Finite Group Databasebeta

← #28348193

Subgroups — #28348193

Order 13696.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812131C212yesmaximal, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20293282C211yesmaximal, characteristic, complemented
4#20293000C211yesmaximal, characteristic, complemented
5#20292856C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292791C212yesmaximal, complemented
9#20292774C212yesmaximal, complemented
10C4×#13133807C211yesmaximal, characteristic, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C2×#13133818C2212yes—
14#16812131—28no—
15C2×Dic428C2212yes—
16C2×#13133807C2212yes—
17C2×#13133807C2212yes—
18C2×#13133807C2211yescharacteristic
19C2×D856C2212yes—
20#16812109C2211yescharacteristic
21#16812106C2211yescharacteristic
22#16812105—21no—
23#16812091—22no—
24#16812088C2212yes—
25#16812083C2212yes—
26#16812082—22no—
27#16812080—21no—
28C4×Dic214C2211yescharacteristic
29C4×D428C2211yescharacteristic
30#16812071—21no—
31C4⋊C4×D107—21no—
32#16812041C2211yescharacteristic
33#16812034C2211yescharacteristic
34#16812034C2212yes—
35#16812031C2211yescharacteristic
36#16812031C2212yes—
37C22×C8×C107C2212yes—
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40C22⋊C8×C107C2212yes—
41C4×C8×C107C2211yescharacteristic
42#13133818—24no—
43#13133818—24no—
44#13133818—42no—
45#13133818—42no—
46C2×Dic214C2312yes—
47C2×Dic214C2311yescharacteristic
48C2×D428—21no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55C2×C4×D107—21no—
56C2×C4×D107—22no—
57C2×C4×D107—22no—
58Dic428—24no—
59#13133807—24no—
60#13133807—42no—
61#13133807D414yescomplemented
62D856—24no—
63C4⋊Dic107—21no—
64C4⋊Dic107—21no—
65C4⋊Dic107C2311yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—22no—
68Dic107⋊C4—21no—
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73C2×D4×C107D411yescharacteristic
74C22×C4×C107C2312yes—
75C2×C8×C107—24no—
76C2×C8×C107C2312yes—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C4⋊C4×C107D411yescharacteristic
80C22⋊C4×C107D412yes—
81C42×C107C2311yescharacteristic
82C23×C107C2×D412yes—
83C22×D107—21no—
84C22×D107—22no—
85D4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—44no—
93C22⋊D107—44no—
94C22⋊D107—82no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99D428—22no—
100D428—21no—
101D428—22no—
102D428—42no—
103D428—81no—
104D428C2×D412yes—
105C4×D107—44no—
106C4×D107—44no—
107C4×D107—42no—
108Dic214—21no—
109Dic214—22no—
110Dic214—22no—
111Dic214C2×D412yes—
112C8×C107—22no—
113C8×C107—21no—
114C8×C107C2×D412yes—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107C22×D411yescharacteristic
119C2×D107—22no—
120C2×D107—41no—
121C2×D107—41no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C4×C107—22no—
126C4×C107—21no—
127C4×C107#13512yescomplemented
128C4×C107C22×D411yescharacteristic
129C4×C107H(5,𝔽2)11yescharacteristic
130Dic107—22no—
131Dic107—42no—
132Dic107—41no—
133Dic107—41no—
134C2×C107—42no—
135C2×C107D4211yescharacteristic
136C2×C107C2×#13511yescharacteristic
137C2×C107#57711yescharacteristic
138D107—42no—
139D107—82no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550—1071no—
144#547—1071no—
145#492—1071no—
146#465—1072no—
147#462—1071no—
148#459—1071no—
149#452—1072no—
