Finite Group Databasebeta

← #28337378

Subgroups — #28337378

Order 13696.

336 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#20293173C211yesmaximal, characteristic
2C2×#16812037C211yesmaximal, characteristic, complemented
3C2×#16812036C211yesmaximal, characteristic, complemented
4C2×#16812035C211yesmaximal, characteristic, complemented
5M16×Dic107C211yesmaximal, characteristic, complemented
6C4×Dic107⋊C4C211yesmaximal, characteristic, complemented
7C2×#104×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10#16812107C411yescharacteristic, complemented
11#16812105C411yescharacteristic, complemented
12#16812098C411yescharacteristic, complemented
13#16812096C411yescharacteristic, complemented
14#16812081—21no—
15#16812081—21no—
16C4×C22⋊D107—21no—
17C4×C22⋊D107—21no—
18C2×Dic107⋊C4—21no—
19C2×C4×Dic107C2211yescharacteristic
20C2×C4×Dic107C2211yescharacteristic
21#16812037—22no—
22#16812036—22no—
23#16812035—22no—
24#16812019—21no—
25C8×Dic107—21no—
26C107⋊#95—21no—
27C2×#41×C107C2211yescharacteristic
28C2×M16×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#104×C107—22no—
31#13133818—21no—
32#13133818—21no—
33#13133818—21no—
34#13133818—21no—
35C2×Dic214C2×C411yescharacteristic
36C2×D428C2×C411yescharacteristic
37C2×C22⋊D107—21no—
38C2×C22⋊D107—21no—
39C2×C22⋊D107—21no—
40C22×Dic107—21no—
41C22×Dic107D411yescharacteristic
42C22×Dic107D411yescharacteristic
43C2×C4×D107—21no—
44#13133802—21no—
45#13133802D412yescomplemented
46C2×C107⋊C8—21no—
47Dic107⋊C4—24no—
48Dic107⋊C4—21no—
49Dic107⋊C4—21no—
50C22⋊Dic107—21no—
51C22⋊Dic107—21no—
52(C2×D107)⋊C4—21no—
53(C2×D107)⋊C4—21no—
54C4×Dic107—21no—
55C4×Dic107—21no—
56C4×Dic107C2×C411yescharacteristic
57C4×Dic107C2×C411yescharacteristic
58C4×Dic107D411yescharacteristic
59C4×Dic107D411yescharacteristic
60#41×C107—21no—
61#41×C107—21no—
62#41×C107—21no—
63#41×C107—21no—
64C2×Q8×C107C2×C411yescharacteristic
65C2×D4×C107—21no—
66C2×D4×C107C2×C411yescharacteristic
67C22×C4×C107—21no—
68C22×C4×C107—21no—
69C22×C4×C107C2311yescharacteristic
70M16×C107—21no—
71M16×C107D412yescomplemented
72C2×C8×C107—21no—
73C42×C107—21no—
74C42×C107—21no—
75C23×C107—21no—
76C23×C107C2×D411yescharacteristic
77C22×D107—21no—
78Q8×C107—21no—
79Q8×C107—21no—
80D4×C107—21no—
81D4×C107—21no—
82D4×C107—41no—
83D4×C107—41no—
84C2×C4×C107—22no—
85C2×C4×C107—21no—
86C2×C4×C107—41no—
87C2×C4×C107—41no—
88C2×C4×C107—41no—
89C2×C4×C107—41no—
90C2×C4×C107C22×C411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107#4111yescharacteristic
95C2×C4×C107#4111yescharacteristic
96C22⋊D107—42no—
97C22⋊D107—41no—
98C22⋊D107—41no—
99C22⋊D107—42no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—41no—
104C2×Dic107—41no—
105C2×Dic107—41no—
106C2×Dic107C22⋊C411yescharacteristic
107C2×Dic107C22⋊C411yescharacteristic
108C2×Dic107C22⋊C411yescharacteristic
109C2×Dic107C22⋊C411yescharacteristic
110D428—21no—
111D428—21no—
112C4×D107—41no—
113C4×D107—41no—
114Dic214—21no—
115Dic214—21no—
116C8×C107—22no—
117C107⋊C8—22no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—41no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107C2×C22⋊C411yescharacteristic
124C22×C107#12411yescharacteristic
125C22×C107#12711yescharacteristic
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—41no—
129C4×C107—42no—
130C4×C107—41no—
131C4×C107C4×D411yescharacteristic
132C4×C107C4×D411yescharacteristic
133C4×C107#12111yescharacteristic
134C4×C107#12111yescharacteristic
135Dic107—21no—
136Dic107—21no—
137Dic107—21no—
138Dic107—42no—
139Dic107—41no—
140Dic107#10412yescomplemented
141C2×C107—21no—
142C2×C107—21no—
143C2×C107—81no—
144C2×C107#39011yescharacteristic
