Finite Group Databasebeta

← #28340114

Subgroups — #28340114

Order 13696.

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

normal subgroups only

This group has 852 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×Q8×D107C211yesmaximal, characteristic, complemented
2C2×#16812095C212yesmaximal, complemented
3C2×#16812082C211yesmaximal, characteristic
4#20292540C211yesmaximal, characteristic, complemented
5#20292539C211yesmaximal, characteristic, complemented
6#20292538C211yesmaximal, characteristic, complemented
7#20292531C212yesmaximal, complemented
8#20292505C212yesmaximal, complemented
9#20292503C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#393×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×Dic214C2212yes—
14C2×Q8×D107—28no—
15C2×Q8×D107—24no—
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C4×D107C2212yes—
20C22×C4×D107C2211yescharacteristic
21#16812095—28no—
22#16812082—24no—
23C2×Dic107⋊C4C2211yescharacteristic
24C2×C22⋊Dic107C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36C2×C22⋊C4×C107C2211yescharacteristic
37#95×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×Dic214—22no—
42C2×Dic214—24no—
43C2×Dic214—24no—
44C2×Dic214—24no—
45C2×Dic214D418yescomplemented
46Q8×D107—232no—
47C2×D428—22no—
48C2×D428—24no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—24no—
51C2×C22⋊D107—22no—
52C2×C22⋊D107—24no—
53C22×Dic107C2312yes—
54C22×Dic107C2311yescharacteristic
55C22×Dic107C2312yes—
56C22×Dic107C2311yescharacteristic
57C2×C4×D107—24no—
58C2×C4×D107—24no—
59C2×C4×D107—24no—
60C2×C4×D107—24no—
61C2×C4×D107D414yes—
62Dic107⋊C4—22no—
63C22⋊Dic107—24no—
64(C2×D107)⋊C4—24no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—24no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—24no—
69C24×C107C2311yescharacteristic
70C2×Q8×C107—24no—
71C2×Q8×C107—22no—
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C22⋊C4×C107—22no—
77C22⋊C4×C107—24no—
78C23×C107—22no—
79C23×C107—21no—
80C23×C107—22no—
81C23×C107—21no—
82C23×C107—21no—
83C23×C107C2411yescharacteristic
84C22×D107—22no—
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—21no—
89C22×D107—22no—
90C22×D107—21no—
91C22×D107—21no—
92C22×D107C2×D412yes—
93C22×D107#4114yes—
94Q8×C107—28no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107C2×D414yes—
104C2×C4×C107C2×D412yes—
105C22⋊D107—44no—
106C22⋊D107—44no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107C2×D414yes—
115C2×Dic107C2×D414yes—
116C2×Dic107C2×D412yes—
117D428—44no—
118C4×D107—216no—
119C4×D107—28no—
120Dic214—28no—
121Dic214—216no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107C22×D411yescharacteristic
128C22×C107C22×D412yes—
129C22×C107C2×#4112yes—
130C22×C1072−1+411yescharacteristic
131C22×C1072−1+411yescharacteristic
132C2×D107—22no—
133C2×D107—22no—
134C2×D107—24no—
135C2×D107—24no—
136C2×D107—41no—
137C2×D107—41no—
138C2×D107—41no—
139C2×D107—42no—
140C2×D107—42no—
141C2×D107#12414yescomplemented
142C4×C107—22no—
143C4×C107—24no—
144C4×C107—41no—
145C4×C107—42no—
146Dic107—24no—
147Dic107—41no—
148Dic107—42no—
149Dic107#12014yes—
150C2×C107—81no—
151C2×C107C2×#12011yescharacteristic
152C2×C107C2×#12411yescharacteristic
153C2×C107#53611yescharacteristic
154C2×C107#54812yes—
155C2×C107#54812yes—
156D107—24no—
157D107—81no—
158#2436—1071nomaximal, Sylow
159C107#243611yescharacteristic, complemented, Sylow
160C23×Q8—1071no—
161C2×#124—1072no—
162C2×#121—1071no—
