Finite Group Databasebeta

← #28339879

Subgroups — #28339879

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×#16812083C211yesmaximal, characteristic, complemented
2C2×#16812083C211yesmaximal, characteristic, complemented
3C2×#16812068C211yesmaximal, characteristic, complemented
4#20292542C212yesmaximal, complemented
5#20292535C212yesmaximal, complemented
6#20292529C211yesmaximal, characteristic, complemented
7#20292527C212yesmaximal, complemented
8#20292525C211yesmaximal, characteristic, complemented
9#20292517C212yesmaximal, complemented
10#20292487C211yesmaximal, characteristic, complemented
11C2×C4×D4×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C22⋊D107C2212yes—
15#16812083—24no—
16#16812083—24no—
17#16812068—24no—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C22×D4×C107C2211yescharacteristic
32C23×C4×C107C2211yescharacteristic
33C23×C4×C107C2211yescharacteristic
34C4×D4×C107—24no—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2211yescharacteristic
38C2×C42×C107C2211yescharacteristic
39C23×D107C2312yes—
40C2×D428—22no—
41C2×D428—22no—
42C2×D428—22no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—42no—
47C2×C22⋊D107—42no—
48C2×C22⋊D107—42no—
49C22×Dic107C2312yes—
50C22×Dic107C2312yes—
51C22×Dic107C2312yes—
52C4⋊Dic107—22no—
53C4⋊Dic107—22no—
54C4⋊Dic107D414yescomplemented
55Dic107⋊C4—42no—
56C22⋊Dic107—24no—
57C22⋊Dic107—42no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—42no—
62C4×Dic107—24no—
63C24×C107C2311yescharacteristic
64C24×C107C2311yescharacteristic
65C2×D4×C107—22no—
66C2×D4×C107—22no—
67C2×D4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107—22no—
70C22×C4×C107—22no—
71C22×C4×C107—22no—
72C22×C4×C107—22no—
73C22×C4×C107D414yescomplemented
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2311yescharacteristic
79C4⋊C4×C107—22no—
80C22⋊C4×C107—22no—
81C22⋊C4×C107—22no—
82C42×C107—22no—
83C23×C107—21no—
84C23×C107—21no—
85C23×C107—21no—
86C23×C107—21no—
87C23×C107—21no—
88C23×C107—21no—
89C23×C107—21no—
90C23×C107—21no—
91C23×C107—21no—
92C23×C107—21no—
93C23×C107—21no—
94C23×C107—21no—
95C23×C107C2×D412yes—
96C23×C107#4112yes—
97C23×C107C2411yescharacteristic
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—22no—
101C22×D107—22no—
102C22×D107—22no—
103C22×D107—22no—
104C22×D107—22no—
105D4×C107—28no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—28no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—44no—
118C2×C4×C107C2×D412yes—
119C2×C4×C107C2×D412yes—
120C2×C4×C107C2×D412yes—
121C2×C4×C107#4112yes—
122C2×C4×C107#4112yes—
123C22⋊D107—48no—
124C22⋊D107—48no—
125C2×Dic107—22no—
126C2×Dic107—22no—
127C2×Dic107—22no—
128C2×Dic107—22no—
129C2×Dic107—22no—
130C2×Dic107—22no—
131C2×Dic107—22no—
132C2×Dic107—22no—
133C2×Dic107C2×D414yes—
134D428—44no—
135C22×C107—22no—
136C22×C107—22no—
137C22×C107—22no—
138C22×C107—22no—
139C22×C107—22no—
140C22×C107—22no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107C22×D411yescharacteristic
150C22×C107C22×D411yescharacteristic
151C22×C107C2×#4111yescharacteristic
152C22×C107C2×#4111yescharacteristic
153C22×C107C2×#4111yescharacteristic
154C22×C107H(5,𝔽2)11yescharacteristic
155C22×C107H(5,𝔽2)11yescharacteristic
156C2×D107—42no—
157C2×D107—42no—
158C2×D107—42no—
159C2×D107—42no—
160C2×D107—42no—
161C2×D107—42no—
162C2×D107—42no—
163C4×C107—22no—
164C4×C107—22no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107#12114yes—
168Dic107—24no—
169Dic107—42no—
170Dic107—42no—
171C2×C107—42no—
172C2×C107—42no—
173C2×C107C2×#12111yescharacteristic
174C2×C107#52511yescharacteristic
175C2×C107#53511yescharacteristic
176C2×C107#54611yescharacteristic
177C2×C107#55011yescharacteristic
