Finite Group Databasebeta

← #28348187

Subgroups — #28348187

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×#16812129C212yesmaximal, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20293001C211yesmaximal, characteristic, complemented
4#20292858C212yesmaximal, complemented
5#20292834C211yesmaximal, characteristic, complemented
6#20292822C211yesmaximal, characteristic, complemented
7#20292787C212yesmaximal, complemented
8#20292772C212yesmaximal, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10D4×C107⋊C8C211yesmaximal, characteristic, complemented
11#547×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13#16812129—28no—
14C2×#13133808C2212yes—
15C2×#13133806C2211yescharacteristic
16C2×#13133805C2212yes—
17C2×#13133805C2212yes—
18C2×#13133803C2212yes—
19C22×C107⋊C8C2212yes—
20#16812106C2211yescharacteristic
21#16812105—21no—
22#16812083C2212yes—
23#16812082—22no—
24C4×Dic214C2211yescharacteristic
25#16812071—21no—
26#16812040C2211yescharacteristic
27C4⋊(C107⋊C8)C2211yescharacteristic
28#16812033C2212yes—
29#16812032C2211yescharacteristic
30#16812030C2211yescharacteristic
31#16812029C2212yes—
32C22⋊(C107⋊C8)C2212yes—
33C4×C107⋊C8C2211yescharacteristic
34C2×#41×C107C2212yes—
35C4⋊Q8×C107C2211yescharacteristic
36#123×C107—22no—
37C22⋊Q8×C107C2212yes—
38#121×C107—21no—
39C4×D4×C107C2211yescharacteristic
40C4×D4×C107C2211yescharacteristic
41C2×C4⋊C4×C107—21no—
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—21no—
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C4×D107—22no—
51#13133808—24no—
52#13133806D414yescomplemented
53#13133805—24no—
54#13133805—42no—
55#13133803—24no—
56C2×C107⋊C8—24no—
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59C2×C107⋊C8C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64#41×C107—24no—
65#41×C107—42no—
66C2×Q8×C107C2312yes—
67C2×D4×C107—22no—
68C2×D4×C107D411yescharacteristic
69C2×D4×C107C2311yescharacteristic
70C22×C4×C107—21no—
71C22×C4×C107—22no—
72C22×C4×C107C2312yes—
73C4⋊C4×C107—21no—
74C4⋊C4×C107—22no—
75C4⋊C4×C107—21no—
76C4⋊C4×C107D411yescharacteristic
77C4⋊C4×C107C2311yescharacteristic
78C22⋊C4×C107—22no—
79C22⋊C4×C107—21no—
80C22⋊C4×C107D412yes—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C23×C107C2×D412yes—
84C22×D107—22no—
85Q8×C107—22no—
86Q8×C107—22no—
87D4×C107—22no—
88D4×C107—21no—
89D4×C107—44no—
90D4×C107C2×D412yes—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—44no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—44no—
102C22⋊D107—82no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105D428—22no—
106D428—22no—
107D428—42no—
108D428—81no—
109C4×D107—44no—
110Dic214—21no—
111Dic214C2×D412yes—
112C107⋊C8—22no—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107—22no—
119C22×C107—41no—
120C22×C107—41no—
121C22×C107C22×D411yescharacteristic
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C4×C107—22no—
126C4×C107—21no—
127C4×C107—42no—
128C4×C107—41no—
129C4×C107#13512yescomplemented
130C4×C107C22×D411yescharacteristic
131C4×C107H(5,𝔽2)11yescharacteristic
132Dic107—22no—
133Dic107—41no—
134C2×C107—42no—
135C2×C107—42no—
136C2×C107D4211yescharacteristic
137C2×C107C2×#13511yescharacteristic
138C2×C107#57711yescharacteristic
139D107—82no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550—1071no—
144#547D10711yescharacteristic, complemented
145#492—1071no—
146#465—1072no—
147#462—1071no—
148#459—1071no—
149#452—1072no—
150#449—1072no—
151C4×SD16—1071no—
152C8×D4—1071no—
153C2×#41—1072no—
154C2×#41C2×D10712yes—
155#135—2148no—
156C2×Q16—1072no—
157C2×SD16—1071no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×D8—1072no—
161C22×C8—1072no—
162C4⋊Q8C2×D10711yescharacteristic
163#127—1071no—
164#127—2141no—
165#123—22no—
166C22⋊Q8C2×D10712yes—
167#121—21no—
168#121—1072no—
169#121—2142no—
170C4×Q8—1071no—
171C4×D4—2141no—
172C4×D4C2×D10711yescharacteristic
173C4×D4C2×D10711yescharacteristic
174C2×C4⋊C4—21no—
175#106—1071no—
176C4⋊C8—1071no—
177#103—1072no—
178#103—1071no—
179#102—1071no—
180#102—1072no—
181C22⋊C8—1072no—
182C4×C8—1071no—
183#41—24no—
184#41—42no—
185#41—4282no—
186#41—2144no—
187C2×Q8—1071no—
188C2×Q8C22×D10712yes—
189C2×D4—22no—
190C2×D4—1072no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4—2141no—
194C2×D4—2141no—
195C2×D4C22⋊D10711yescharacteristic
196C2×D4C22×D10711yescharacteristic
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—2142no—
200C22×C4C22×D10712yes—
201Q16—2144no—
202SD16—1074no—
203SD16—4282no—
204SD16—2144no—
205D8—2144no—
206C2×C8—1071no—
207C2×C8—1071no—
208C2×C8—1072no—
209C2×C8—2144no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—1071no—
214C4⋊C4—2141no—
215C4⋊C4C22⋊D10711yescharacteristic
216C4⋊C4C22×D10711yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—2142no—
220C22⋊C4C22⋊D10712yes—
221C42—2141no—
222C42C22×D10711yescharacteristic
223C23—21no—
224C23—2142no—
225C23C2×C22⋊D10712yes—
226Q8—22no—
227Q8—22no—
228Q8—1072no—
229Q8—2141no—
230D4—22no—
231D4—21no—
232D4—44no—
233D4—8561no—
234D4—8562no—
235D4—4284no—
236D4—4282no—
237D4—2142no—
238D4—2142no—
239D4D4×D10712yes—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—4284no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4C2×C22⋊D10711yescharacteristic
250C2×C4C2×C22⋊D10712yes—
251C2×C4C2×C22⋊D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—1072no—
254C8—2142no—
255C8—2141no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—41no—
261C22—41no—
262C22—4282no—
263C22—4282no—
264C22—4282no—
265C22C22×C22⋊D10711yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—4281no—
271C4—2142no—
272C4#1681212912yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215411yescharacteristic
275C2—42no—
276C2—42no—
277C2—8562no—
278C2D4×C22⋊D10711yescharacteristic
279C2C2×#1681212911yescharacteristic
280C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3284
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. #20293298C2semidirect
3. #20293001C2semidirect
4. #20292858C2semidirect
5. #20292834C2semidirect
6. #20292822C2semidirect
7. #20292787C2semidirect
8. #20292772C2semidirect
9. C4×#13133806C2semidirect
10. D4×C107⋊C8C2semidirect
11. #547×C107C2semidirect
52. #13133806D4semidirect
129. C4×C107#135semidirect
141. C107#3284semidirect
144. #547D107semidirect
272. C4#16812129semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202932982
#202930012
#202928582
#202928342
#202928222
#202927872
#202927722
C4×#131338062
D4×C107⋊C82
#547×C1072
#3284107