Finite Group Databasebeta

← #28339899

Subgroups — #28339899

Order 13696.

788 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812084C212yesmaximal, complemented
2C2×#16812084C211yesmaximal, characteristic, complemented
3C2×#16812084C211yesmaximal, characteristic, complemented
4C2×C4×D428C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
6#20292549C212yesmaximal, complemented
7#20292545C212yesmaximal, complemented
8#20292529C211yesmaximal, characteristic, complemented
9#20292505C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11#389×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C22×D428C2211yescharacteristic
14C22×D428C2212yes—
15C22×D428C2211yescharacteristic
16C22×C4×D107C2211yescharacteristic
17C22×C4×D107C2212yes—
18C22×C4×D107C2211yescharacteristic
19#16812084—28no—
20#16812084—24no—
21#16812084—24no—
22C4×D428—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4C2212yes—
26C4⋊C4×D107—24no—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2212yes—
34C107⋊#95C2212yes—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107C2212yes—
37C2×C4⋊C4×C107C2211yescharacteristic
38C2×C42×C107C2211yescharacteristic
39#95×C107C2212yes—
40C23×D107C2312yes—
41C23×D107C2311yescharacteristic
42C23×D107C2311yescharacteristic
43C2×D428—22no—
44C2×D428—22no—
45C2×D428—24no—
46C2×D428—24no—
47C2×D428—24no—
48C2×D428—24no—
49C2×D428—22no—
50C2×D428—24no—
51C2×D428—22no—
52C2×D428—22no—
53C2×D428—42no—
54C2×D428D414yescomplemented
55C22×Dic107C2312yes—
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2311yescharacteristic
58C2×C4×D107—22no—
59C2×C4×D107—22no—
60C2×C4×D107—24no—
61C2×C4×D107—24no—
62C2×C4×D107—22no—
63C2×C4×D107—22no—
64C2×C4×D107—22no—
65C2×C4×D107—42no—
66C2×C4×D107D414yescomplemented
67C4⋊Dic107—22no—
68C4⋊Dic107—22no—
69Dic107⋊C4—24no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—24no—
72(C2×D107)⋊C4—24no—
73(C2×D107)⋊C4—24no—
74(C2×D107)⋊C4—22no—
75(C2×D107)⋊C4—24no—
76(C2×D107)⋊C4—42no—
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2312yes—
79C22×C4×C107C2311yescharacteristic
80C22×C4×C107C2312yes—
81C22×C4×C107C2311yescharacteristic
82C4⋊C4×C107—24no—
83C4⋊C4×C107—22no—
84C4⋊C4×C107D414yescomplemented
85C42×C107—22no—
86C23×C107C2411yescharacteristic
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107—21no—
92C22×D107—21no—
93C22×D107—21no—
94C22×D107—21no—
95C22×D107—21no—
96C22×D107—21no—
97C22×D107—22no—
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—22no—
101C22×D107—22no—
102C22×D107—22no—
103C22×D107—22no—
104C22×D107C2×D412yes—
105C22×D107C2×D412yes—
106C22×D107C2×D412yes—
107C22×D107#4112yes—
108C2×C4×C107—21no—
109C2×C4×C107—22no—
110C2×C4×C107—21no—
111C2×C4×C107—22no—
112C2×C4×C107—21no—
113C2×C4×C107—22no—
114C2×C4×C107—22no—
115C2×C4×C107—22no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107C2×D412yes—
119C2×C4×C107C2×D412yes—
120C2×C4×C107C2×D414yes—
121C2×C4×C107C2×D412yes—
122C2×C4×C107#4112yes—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—22no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—22no—
129C2×Dic107—21no—
130C2×Dic107—22no—
131C2×Dic107C2×D412yes—
132D428—28no—
133D428—48no—
134D428—48no—
135D428—44no—
136C4×D107—28no—
137C4×D107—48no—
138C4×D107—44no—
139C22×C107C22×D411yescharacteristic
140C22×C107C22×D411yescharacteristic
141C22×C107C22×D411yescharacteristic
142C22×C107C2×#4111yescharacteristic
143C22×C107C2×#4111yescharacteristic
144C22×C107H(5,𝔽2)11yescharacteristic
145C22×C1072−1+411yescharacteristic
146C2×D107—22no—
147C2×D107—22no—
148C2×D107—22no—
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—42no—
160C2×D107—42no—
161C2×D107—42no—
162C2×D107—42no—
163C2×D107—42no—
164C2×D107—42no—
165C2×D107—42no—
166C2×D107#12114yescomplemented
167C4×C107—22no—
168C4×C107—24no—
169C4×C107—42no—
170C4×C107—41no—
171C4×C107#12114yescomplemented
172Dic107—22no—
173Dic107—41no—
174Dic107—42no—
175C2×C107C2×#12111yescharacteristic
176C2×C107C2×#12111yescharacteristic
177C2×C107#53711yescharacteristic
178C2×C107D4211yescharacteristic
179C2×C107#54711yescharacteristic
180C2×C107#54811yescharacteristic
181C2×C107#55311yescharacteristic
182D107—24no—
183D107—42no—
184D107—82no—
185#2416—1071nomaximal, Sylow
186C107#241611yescharacteristic, complemented, Sylow
187C2×#121—1072no—
188C2×#121—1071no—