163#393—1071no—
164#393—1071no—
165#393—1071no—
166#393D10711yescharacteristic, complemented
167#391—1072no—
168#386—1072no—
169#386—1072no—
170C4×C22⋊C4—1071no—
171C22×Q8—1072no—
172C22×Q8—1071no—
173C22×Q8—2144no—
174C22×Q8—2148no—
175C22×Q8C2×D10711yescharacteristic
176C22×D4—1071no—
177C22×D4—1071no—
178C22×D4—1071no—
179C23×C4—1071no—
180C23×C4—1072no—
181#124—2148no—
182#121—2144no—
183C2×C4⋊C4—1071no—
184C2×C22⋊C4—1072no—
185C2×C22⋊C4—1071no—
186C2×C22⋊C4—1072no—
187C2×C22⋊C4—1072no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4C2×D10711yescharacteristic
190C2×C22⋊C4C2×D10712yes—
191C2×C42—1072no—
192#95—1071no—
193#95—1072no—
194#95—1072no—
195#95—1072no—
196#95C2×D10711yescharacteristic
197#95C2×D10712yes—
198C24—1071no—
199C24—1071no—
200C24C22×D10711yescharacteristic
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—1078no—
204C2×Q8—2144no—
205C2×Q8—2144no—
206C2×Q8—2144no—
207C2×Q8—21432no—
208C2×Q8—2142no—
209C2×D4—2142no—
210C2×D4—2144no—
211C2×D4—2142no—
212C2×D4—2144no—
213C2×D4—2142no—
214C2×D4—2144no—
215C22×C4—1071no—
216C22×C4—1072no—
217C22×C4—1074no—
218C22×C4—1071no—
219C22×C4—1072no—
220C22×C4—2144no—
221C22×C4—2144no—
222C22×C4—2144no—
223C22×C4—2144no—
224C22×C4C22×D10711yescharacteristic
225C22×C4C22×D10712yes—
226C22×C4C22×D10711yescharacteristic
227C22×C4C22×D10712yes—
228C4⋊C4—2142no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—2144no—
232C22⋊C4—2144no—
233C22⋊C4—2144no—
234C22⋊C4—2142no—
235C22⋊C4—2144no—
236C42—2144no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—1072no—
243C23—1074no—
244C23—2142no—
245C23—2141no—
246C23—2142no—
247C23—2141no—
248C23—2141no—
249C23—2142no—
250C23—2141no—
251C23—2141no—
252C23C23×D10711yescharacteristic
253Q8—28no—
254Q8—21416no—
255Q8—2148no—
256D4—4284no—
257D4—4284no—
258D4—4284no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—1074no—
268C2×C4—1072no—
269C2×C4—1074no—
270C2×C4—2148no—
271C2×C4—21416no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4—2142no—
275C2×C4—2142no—
276C2×C4—2142no—
277C2×C4—2142no—
278C2×C4—2142no—
279C2×C4D4×D10712yes—
280C2×C4D4×D10714yes—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—1074no—
287C22—4281no—
288C22—4281no—
289C22—4282no—
290C22—4282no—
291C22—4281no—
292C22—2142no—
293C22—2144no—
294C22—2144no—
295C22—2142no—
296C22C2×D4×D10711yescharacteristic
297C22C2×D4×D10712yes—
298C22#41×D10712yes—
299C22#1681215611yescharacteristic
300C22#1681215711yescharacteristic
301C4—22no—
302C4—24no—
303C4—41no—
304C4—42no—
305C4—1074no—
306C4—4281no—
307C4—4282no—
308C4—2144no—
309C2—81no—
310C2—8561no—
311C2—2144no—
312C2#120×D10711yescharacteristic
313C2#124×D10711yescharacteristic
314C2#2029320011yescharacteristic
315C2#2029328512yes—
316C2#2029328712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#393×C107C2
prime radicalC2×#16812082C2
hypercenter#393D107
hypocenterC107#2436
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812095C2semidirect
4. #20292540C2semidirect
5. #20292539C2semidirect
6. #20292538C2semidirect
7. #20292531C2semidirect
8. #20292505C2semidirect
9. #20292503C2semidirect
10. #20292474C2semidirect
11. #393×C107C2semidirect
45. C2×Dic214D4semidirect
141. C2×D107#124semidirect
159. C107#2436semidirect
166. #393D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168120952
C2×#168120822
#202925402
#202925392
#202925382
#202925312
#202925052
#202925032
#202924742
#393×C1072
#2436107