Finite Group Databasebeta

← #28339882

Subgroups — #28339882

Order 13696.

704 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812080C211yesmaximal, characteristic, complemented
2C2×#16812080C211yesmaximal, characteristic, complemented
3C2×C4×D428C211yesmaximal, characteristic, complemented
4C2×#16812067C211yesmaximal, characteristic, complemented
5#20292542C212yesmaximal, complemented
6#20292536C212yesmaximal, complemented
7#20292527C212yesmaximal, complemented
8#20292525C211yesmaximal, characteristic, complemented
9#20292520C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11#390×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C22⋊D107C2212yes—
15C22×C4×D107C2211yescharacteristic
16C22×C4×D107C2211yescharacteristic
17#16812080—24no—
18#16812080—24no—
19C4×D428—24no—
20#16812067—24no—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4C2212yes—
25C2×C22⋊Dic107C2212yes—
26C2×C22⋊Dic107C2212yes—
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2212yes—
31C2×C4×Dic107C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2212yes—
38C2×C42×C107C2211yescharacteristic
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×D428—22no—
42C2×D428—22no—
43C2×D428—22no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—42no—
48C2×C22⋊D107—42no—
49C2×C22⋊D107—42no—
50C22×Dic107C2311yescharacteristic
51C22×Dic107C2312yes—
52C22×Dic107C2311yescharacteristic
53C22×Dic107C2312yes—
54C2×C4×D107—22no—
55C2×C4×D107—22no—
56C2×C4×D107—22no—
57C2×C4×D107—22no—
58C2×C4×D107—22no—
59C2×C4×D107D414yescomplemented
60C4⋊Dic107—22no—
61C4⋊Dic107—22no—
62C4⋊Dic107D414yescomplemented
63Dic107⋊C4—42no—
64C22⋊Dic107—24no—
65C22⋊Dic107—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—24no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—42no—
70C4×Dic107—24no—
71C24×C107C2312yes—
72C2×D4×C107—22no—
73C2×D4×C107—22no—
74C2×D4×C107—22no—
75C22×C4×C107C2312yes—
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2311yescharacteristic
79C4⋊C4×C107—22no—
80C22⋊C4×C107—24no—
81C22⋊C4×C107—42no—
82C42×C107—22no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—22no—
88C23×C107—22no—
89C23×C107—22no—
90C23×C107C2411yescharacteristic
91C22×D107—21no—
92C22×D107—21no—
93C22×D107—21no—
94C22×D107—21no—
95C22×D107—21no—
96C22×D107—21no—
97C22×D107—21no—
98C22×D107—21no—
99C22×D107—21no—
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—21no—
103C22×D107C2×D412yes—
104C22×D107#4112yes—
105D4×C107—44no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—22no—
110C2×C4×C107—22no—
111C2×C4×C107—21no—
112C2×C4×C107—22no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D412yes—
115C2×C4×C107C2×D412yes—
116C2×C4×C107#4112yes—
117C2×C4×C107#4112yes—
118C22⋊D107—48no—
119C22⋊D107—48no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—22no—
123C2×Dic107—22no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—22no—
127C2×Dic107—21no—
128C2×Dic107—22no—
129C2×Dic107—22no—
130C2×Dic107C2×D414yes—
131C2×Dic107C2×D412yes—
132D428—28no—
133C4×D107—28no—
134C4×D107—44no—
135C22×C107—42no—
136C22×C107—42no—
137C22×C107—42no—
138C22×C107—42no—
139C22×C107—42no—
140C22×C107—42no—
141C22×C107—42no—
142C22×C107C22×D411yescharacteristic
143C22×C107C22×D411yescharacteristic
144C22×C107C2×#4111yescharacteristic
145C22×C107C2×#4111yescharacteristic
146C22×C107C2×#4111yescharacteristic
147C22×C107H(5,𝔽2)11yescharacteristic
148C22×C107H(5,𝔽2)11yescharacteristic
149C2×D107—22no—
150C2×D107—22no—
151C2×D107—22no—
152C2×D107—22no—
153C2×D107—22no—
154C2×D107—22no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C4×C107—22no—
164C4×C107—41no—
165C4×C107—42no—
166C4×C107#12114yes—
167Dic107—22no—
168Dic107—24no—
169Dic107—41no—
170Dic107—42no—
171C2×C107—82no—
172C2×C107C2×#12111yescharacteristic
173C2×C107#52511yescharacteristic
174C2×C107#53511yescharacteristic
175C2×C107#54611yescharacteristic
176C2×C107#55011yescharacteristic
177C2×C107#55311yescharacteristic
178C2×C107#55511yescharacteristic
179D107—42no—
180D107—42no—
