Finite Group Databasebeta

← #28340117

Subgroups — #28340117

Order 13696.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic214C211yesmaximal, characteristic, complemented
2C2×#16812095C212yesmaximal, complemented
3C2×#16812081C211yesmaximal, characteristic
4#20292540C212yesmaximal, complemented
5#20292539C212yesmaximal, complemented
6#20292531C212yesmaximal, complemented
7#20292507C211yesmaximal, characteristic, complemented
8#20292506C211yesmaximal, characteristic, complemented
9#20292504C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic107C211yesmaximal, characteristic, complemented
11#386×C107C211yesmaximal, characteristic, complemented
12C22×Dic214—28no—
13C22×Dic214—24no—
14C22×Dic214C2212yes—
15C22×Dic214C2212yes—
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic
20#16812095—28no—
21#16812081—24no—
22C2×Dic107⋊C4C2211yescharacteristic
23C2×C22⋊Dic107C2212yes—
24C2×C22⋊Dic107C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C23×C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×Dic214—232no—
41C2×Dic214—24no—
42C2×Dic214—24no—
43C2×Dic214—24no—
44C2×Dic214—24no—
45C2×Dic214—24no—
46C2×Dic214D418yescomplemented
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—24no—
52C22×Dic107—22no—
53C22×Dic107—24no—
54C22×Dic107—24no—
55C22×Dic107D414yes—
56C22×Dic107C2312yes—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2312yes—
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107—22no—
64C22⋊Dic107—24no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—24no—
69C24×C107C2311yescharacteristic
70C24×C107C2311yescharacteristic
71C2×D4×C107—22no—
72C2×D4×C107—22no—
73C2×D4×C107—22no—
74C22×C4×C107—22no—
75C22×C4×C107—24no—
76C22×C4×C107C2312yes—
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2312yes—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—24no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—21no—
85C23×C107—21no—
86C23×C107—21no—
87C23×C107—22no—
88C23×C107C2×D412yes—
89C23×C107#4112yes—
90C23×C107#4112yes—
91C23×C107C2411yescharacteristic
92C22×D107—22no—
93C22×D107—21no—
94C22×D107—21no—
95C22×D107—21no—
96C22×D107—22no—
97D4×C107—44no—
98C2×C4×C107—28no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107—21no—
103C2×C4×C107—22no—
104C2×C4×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107—22no—
107C2×C4×C107C2×D414yes—
108C22⋊D107—44no—
109C22⋊D107—44no—
110C2×Dic107—28no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—28no—
115C2×Dic107—22no—
116C2×Dic107—21no—
117C2×Dic107—22no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—22no—
121C2×Dic107C2×D414yes—
122C2×Dic107C2×D412yes—
123C2×Dic107C2×D414yes—
124C2×Dic107C2×D412yes—
125Dic214—216no—
126Dic214—216no—
127C22×C107—22no—
128C22×C107—22no—
129C22×C107—24no—
130C22×C107—24no—
131C22×C107—41no—
132C22×C107—41no—
133C22×C107—41no—
134C22×C107—42no—
135C22×C107—42no—
136C22×C107#12414yescomplemented
137C22×C107C22×D411yescharacteristic
138C22×C107C22×D412yes—
139C22×C107C2×#4111yescharacteristic
140C22×C107C2×#4111yescharacteristic
141C22×C1072−1+412yes—
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C2×D107—42no—
146C2×D107—42no—
147C4×C107—24no—
148C4×C107—41no—
149C4×C107—42no—
150Dic107—22no—
151Dic107—24no—
152Dic107—41no—
153Dic107—42no—
154Dic107#12014yes—
155C2×C107—24no—
156C2×C107—81no—
157C2×C107C2×#12011yescharacteristic
158C2×C107C2×#12411yescharacteristic
159C2×C107#53611yescharacteristic
160C2×C107#54812yes—
161C2×C107#54812yes—
162D107—81no—
163#2436—1071nomaximal, Sylow
164C107#243611yescharacteristic, complemented, Sylow