178C2×C107#55311yescharacteristic
179C2×C107#55511yescharacteristic
180D107—82no—
181#2414—1071nomaximal, Sylow
182C107#241411yescharacteristic, complemented, Sylow
183C2×#121—1071no—
184C2×#121—1071no—
185C2×C4×D4D10711yescharacteristic, complemented
186C2×#117—1071no—
187#394—1072no—
188#392—1072no—
189#390—1072no—
190#390—1071no—
191#389—1071no—
192#388—1072no—
193#380—1071no—
194C22×D4—1072no—
195C22×D4—1072no—
196C22×D4—1071no—
197C22×D4C2×D10711yescharacteristic
198C23×C4C2×D10711yescharacteristic
199C23×C4C2×D10711yescharacteristic
200#121—2144no—
201#121—2144no—
202C4×D4—24no—
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—1072no—
214C2×C22⋊C4—1072no—
215C2×C22⋊C4C2×D10711yescharacteristic
216C2×C22⋊C4C2×D10711yescharacteristic
217C2×C42—1072no—
218C2×C42C2×D10711yescharacteristic
219#95—1072no—
220#95—1072no—
221C24—1072no—
222C24C22×D10711yescharacteristic
223C24C22×D10711yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—4282no—
228C2×D4—4282no—
229C2×D4—4282no—
230C2×D4—2144no—
231C2×D4—2144no—
232C2×D4—2144no—
233C2×D4—2142no—
234C2×D4—2142no—
235C2×D4—2142no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—1072no—
242C22×C4—1072no—
243C22×C4—1072no—
244C22×C4C22⋊D10714yescomplemented
245C22×C4C22×D10711yescharacteristic
246C22×C4C22×D10711yescharacteristic
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—22no—
256C22⋊C4—22no—
257C22⋊C4—4282no—
258C22⋊C4—4282no—
259C22⋊C4—2144no—
260C22⋊C4—2144no—
261C22⋊C4—2144no—
262C22⋊C4—2144no—
263C42—22no—
264C42—2144no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—2142no—
278C23—2142no—
279C23—2142no—
280C23—2142no—
281C23—2142no—
282C23—2142no—
283C23—2142no—
284C23C2×C22⋊D10712yes—
285C23#1313381812yes—
286C23C23×D10711yescharacteristic
287D4—28no—
288D4—4288no—
289D4—4288no—
290D4—4284no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—28no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—1074no—
304C2×C4—2142no—
305C2×C4—2142no—
306C2×C4—2142no—
307C2×C4—2142no—
308C2×C4—2142no—
309C2×C4—2142no—
310C2×C4—2142no—
311C2×C4—2142no—
312C2×C4D4×D10712yes—
313C2×C4C2×C22⋊D10712yes—
314C2×C4C2×C22⋊D10712yes—
315C2×C4#1313381712yes—
316C2×C4#1313381812yes—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—4282no—
332C22—4282no—
333C22—4282no—
334C22—4282no—
335C22—4282no—
336C22—4282no—
337C22—4282no—
338C22C2×D4×D10711yescharacteristic
339C22C22×C22⋊D10711yescharacteristic
340C22C2×#1313381711yescharacteristic
341C22C2×#1313381811yescharacteristic
342C22C2×#1313381811yescharacteristic
343C22#1681215511yescharacteristic
344C22#1681215511yescharacteristic
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—4282no—
350C4—4282no—
351C4—2144no—
352C4#1681208114yes—
353C2—42no—
354C2—42no—
355C2—8562no—
356C2C2×#1681208111yescharacteristic
357C2#2029313811yescharacteristic
358C2#2029319111yescharacteristic
359C2#2029327311yescharacteristic
360C2#2029329811yescharacteristic
361C2#2029332311yescharacteristic
362C2#2029334211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4×D4×C107C2
hypercenterC2×C4×D4D107
hypocenterC107#2414
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812083C2semidirect
2. C2×#16812083C2semidirect
3. C2×#16812068C2semidirect
4. #20292542C2semidirect
5. #20292535C2semidirect
6. #20292529C2semidirect
7. #20292527C2semidirect
8. #20292525C2semidirect
9. #20292517C2semidirect
10. #20292487C2semidirect
11. C2×C4×D4×C107C2semidirect
54. C4⋊Dic107D4semidirect
73. C22×C4×C107D4semidirect
182. C107#2414semidirect
185. C2×C4×D4D107semidirect
244. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120682
#202925422
#202925352
#202925292
#202925272
#202925252
#202925172
#202924872
C2×C4×D4×C1072
#2414107