Finite Group Databasebeta

← #28340107

Subgroups — #28340107

Order 13696.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×D4×D107C211yesmaximal, characteristic
2C2×#16812095C212yesmaximal, complemented
3C2×#16812087C211yesmaximal, characteristic, complemented
4#20292545C211yesmaximal, characteristic, complemented
5#20292544C211yesmaximal, characteristic, complemented
6#20292541C211yesmaximal, characteristic, complemented
7#20292527C212yesmaximal, complemented
8#20292482C212yesmaximal, complemented
9#20292482C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic, complemented
12C24×D107C2212yes—
13C22×Dic214C2211yescharacteristic, complemented
14C22×D428C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C2×D4×D107—28no—
17C2×D4×D107—24no—
18C22×C4×D107C2211yescharacteristic
19#16812095—28no—
20#16812087—24no—
21C2×Dic107⋊C4C2211yescharacteristic, complemented
22C2×Dic107⋊C4C2211yescharacteristic, complemented
23C2×C22⋊Dic107C2212yescomplemented
24C2×C22⋊Dic107C2212yescomplemented
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yescomplemented
29C2×(C2×D107)⋊C4C2212yescomplemented
30C2×(C2×D107)⋊C4C2212yescomplemented
31C2×(C2×D107)⋊C4C2212yescomplemented
32C2×C4×Dic107C2212yescomplemented
33C107⋊#95C2211yescharacteristic, complemented
34C22×D4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic, complemented
36C2×C22⋊C4×C107C2212yescomplemented
37C2×C22⋊C4×C107C2212yescomplemented
38#95×C107C2211yescharacteristic, complemented
39C23×D107—24no—
40C23×D107—24no—
41C23×D107—24no—
42C23×D107—24no—
43C23×D107—24no—
44C23×D107—42no—
45C23×D107—42no—
46C23×D107C2312yes—
47C23×D107C2311yescharacteristic
48C2×Dic214—22no—
49C2×Dic214—24no—
50C2×D428—24no—
51C2×D428—22no—
52C2×C22⋊D107—24no—
53C2×C22⋊D107—24no—
54C2×C22⋊D107D418yes—
55D4×D107—232no—
56C22×Dic107C2312yes—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C2×C4×D107—24no—
61C2×C4×D107D414yes—
62Dic107⋊C4—22no—
63Dic107⋊C4—22no—
64C22⋊Dic107—24no—
65C22⋊Dic107—42no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—24no—
69(C2×D107)⋊C4—24no—
70(C2×D107)⋊C4—24no—
71(C2×D107)⋊C4—42no—
72(C2×D107)⋊C4—42no—
73C4×Dic107—24no—
74C24×C107C2312yes—
75C2×D4×C107—24no—
76C2×D4×C107—22no—
77C22×C4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79C22×C4×C107C2311yescharacteristic
80C22×C4×C107C2311yescharacteristic
81C4⋊C4×C107—22no—
82C22⋊C4×C107—24no—
83C22⋊C4×C107—42no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—22no—
88C23×C107—22no—
89C23×C107—22no—
90C23×C107C2×D414yes—
91C23×C107C2411yescharacteristic
92C22×D107—22no—
93C22×D107—21no—
94C22×D107—21no—
95C22×D107—216no—
96C22×D107—22no—
97C22×D107—22no—
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—22no—
101C22×D107—22no—
102C22×D107—48no—
103C22×D107—48no—
104C22×D107—48no—
105C22×D107—48no—
106C22×D107—48no—
107C22×D107—48no—
108C22×D107C2×D414yes—
109C22×D107C2×D412yes—
110C22×D107#4114yes—
111D4×C107—28no—
112C2×C4×C107—22no—
113C2×C4×C107—22no—
114C2×C4×C107—22no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—22no—
118C2×C4×C107—21no—
119C2×C4×C107—22no—
120C2×C4×C107—21no—
121C2×C4×C107C2×D412yes—
122C22⋊D107—216no—
123C2×Dic107—22no—
124C2×Dic107—22no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—22no—
128C2×Dic107—21no—
129C2×Dic107—22no—
130C2×Dic107—21no—
131C2×Dic107C2×D414yes—
132C2×Dic107C2×D412yes—
133D428—28no—
134C4×D107—28no—
135Dic214—44no—
136C22×C107—24no—
137C22×C107—24no—
138C22×C107—24no—
139C22×C107—42no—
140C22×C107—42no—
141C22×C107—42no—
142C22×C107—42no—
143C22×C107C22×D411yescharacteristic
144C22×C107C22×D412yes—
145C22×C107C2×#4112yes—
146C22×C107H(5,𝔽2)11yescharacteristic
147C22×C107H(5,𝔽2)11yescharacteristic
148C2×D107—22no—
149C2×D107—22no—
150C2×D107—24no—
151C2×D107—24no—
152C2×D107—24no—
153C2×D107—24no—
154C2×D107—24no—
155C2×D107—432no—
156C2×D107—42no—
157C2×D107—42no—
158C2×D107—42no—
159C2×D107—42no—
160C2×D107#12414yescomplemented
161C4×C107—22no—
162C4×C107—41no—
163C4×C107—41no—
164C4×C107—42no—
165Dic107—41no—
166Dic107—41no—
167Dic107—42no—
168Dic107#12014yes—
169C2×C107—44no—
170C2×C107C2×#12011yescharacteristic
171C2×C107C2×#12411yescharacteristic
172C2×C107#53511yescharacteristic
173C2×C107#54612yes—
174C2×C107#54612yes—
175D107—24no—
176D107—44no—
177#2435—1071nomaximal, Sylow
178C107#243511yescharacteristic, complemented, Sylow
