Finite Group Databasebeta

← #28339846

Subgroups — #28339846

Order 13696.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812092C211yesmaximal, characteristic, complemented
2C2×#16812078C211yesmaximal, characteristic
3C2×#16812068C211yesmaximal, characteristic, complemented
4#20292550C212yesmaximal, complemented
5#20292543C212yesmaximal, complemented
6#20292527C212yesmaximal, complemented
7#20292521C212yesmaximal, complemented
8#20292506C211yesmaximal, characteristic, complemented
9#20292500C211yesmaximal, characteristic, complemented
10#20292483C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14#16812092—24no—
15#16812078—24no—
16#16812068—24no—
17C2×C4⋊Dic107C2212yescomplemented
18C2×Dic107⋊C4C2211yescharacteristic, complemented
19C2×Dic107⋊C4C2211yescharacteristic, complemented
20C2×Dic107⋊C4C2212yescomplemented
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C22⋊Dic107C2212yes—
24C2×C22⋊Dic107C2212yes—
25C2×C22⋊Dic107C2212yes—
26C2×C22⋊Dic107C2211yescharacteristic
27C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D107)⋊C4C2212yescomplemented
30C2×C4×Dic107C2212yescomplemented
31C107⋊#95C2212yescomplemented
32C107⋊#95C2212yescomplemented
33C25×C107C2211yescharacteristic
34C23×C4×C107C2211yescharacteristic
35C23×C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107—24no—
37C2×C22⋊C4×C107C2212yes—
38C2×C22⋊C4×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×C22⋊D107—22no—
41C2×C22⋊D107—22no—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—42no—
46C22×Dic107C2312yes—
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2312yes—
51C4⋊Dic107—24no—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—42no—
55C22⋊Dic107—22no—
56C22⋊Dic107—22no—
57C22⋊Dic107—24no—
58C22⋊Dic107—24no—
59C22⋊Dic107—42no—
60C22⋊Dic107D414yes—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—42no—
64C4×Dic107—24no—
65C24×C107—22no—
66C24×C107—24no—
67C24×C107—22no—
68C24×C107—22no—
69C24×C107—42no—
70C24×C107C2312yes—
71C24×C107C2311yescharacteristic
72C22×C4×C107—22no—
73C22×C4×C107—22no—
74C22×C4×C107—22no—
75C22×C4×C107—42no—
76C22×C4×C107D414yescomplemented
77C22×C4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79C22×C4×C107C2311yescharacteristic
80C22⋊C4×C107—24no—
81C22⋊C4×C107—24no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—28no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—22no—
88C23×C107—22no—
89C23×C107—22no—
90C23×C107—22no—
91C23×C107—48no—
92C23×C107—48no—
93C23×C107—44no—
94C23×C107—44no—
95C23×C107C2×D412yes—
96C23×C107C2×D412yes—
97C23×C107#4112yes—
98C23×C107#4114yes—
99C23×C107C2411yescharacteristic
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—21no—
103C22×D107—22no—
104C22×D107—22no—
105C2×C4×C107—28no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—22no—
111C2×C4×C107—22no—
112C2×C4×C107—22no—
113C2×C4×C107—44no—
114C2×C4×C107C2×D412yes—
115C2×C4×C107C2×D412yes—
116C22⋊D107—48no—
117C22⋊D107—44no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107—22no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—22no—
126C2×Dic107—22no—
127C2×Dic107—21no—
128C2×Dic107—22no—
129C2×Dic107—22no—
130C2×Dic107C2×D414yes—
131C22×C107—22no—
132C22×C107—22no—
133C22×C107—24no—
134C22×C107—24no—
135C22×C107—24no—
136C22×C107—24no—
137C22×C107—24no—
138C22×C107—416no—
139C22×C107—42no—
140C22×C107—42no—
141C22×C107—42no—
142C22×C107—42no—
143C22×C107#12114yes—
144C22×C107C22×D411yescharacteristic
145C22×C107C22×D411yescharacteristic
146C22×C107C2×#4111yescharacteristic
147C22×C107C2×#4112yes—
148C22×C107H(5,𝔽2)12yes—
149C2×D107—41no—
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—42no—
153C2×D107—42no—
154C4×C107—22no—
155C4×C107—22no—
156C4×C107—42no—
157Dic107—24no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161Dic107—42no—
162C2×C107—24no—
163C2×C107—44no—
164C2×C107C2×#12111yescharacteristic
165C2×C107#52511yescharacteristic
166C2×C107#53511yescharacteristic
167C2×C107#54612yes—
168C2×C107#55112yes—