165C23×Q8—1071no—
166C2×#124—1072no—
167C2×#121—1071no—
168#393—1072no—
169#393—1072no—
170#391—1072no—
171#386—1071no—
172#386—1071no—
173#386—1071no—
174#386D10711yescharacteristic, complemented
175C4×C22⋊C4—1071no—
176C22×Q8—1072no—
177C22×Q8—1072no—
178C22×Q8—2144no—
179C22×Q8—2148no—
180C22×D4—1071no—
181C22×D4—1071no—
182C22×D4C2×D10711yescharacteristic
183C23×C4—1071no—
184C23×C4—1071no—
185C23×C4C2×D10711yescharacteristic
186#124—2148no—
187#121—2144no—
188C2×C4⋊C4—1071no—
189C2×C22⋊C4—1071no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1071no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4C2×D10712yes—
196C2×C22⋊C4C2×D10711yescharacteristic
197C2×C42—1072no—
198#95—1072no—
199#95—1072no—
200#95—1072no—
201#95—1072no—
202#95C2×D10712yes—
203C24—1071no—
204C24C22×D10711yescharacteristic
205C24C22×D10711yescharacteristic
206C2×Q8—1078no—
207C2×Q8—2144no—
208C2×Q8—2144no—
209C2×Q8—2144no—
210C2×Q8—2144no—
211C2×Q8—21432no—
212C2×Q8—2144no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—2142no—
217C2×D4—2144no—
218C2×D4—2142no—
219C2×D4—2142no—
220C2×D4—2142no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—1071no—
224C22×C4—1074no—
225C22×C4—1071no—
226C22×C4—1072no—
227C22×C4—1071no—
228C22×C4—1072no—
229C22×C4—2144no—
230C22×C4—2144no—
231C22×C4—2142no—
232C22×C4C22×D10711yescharacteristic
233C22×C4C22×D10712yes—
234C22×C4C22×D10712yes—
235C4⋊C4—2142no—
236C22⋊C4—22no—
237C22⋊C4—24no—
238C22⋊C4—2142no—
239C22⋊C4—2144no—
240C22⋊C4—2142no—
241C22⋊C4—2144no—
242C22⋊C4—2142no—
243C22⋊C4—2144no—
244C42—2144no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—2142no—
253C23—2141no—
254C23—2141no—
255C23—2141no—
256C23—2142no—
257C23D4×D10712yes—
258C23#1313381712yes—
259C23#1313381812yes—
260C23C23×D10711yescharacteristic
261Q8—21416no—
262Q8—21416no—
263D4—44no—
264D4—4284no—
265D4—4284no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—1074no—
276C2×C4—1072no—
277C2×C4—1072no—
278C2×C4—1074no—
279C2×C4—2148no—
280C2×C4—2142no—
281C2×C4—2141no—
282C2×C4—2141no—
283C2×C4—2142no—
284C2×C4—2141no—
285C2×C4—2142no—
286C2×C4—2148no—
287C2×C4—2142no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4D4×D10714yes—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—4281no—
301C22—4281no—
302C22—4282no—
303C22—4282no—
304C22—4281no—
305C22#1681209514yescomplemented
306C22C2×D4×D10711yescharacteristic
307C22C2×D4×D10712yes—
308C22C2×#1313381711yescharacteristic
309C22C2×#1313381811yescharacteristic
310C22#1681215612yes—
311C4—24no—
312C4—41no—
313C4—42no—
314C4—1074no—
315C4—4281no—
316C4—4282no—
317C4—2142no—
318C4—2144no—
319C2—24no—
320C2—81no—
321C2—8561no—
322C2#120×D10711yescharacteristic
323C2C2×#1681209511yescharacteristic
324C2#2029319911yescharacteristic
325C2#2029328312yes—
326C2#2029328612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC2×#16812081C2
hypercenter#386D107
hypocenterC107#2436
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic214C2semidirect
2. C2×#16812095C2semidirect
4. #20292540C2semidirect
5. #20292539C2semidirect
6. #20292531C2semidirect
7. #20292507C2semidirect
8. #20292506C2semidirect
9. #20292504C2semidirect
10. C22⋊C4×Dic107C2semidirect
11. #386×C107C2semidirect
46. C2×Dic214D4semidirect
136. C22×C107#124semidirect
164. C107#2436semidirect
174. #386D107semidirect
305. C22#16812095semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2142
C2×#168120952
C2×#168120812
#202925402
#202925392
#202925312
#202925072
#202925062
#202925042
C22⋊C4×Dic1072
#386×C1072
#2436107