179C23×D4—1071no—
180C2×#124—1072no—
181C2×C22⋊Q8—1071no—
182#394—1071no—
183#394—1071no—
184#394—1071no—
185#394D10711yescharacteristic, complemented
186#390—1072no—
187#379—1072no—
188#379—1072no—
189C4×C22⋊C4—1071no—
190C25—1072no—
191C22×Q8—1071no—
192C22×D4—1072no—
193C22×D4—1071no—
194C22×D4—2144no—
195C22×D4—2148no—
196C22×D4C2×D10711yescharacteristic
197C23×C4—1071no—
198#124—2148no—
199C22⋊Q8—2144no—
200C2×C4⋊C4—1071no—
201C2×C4⋊C4—1071no—
202C2×C4⋊C4C2×D10711yescharacteristic, complemented
203C2×C22⋊C4—1072no—
204C2×C22⋊C4—1072no—
205C2×C22⋊C4—1072no—
206C2×C22⋊C4—1072no—
207C2×C22⋊C4—1071no—
208C2×C22⋊C4—1071no—
209C2×C22⋊C4—1072no—
210C2×C22⋊C4—1072no—
211C2×C22⋊C4—1072no—
212C2×C22⋊C4C2×D10712yescomplemented
213C2×C22⋊C4C2×D10712yescomplemented
214C2×C42—1072no—
215#95—1071no—
216#95C2×D10711yescharacteristic, complemented
217C24—1071no—
218C24—1072no—
219C24—4282no—
220C24—4282no—
221C24—2144no—
222C24—2144no—
223C24—2144no—
224C24—2144no—
225C24—2144no—
226C24C22×D10712yes—
227C2×Q8—2142no—
228C2×Q8—2144no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—1078no—
232C2×D4—2144no—
233C2×D4—2144no—
234C2×D4—2144no—
235C2×D4—21432no—
236C2×D4—2142no—
237C22×C4—1071no—
238C22×C4—1074no—
239C22×C4—1072no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—2144no—
243C22×C4C22×D10712yes—
244C22×C4C22×D10711yescharacteristic
245C22×C4C22×D10711yescharacteristic
246C22×C4C22×D10711yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—2142no—
249C4⋊C4—2142no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—4282no—
253C22⋊C4—4282no—
254C22⋊C4—4282no—
255C22⋊C4—2142no—
256C22⋊C4—2142no—
257C22⋊C4—2144no—
258C22⋊C4—2144no—
259C22⋊C4—2144no—
260C22⋊C4—2144no—
261C42—2144no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—1072no—
269C23—1074no—
270C23—1074no—
271C23—4288no—
272C23—4288no—
273C23—4288no—
274C23—4288no—
275C23—4288no—
276C23—4288no—
277C23—21416no—
278C23—2142no—
279C23—2142no—
280C23—2142no—
281C23—2142no—
282C23—2142no—
283C23—2142no—
284C23—2142no—
285C23—2141no—
286C23—2141no—
287C23D4×D10714yes—
288C23C23×D10711yescharacteristic
289Q8—4284no—
290D4—28no—
291D4—2148no—
292D4—21416no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—1074no—
303C2×C4—1072no—
304C2×C4—2148no—
305C2×C4—2142no—
306C2×C4—2142no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2142no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2142no—
313C2×C4D4×D10712yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—1074no—
322C22—42832no—
323C22—4282no—
324C22—4282no—
325C22—4282no—
326C22—4282no—
327C22—2144no—
328C22—2144no—
329C22—2144no—
330C22—2142no—
331C22—2144no—
332C22—2144no—
333C22—2142no—
334C22C2×D4×D10711yescharacteristic
335C22C2×D4×D10712yes—
336C22#41×D10712yes—
337C22#1681215411yescharacteristic
338C22#1681215511yescharacteristic
339C4—22no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—1074no—
344C4—4281no—
345C4—4282no—
346C4—4281no—
347C2—44no—
348C2—4284no—
349C2—2144no—
350C2#120×D10711yescharacteristic
351C2#124×D10711yescharacteristic
352C2#2029319311yescharacteristic
353C2#2029326612yes—
354C2#2029326912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
prime radicalC22×D4×D107C2
hypercenter#394D107
hypocenterC107#2435
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#16812095C2semidirect
3. C2×#16812087C2semidirect
4. #20292545C2semidirect
5. #20292544C2semidirect
6. #20292541C2semidirect
7. #20292527C2semidirect
8. #20292482C2semidirect
9. #20292482C2semidirect
10. #20292474C2semidirect
11. #394×C107C2semidirect
13. C22×Dic214C22semidirect
21. C2×Dic107⋊C4C22semidirect
22. C2×Dic107⋊C4C22semidirect
23. C2×C22⋊Dic107C22semidirect
24. C2×C22⋊Dic107C22semidirect
28. C2×(C2×D107)⋊C4C22semidirect
29. C2×(C2×D107)⋊C4C22semidirect
30. C2×(C2×D107)⋊C4C22semidirect
31. C2×(C2×D107)⋊C4C22semidirect
32. C2×C4×Dic107C22semidirect
33. C107⋊#95C22semidirect
35. C2×C4⋊C4×C107C22semidirect
36. C2×C22⋊C4×C107C22semidirect
37. C2×C22⋊C4×C107C22semidirect
38. #95×C107C22semidirect
160. C2×D107#124semidirect
178. C107#2435semidirect
185. #394D107semidirect
202. C2×C4⋊C4C2×D107semidirect
212. C2×C22⋊C4C2×D107semidirect
213. C2×C22⋊C4C2×D107semidirect
216. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168120952
C2×#168120872
#202925452
#202925442
#202925412
#202925272
#202924822
#202924742
#394×C1072
#2435107