181#2414—1071nomaximal, Sylow
182C107#241411yescharacteristic, complemented, Sylow
183C2×#121—1071no—
184C2×#121—1071no—
185C2×C4×D4—1071no—
186C2×#117—1071no—
187#394—1072no—
188#392—1072no—
189#390—1072no—
190#390D10711yescharacteristic, complemented
191#389—1071no—
192#388—1072no—
193#380—1071no—
194C22×D4—1072no—
195C22×D4—1072no—
196C22×D4—1071no—
197C22×D4C2×D10711yescharacteristic
198C23×C4—1071no—
199C23×C4—1071no—
200#121—2144no—
201#121—2144no—
202C4×D4—2144no—
203#117—2144no—
204C2×C4⋊C4—1071no—
205C2×C4⋊C4—1072no—
206C2×C4⋊C4—1071no—
207C2×C4⋊C4—1071no—
208C2×C4⋊C4C2×D10711yescharacteristic
209C2×C22⋊C4—1072no—
210C2×C22⋊C4—1072no—
211C2×C22⋊C4—1072no—
212C2×C22⋊C4—1072no—
213C2×C22⋊C4—1071no—
214C2×C22⋊C4—1071no—
215C2×C22⋊C4C2×D10712yes—
216C2×C22⋊C4C2×D10712yes—
217C2×C42—1072no—
218C2×C42C2×D10711yescharacteristic
219#95—1072no—
220#95—1072no—
221C24—1071no—
222C24—1071no—
223C24C22×D10712yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—4282no—
228C2×D4—4282no—
229C2×D4—4282no—
230C2×D4—2142no—
231C2×D4—2142no—
232C2×D4—2144no—
233C2×D4—2144no—
234C2×D4—2144no—
235C2×D4—2142no—
236C22×C4—1072no—
237C22×C4—1071no—
238C22×C4—1074no—
239C22×C4—1071no—
240C22×C4—1072no—
241C22×C4—2142no—
242C22×C4—2142no—
243C22×C4—2142no—
244C22×C4—2142no—
245C22×C4—2142no—
246C22×C4C22×D10712yes—
247C22×C4C22×D10711yescharacteristic
248C22×C4C22×D10711yescharacteristic
249C22×C4C22×D10711yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—1074no—
252C4⋊C4—4282no—
253C4⋊C4—2142no—
254C4⋊C4—2142no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—4282no—
258C22⋊C4—2144no—
259C22⋊C4—2142no—
260C22⋊C4—2142no—
261C22⋊C4—2144no—
262C22⋊C4—2144no—
263C42—22no—
264C42—2144no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—1072no—
273C23—1072no—
274C23—2141no—
275C23—2141no—
276C23—2141no—
277C23—2141no—
278C23—2141no—
279C23—2141no—
280C23—2141no—
281C23—2141no—
282C23—2141no—
283C23—2141no—
284C23—2141no—
285C23—2141no—
286C23C23×D10711yescharacteristic
287D4—44no—
288D4—4288no—
289D4—4288no—
290D4—2148no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—1074no—
300C2×C4—1072no—
301C2×C4—4284no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2142no—
305C2×C4—2141no—
306C2×C4—2142no—
307C2×C4—2142no—
308C2×C4—2141no—
309C2×C4—2142no—
310C2×C4—2141no—
311C2×C4—2142no—
312C2×C4—2148no—
313C2×C4D4×D10712yes—
314C2×C4D4×D10712yes—
315C2×C4#1313381712yes—
316C2×C4#1313381812yes—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—4281no—
325C22—4281no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22—4281no—
330C22—4281no—
331C22—4281no—
332C22—2142no—
333C22—2142no—
334C22—2142no—
335C22—2142no—
336C22—2142no—
337C22—2142no—
338C22C2×D4×D10711yescharacteristic
339C22C2×D4×D10711yescharacteristic
340C22#41×D10711yescharacteristic
341C22C2×#1313381711yescharacteristic
342C22C2×#1313381811yescharacteristic
343C22#1681215411yescharacteristic
344C22#1681215511yescharacteristic
345C4—22no—
346C4—41no—
347C4—42no—
348C4—4282no—
349C4—4281no—
350C4—2144no—
351C4—2142no—
352C4#1681208214yes—
353C2—82no—
354C2—4282no—
355C2—4282no—
356C2C2×#1681208211yescharacteristic
357C2#2029313711yescharacteristic
358C2#2029319311yescharacteristic
359C2#2029327011yescharacteristic
360C2#2029329711yescharacteristic
361C2#2029332711yescharacteristic
362C2#2029334211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
hypercenter#390D107
hypocenterC107#2414
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812080C2semidirect
2. C2×#16812080C2semidirect
3. C2×C4×D428C2semidirect
4. C2×#16812067C2semidirect
5. #20292542C2semidirect
6. #20292536C2semidirect
7. #20292527C2semidirect
8. #20292525C2semidirect
9. #20292520C2semidirect
10. #20292484C2semidirect
11. #390×C107C2semidirect
59. C2×C4×D107D4semidirect
62. C4⋊Dic107D4semidirect
182. C107#2414semidirect
190. #390D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120802
C2×C4×D4282
C2×#168120672
#202925422
#202925362
#202925272
#202925252
#202925202
#202924842
#390×C1072
#2414107