145C2×C107C2×#10411yescharacteristic
146C2×C107#42111yescharacteristic
147D107—81no—
148#1980—1071nomaximal, Sylow
149C107#198011yescharacteristic, complemented, Sylow
150#532—1071no—
151C2×#104—1071no—
152C2×#104—1071no—
153C2×#104—1071no—
154C2×#104D10711yescharacteristic, complemented
155C4×M16—1071no—
156C4×C4⋊C4—1071no—
157C2×#41—1071no—
158C2×#41C2×D10711yescharacteristic
159C2×M16—1071no—
160C2×M16C2×D10711yescharacteristic
161C4⋊Q8—1071no—
162#127—1071no—
163#124—1071no—
164#124—1071no—
165#121—2141no—
166#121—2141no—
167C4×D4—2141no—
168C4×D4—2141no—
169C2×C4⋊C4—2141no—
170C2×C42—1071no—
171C2×C42—1071no—
172C2×C42C2×D10711yescharacteristic
173#104—22no—
174#104—2142no—
175#104—2142no—
176#104—2142no—
177#97—2141no—
178C4×C8—2141no—
179#95—2141no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—2141no—
185#41—2141no—
186#41—2141no—
187#41—2141no—
188C2×Q8—1071no—
189C2×Q8C4×D10711yescharacteristic
190C2×D4—21no—
191C2×D4—1071no—
192C2×D4—2141no—
193C2×D4—2141no—
194C2×D4—2141no—
195C2×D4C4×D10711yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—2141no—
201C22×C4—2141no—
202C22×C4C22×D10711yescharacteristic
203M16—21no—
204M16—1072no—
205M16—2141no—
206M16C22⋊D10712yescomplemented
207C2×C8—21no—
208C2×C8—2141no—
209C4⋊C4—2141no—
210C4⋊C4—2141no—
211C4⋊C4—2144no—
212C22⋊C4—2141no—
213C22⋊C4—2141no—
214C22⋊C4—2141no—
215C22⋊C4—2141no—
216C42—21no—
217C42—21no—
218C42—1071no—
219C42—1071no—
220C42—1071no—
221C42—1071no—
222C42—2141no—
223C42—2141no—
224C23—21no—
225C23—2141no—
226C23D4×D10711yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—2141no—
230Q8—2141no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—4282no—
236D4—4281no—
237D4—4281no—
238D4—4282no—
239D4—2141no—
240D4—2141no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—1071no—
248C2×C4—1071no—
249C2×C4—1071no—
250C2×C4—1071no—
251C2×C4—4281no—
252C2×C4—4281no—
253C2×C4—4281no—
254C2×C4—4281no—
255C2×C4—4281no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4—2141no—
259C2×C4C2×C4×D10711yescharacteristic
260C2×C4D4×D10711yescharacteristic
261C2×C4D4×D10711yescharacteristic
262C2×C4C2×C22⋊D10711yescharacteristic
263C2×C4#1313381711yescharacteristic
264C2×C4#1313381811yescharacteristic
265C8—22no—
266C8—2142no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—4281no—
273C22—4281no—
274C22—4281no—
275C22C22⋊C4×D10711yescharacteristic
276C22#1681209511yescharacteristic
277C22#1681210511yescharacteristic
278C4—42no—
279C4—41no—
280C4—1072no—
281C4—4282no—
282C4—4281no—
283C4—2141no—
284C4—2141no—
285C4—2141no—
286C4(C22⋊D107)⋊C411yescharacteristic
287C4C4×C22⋊D10711yescharacteristic
288C4#1681208111yescharacteristic
289C4#1681208211yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—8561no—
294C2#2029252711yescharacteristic
295C2#104×D10711yescharacteristic
296C2#2029264211yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#104×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293173C2
cosocleC22×C4×C107C23
hypercenterC2×#104D107
hypocenterC107#1980
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16812037C2semidirect
3. C2×#16812036C2semidirect
4. C2×#16812035C2semidirect
5. M16×Dic107C2semidirect
6. C4×Dic107⋊C4C2semidirect
7. C2×#104×C107C2semidirect
10. #16812107C4semidirect
11. #16812105C4semidirect
12. #16812098C4semidirect
13. #16812096C4semidirect
45. #13133802D4semidirect
71. M16×C107D4semidirect
140. Dic107#104semidirect
149. C107#1980semidirect
154. C2×#104D107semidirect
206. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
C2×#168120372
C2×#168120362
C2×#168120352
M16×Dic1072
C4×Dic107⋊C42
C2×#104×C1072
#1980107