150#449—1072no—
151C4×SD16—1071no—
152C8×D4D10711yescharacteristic, complemented
153C2×#41—1072no—
154C2×#41—1072no—
155#135—2148no—
156C2×Q16—1072no—
157C2×SD16—1071no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×D8—1072no—
161C22×C8C2×D10712yes—
162C4⋊Q8—1071no—
163#127—1071no—
164#127—2141no—
165#123—2142no—
166C22⋊Q8—1072no—
167#121—1072no—
168#121—2141no—
169#121—2142no—
170C4×Q8—1071no—
171C4×D4—1071no—
172C4×D4—2141no—
173C4×D4C2×D10711yescharacteristic
174C2×C4⋊C4—2141no—
175#106—1071no—
176C4⋊C8C2×D10711yescharacteristic
177#103—1071no—
178#103—1072no—
179#102—1071no—
180#102—1072no—
181C22⋊C8C2×D10712yes—
182C4×C8C2×D10711yescharacteristic
183#41—4282no—
184#41—4282no—
185#41—2144no—
186#41—2144no—
187C2×Q8—1071no—
188C2×Q8—1072no—
189C2×D4—1071no—
190C2×D4—1072no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2141no—
195C2×D4—2141no—
196C2×D4D42811yescharacteristic
197C22×C4—2141no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4C22×D10712yes—
201Q16—2144no—
202SD16—1074no—
203SD16—4282no—
204SD16—2144no—
205D8—2144no—
206C2×C8—24no—
207C2×C8C22×D10711yescharacteristic
208C2×C8C22×D10712yes—
209C2×C8C22×D10711yescharacteristic
210C4⋊C4—1071no—
211C4⋊C4—1071no—
212C4⋊C4—2141no—
213C4⋊C4—2142no—
214C4⋊C4—2141no—
215C4⋊C4—2141no—
216C4⋊C4D42811yescharacteristic
217C22⋊C4—2142no—
218C22⋊C4—2141no—
219C22⋊C4—2142no—
220C22⋊C4D42812yes—
221C42—2141no—
222C42C22×D10711yescharacteristic
223C23—2142no—
224C23—2141no—
225C23C2×D42812yes—
226Q8—1072no—
227Q8—2142no—
228Q8—2142no—
229Q8—2141no—
230D4—22no—
231D4—8561no—
232D4—8562no—
233D4—1072no—
234D4—4284no—
235D4—4284no—
236D4—4282no—
237D4—2142no—
238D4—2142no—
239D4—2141no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—4284no—
243C2×C4—4284no—
244C2×C4—4282no—
245C2×C4—2141no—
246C2×C4—2141no—
247C2×C4—2142no—
248C2×C4—2141no—
249C2×C4C2×D42811yescharacteristic
250C2×C4C2×D42812yes—
251C2×C4C2×D42811yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—22no—
254C8—21no—
255C8D4×D10712yes—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—4282no—
260C22—4282no—
261C22—4282no—
262C22—4281no—
263C22—4281no—
264C22—2142no—
265C22C22×D42811yescharacteristic
266C4—21no—
267C4—22no—
268C4—4281no—
269C4—4281no—
270C4—4282no—
271C4—2142no—
272C4#1681213112yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215511yescharacteristic
275C2—42no—
276C2—8562no—
277C2—4282no—
278C2D4×D42811yescharacteristic
279C2C2×#1681213111yescharacteristic
280C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107#3284
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812131C2semidirect
2. #20293298C2semidirect
3. #20293282C2semidirect
4. #20293000C2semidirect
5. #20292856C2semidirect
6. #20292838C2semidirect
7. #20292823C2semidirect
8. #20292791C2semidirect
9. #20292774C2semidirect
10. C4×#13133807C2semidirect
11. C8×D4×C107C2semidirect
61. #13133807D4semidirect
127. C4×C107#135semidirect
141. C107#3284semidirect
152. C8×D4D107semidirect
272. C4#16812131semidirect

Maximal subgroups

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

subgroupindex
C2×#168121312
#202932982
#202932822
#202930002
#202928562
#202928382
#202928232
#202927912
#202927742
C4×#131338072
C8×D4×C1072
#3284107