189C2×#121—1071no—
190C2×C4×D4—1071no—
191C22×C4⋊C4—1071no—
192#396—1072no—
193#394—1072no—
194#390—1071no—
195#389D10711yescharacteristic, complemented
196#386—1072no—
197#380—1071no—
198C22×D4—1071no—
199C22×D4—1072no—
200C22×D4—1072no—
201C22×D4—1071no—
202C23×C4—1071no—
203C23×C4—1072no—
204C23×C4—1071no—
205#121—2144no—
206#121—2148no—
207#121—2144no—
208C4×D4—2144no—
209C2×C4⋊C4—1072no—
210C2×C4⋊C4—1071no—
211C2×C4⋊C4—1071no—
212C2×C4⋊C4—2144no—
213C2×C4⋊C4C2×D10711yescharacteristic
214C2×C4⋊C4C2×D10712yes—
215C2×C4⋊C4C2×D10711yescharacteristic
216C2×C22⋊C4—1072no—
217C2×C22⋊C4—1071no—
218C2×C22⋊C4—1071no—
219C2×C22⋊C4—1072no—
220C2×C22⋊C4—1072no—
221C2×C22⋊C4—1072no—
222C2×C22⋊C4—1072no—
223C2×C42C2×D10711yescharacteristic
224#95—1072no—
225#95C2×D10712yes—
226C24—1071no—
227C24—1071no—
228C24—1072no—
229C2×D4—1074no—
230C2×D4—4282no—
231C2×D4—2144no—
232C2×D4—2144no—
233C2×D4—2144no—
234C2×D4—2144no—
235C2×D4—2142no—
236C2×D4—2142no—
237C2×D4—2142no—
238C2×D4—2144no—
239C2×D4—2142no—
240C2×D4—2142no—
241C22×C4—1071no—
242C22×C4—1074no—
243C22×C4—1071no—
244C22×C4—1072no—
245C22×C4—4282no—
246C22×C4—2144no—
247C22×C4—2142no—
248C22×C4—2144no—
249C22×C4—2142no—
250C22×C4—2142no—
251C22×C4—2142no—
252C22×C4—2142no—
253C22×C4C22×D10712yes—
254C22×C4C22×D10711yescharacteristic
255C22×C4C22×D10711yescharacteristic
256C22×C4C22×D10712yes—
257C22×C4C22×D10711yescharacteristic
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—2142no—
261C4⋊C4—2144no—
262C4⋊C4—2142no—
263C4⋊C4D42814yescomplemented
264C22⋊C4—4282no—
265C22⋊C4—2142no—
266C22⋊C4—2144no—
267C22⋊C4—2142no—
268C22⋊C4—2144no—
269C22⋊C4—2144no—
270C22⋊C4—2144no—
271C42—22no—
272C23—1072no—
273C23—1072no—
274C23—1072no—
275C23—1072no—
276C23—2142no—
277C23—2142no—
278C23—2142no—
279C23—2142no—
280C23—2142no—
281C23—2142no—
282C23—2142no—
283C23—2141no—
284C23—2141no—
285C23—2141no—
286C23—2141no—
287C23—2141no—
288C23—2141no—
289C23—2141no—
290C23—2141no—
291C23—2141no—
292C23—2141no—
293C23C23×D10711yescharacteristic
294D4—4288no—
295D4—4288no—
296D4—4284no—
297D4—2148no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—1072no—
309C2×C4—4288no—
310C2×C4—4284no—
311C2×C4—2142no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2142no—
315C2×C4—2141no—
316C2×C4—2142no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2148no—
320C2×C4D4×D10712yes—
321C2×C4D4×D10712yes—
322C2×C4C2×D42814yes—
323C2×C4C2×D42812yes—
324C2×C4#1313381912yes—
325C22—1074no—
326C22—4282no—
327C22—4282no—
328C22—4282no—
329C22—4282no—
330C22—4282no—
331C22—4282no—
332C22—4282no—
333C22—4281no—
334C22—4281no—
335C22—4281no—
336C22—4281no—
337C22—2142no—
338C22—2142no—
339C22—2142no—
340C22—2142no—
341C22—2142no—
342C22—2142no—
343C22—2142no—
344C22—2142no—
345C22—2142no—
346C22C2×D4×D10711yescharacteristic
347C22C2×D4×D10711yescharacteristic
348C22C22×D42811yescharacteristic
349C22#41×D10711yescharacteristic
350C22C2×#1313381911yescharacteristic
351C22#1681215511yescharacteristic
352C22#1681215711yescharacteristic
353C4—24no—
354C4—22no—
355C4—41no—
356C4—42no—
357C4—4281no—
358C4—4282no—
359C4—2142no—
360C4#1681208414yescomplemented
361C2—8562no—
362C2—4282no—
363C2—2144no—
364C2#121×D10711yescharacteristic
365C2C2×#1681208411yescharacteristic
366C2#2029320711yescharacteristic
367C2D4×D42811yescharacteristic
368C2#2029328011yescharacteristic
369C2#2029328911yescharacteristic
370C2#2029333011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#389×C107C2
hypercenter#389D107
hypocenterC107#2416
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#16812084C2semidirect
2. C2×#16812084C2semidirect
3. C2×#16812084C2semidirect
4. C2×C4×D428C2semidirect
5. C2×C4⋊C4×D107C2semidirect
6. #20292549C2semidirect
7. #20292545C2semidirect
8. #20292529C2semidirect
9. #20292505C2semidirect
10. #20292484C2semidirect
11. #389×C107C2semidirect
54. C2×D428D4semidirect
66. C2×C4×D107D4semidirect
84. C4⋊C4×C107D4semidirect
166. C2×D107#121semidirect
171. C4×C107#121semidirect
186. C107#2416semidirect
195. #389D107semidirect
263. C4⋊C4D428semidirect
360. C4#16812084semidirect

Maximal subgroups

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

subgroupindex
C2×#168120842
C2×C4×D4282
C2×C4⋊C4×D1072
#202925492
#202925452
#202925292
#202925052
#202924842
#389×C1072
#2416107