
221C2×C8—2141no—
222C4⋊C4—21no—
223C4⋊C4—2141no—
224C22⋊C4—21no—
225C22⋊C4—2141no—
226C42—21no—
227C42—2141no—
228C23—21no—
229C23—1072no—
230C23D4×D10711yescharacteristic
231Q8—21no—
232Q8—1074no—
233Q8—2148no—
234Q8—2142no—
235Q8—2141no—
236Q8—2142no—
237Q8—2141no—
238Q8—2142no—
239Q8(C2×D107)⋊C412yes—
240D4—22no—
241D4—22no—
242D4—21no—
243D4—2148no—
244D4—2142no—
245D4—2142no—
246D4—2141no—
247D4—2141no—
248D4(C2×D107)⋊C412yes—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—1072no—
254C2×C4—4281no—
255C2×C4—2148no—
256C2×C4—2148no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4C2×C4×D10711yescharacteristic
263C2×C4D4×D10711yescharacteristic
264C2×C4D4×D10711yescharacteristic
265C2×C4D4×D10711yescharacteristic
266C2×C4C2×C22⋊D10711yescharacteristic
267C2×C4C2×D42811yescharacteristic
268C8—41no—
269C8—4281no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—4282no—
274C22—2142no—
275C22C22⋊C4×D10711yescharacteristic
276C22#1681207711yescharacteristic
277C22#1681207911yescharacteristic
278C4—22no—
279C4—81no—
280C4—8561no—
281C4—2142no—
282C4—2144no—
283C4C22⋊C4×D10711yescharacteristic
284C4C2×(C2×D107)⋊C411yescharacteristic
285C4#1681207711yescharacteristic
286C4#1681207911yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—42no—
291C2—4282no—
292C2#2029248211yescharacteristic
subgroupG/H
centerC2#20292482
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#421×C107C2
socleC2×C107#379
prime radicalC2×#16812156C2
cosocleC22×C4×C107C23
hypercenter#421D107
hypocenterC107#1796
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20292644C2semidirect
3. #20292637C2semidirect
4. #20292635C2semidirect
5. #20292609C2semidirect
6. #20292600C2semidirect
7. #421×C107C2semidirect
15. C2×#13133817C4semidirect
18. C2×Q8×D107C4semidirect
147. C107#1796semidirect
150. #421D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
#202926442
#202926372
#202926352
#202926092
#202926002
#421×C1072
#1796107