169D107—81no—
170#2411—1071nomaximal, Sylow
171C107#241111yescharacteristic, complemented, Sylow
172C2×#123—1071no—
173C2×#120—1071no—
174C2×#117—1071no—
175C22×C22⋊C4D10711yescharacteristic, complemented
176#396—1072no—
177#394—1072no—
178#390—1072no—
179#388—1072no—
180#386—1071no—
181#385—1071no—
182#379—1071no—
183C25C2×D10711yescharacteristic
184C22×D4—1072no—
185C22×D4—1071no—
186C23×C4C2×D10711yescharacteristic
187C23×C4C2×D10711yescharacteristic
188#123—2144no—
189#120—2144no—
190#117—2144no—
191C2×C4⋊C4—1072no—
192C2×C4⋊C4—1072no—
193C2×C4⋊C4—1071no—
194C2×C4⋊C4—1071no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—1071no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4—1072no—
199C2×C22⋊C4—1071no—
200C2×C22⋊C4—1072no—
201C2×C22⋊C4—1071no—
202C2×C22⋊C4—1072no—
203C2×C22⋊C4—1071no—
204C2×C22⋊C4—1071no—
205C2×C22⋊C4C2×D10712yes—
206C2×C22⋊C4C2×D10712yes—
207C2×C42—1072no—
208#95—1072no—
209#95—1072no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—1071no—
216C24C22×D10712yes—
217C24C22×D10711yescharacteristic
218C2×D4—4282no—
219C2×D4—2144no—
220C2×D4—2142no—
221C2×D4—2142no—
222C2×D4—2142no—
223C2×D4—2144no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—1072no—
229C22×C4—1071no—
230C22×C4—1071no—
231C22×C4—1072no—
232C22×C4—1071no—
233C22×C4C22⋊D10714yescomplemented
234C22×C4C22×D10712yes—
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10711yescharacteristic
237C4⋊C4—4282no—
238C4⋊C4—2144no—
239C4⋊C4—2142no—
240C4⋊C4—2142no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—1074no—
244C22⋊C4—4282no—
245C22⋊C4—4282no—
246C22⋊C4—2144no—
247C22⋊C4—2142no—
248C22⋊C4—2142no—
249C22⋊C4—2144no—
250C22⋊C4—2142no—
251C22⋊C4—2142no—
252C42—2144no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—2141no—
267C23—2141no—
268C23—2141no—
269C23—2142no—
270C23—2142no—
271C23D4×D10712yes—
272C23C2×C22⋊D10712yes—
273C23#1313381712yes—
274C23#1313381814yes—
275C23C23×D10711yescharacteristic
276D4—4284no—
277D4—4288no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—1074no—
288C2×C4—2142no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4—2141no—
292C2×C4—2141no—
293C2×C4—2141no—
294C2×C4—2142no—
295C2×C4—2142no—
296C2×C4—2142no—
297C2×C4—2142no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4C2×C22⋊D10712yes—
301C2×C4C2×C22⋊D10712yes—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—4281no—
315C22—4281no—
316C22—4282no—
317C22—4282no—
318C22—4281no—
319C22#1681208114yes—
320C22C2×D4×D10711yescharacteristic
321C22C22×C22⋊D10711yescharacteristic
322C22C2×#1313381711yescharacteristic
323C22C2×#1313381812yes—
324C22#1681215412yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—4281no—
329C4—4281no—
330C4—4282no—
331C4—4281no—
332C4—2144no—
333C2—24no—
334C2—44no—
335C2—8561no—
336C2C2×#1681208111yescharacteristic
337C2#2029313811yescharacteristic
338C2#2029319411yescharacteristic
339C2#2029326712yes—
340C2#2029330312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C22⋊C4×C107C2
prime radicalC2×#16812078C2
hypercenterC22×C22⋊C4D107
hypocenterC107#2411
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16812092C2semidirect
3. C2×#16812068C2semidirect
4. #20292550C2semidirect
5. #20292543C2semidirect
6. #20292527C2semidirect
7. #20292521C2semidirect
8. #20292506C2semidirect
9. #20292500C2semidirect
10. #20292483C2semidirect
11. C22×C22⋊C4×C107C2semidirect
17. C2×C4⋊Dic107C22semidirect
18. C2×Dic107⋊C4C22semidirect
19. C2×Dic107⋊C4C22semidirect
20. C2×Dic107⋊C4C22semidirect
27. C2×(C2×D107)⋊C4C22semidirect
28. C2×(C2×D107)⋊C4C22semidirect
29. C2×(C2×D107)⋊C4C22semidirect
30. C2×C4×Dic107C22semidirect
31. C107⋊#95C22semidirect
32. C107⋊#95C22semidirect
76. C22×C4×C107D4semidirect
171. C107#2411semidirect
175. C22×C22⋊C4D107semidirect
233. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120922
C2×#168120782
C2×#168120682
#202925502
#202925432
#202925272
#202925212
#202925062
#202925002
#202924832
C22×C22⋊C4×C1072
#2411107